E=mc^2

Derived in ten minutes while I was on the toilet:

Edit: 11-16-16

There was a slight error in the set-up of the center of mass calculation. Light appears to move an effective mass from -L/2 to L/2-l, the starting and stopping points of light. Here’s the reformulation to capture that.

Time spent on this derivation: ten minutes on the toilet last night and ten minutes before breakfast writing the correction (yeah, that’s what I get for going really fast).

Don’t thank me, this is Einstein’s calculation.

Edit: 11-18-17

As I’m still thinking about this post, I figure it might be beneficial to flesh out the reason that it was written. This post was used as a response to a comment on another blog… if you want to back up a statement you made about something and somebody is accusing you of not providing evidence, most people provide citations, net links, references, etc. In this particular case, the argument was about a piece of math and I was being accused of lying about said piece of math by someone who clearly likes to believe he knows everything without actually knowing practically anything. Yes, skeptics are guilty of Dunning-Kruger, just like everyone else (This is an unfair statement and I apologize for it.) What better way to slam the textbook in someone’s face than to actually work the problem? If you want the final word on what Einstein said about something, quote Einstein’s work! And so, a piece of Einstein’s work is posted above.

The argument in question started with a fellow suggesting to me that mass-energy equivalence can be derived but not proven with classical physics. I beg to differ; energy is a classical concept, from Thermo, E&M and classical mechanics… all three! You don’t need relativity or quantum mechanics to justify statements about how energy works; measurements of kinematics and force are sufficient to show that energy as a concept works. Mass-energy equivalence arose from Einstein’s notion that the newly completed classical field of electromagnetism must be consistent with the older fields of classical mechanics. The equation E=pc is not relativistic: it came directly out of electromagnetism (and, believe me, I’ve been through that calculation too because I didn’t believe it at first.) Imposing that these two fields must be cross-consistent is the origin of mass-energy equivalence…. light carries momentum (by Poynting’s vector and well defined in the electromagnetic stress-energy tensor) and light interacts with mass, therefore conservation of momentum (and consequently conservation of center of mass in absence of external forces) requires that light carry an equivalent of mass in order for forces to add up in a situation where light interacts with matter but no forces interact externally on the system comprised by the light and the matter. Mass-energy equivalence is required by this, no ifs, ands, buts or “yeah, but you didn’t proves…”

Einstein’s thought experiment validating this set-up is an exceptionally elegant one. It’s called “Einstein’s Box.” Everybody loves Schrodinger’s cat-in-a-box… well, Einstein had a box too and this box is older than Schrodinger’s. Einstein’s box is a closed box sitting out in space where it feels no external forces. A flash of light is emitted inside the box from one wall and travels across the box to strike the opposite wall. E&M states that light must carry momentum. If the system has no external forces acting on it, the emission of the light inside the box requires that momentum of the system be conserved, which requires that the box recoils with a momentum equal to that carried by the light, causing the box to move at some velocity consistent with the momentum carried by the light (which turns out to be directly proportional to the energy carried by that light as stated by E=pc). Net momentum of the system must remain zero by conservation of momentum. When the light travels across the box and collides with the opposite wall, the momentum of the light cancels the momentum of the box and the box stops moving. Thing about this is that center of mass, as a consequence of momentum conservation, could not have moved. No forces on the outside of the box.

Center of mass is a damn well classical concept, well worked out in the 1700s and 1800s… and since the box moved, the box’s center of mass moved! But no forces acted on the outside of the system, so the overall center of mass of the system could not have moved. This requires the light to have carried with it a value of mass, taken from the location where the light was emitted and deposited again at the location where the light was absorbed. But light is known not to carry mass since it is a wave-like solution of immaterial fields in the form of Maxwell’s equations. If you set up this situation and work through the calculation, Newtonian mechanics and electromagnetism –nothing more–, this turns out the classical requirement that energy and mass have an equivalence in the form of E=mc^2. No quantization or probability relations from quantum mechanics, no frame of reference shifting from relativity, not even delineating that light is some package of photons… this is purely classical. Moreover, energy is a tabular result to begin with: it is not something that is by itself ever directly observed and it must always be carried by something else, a field, a heat, a potential, a motion or what have you. The statement that this tabular relationship extends to something else that is technically only indirectly observed, mass, is a proof. And yes, mass is indirect since you can only know mass from weight, which is a force!

If you have concepts of weight, momentum and light together in the same model as expressed by classical physics, mass-energy equivalence is required for self-consistency.

Granted, special relativity quite naturally produces this result as well, but special relativity is not required to produce mass-energy equivalence. Had Einstein not discovered it, someone else damned well would’ve and it would not have required relativity to do –at all!

Now, the thing that doubly made me angry about this conversation is that it was with a fellow who absolutely craved physicist street cred: he name dropped Arxiv and seemed to want to chase around details. Sadly, his whole argument ultimately amounted to insulting someone and not backing up his ability to absolutely know what he was claiming to know. Does it matter that you don’t believe my statement if you aren’t competent to evaluate the field in question? Not at all: such a person has no place at the table to start with. This is why it’s possible for a Nobel Laureate to descend in to crankery… just because you have a big prize doesn’t mean you are always equally competent at everything! I’m guessing the guy was a surgeon given the ego and the blog, but if he was a physicist, I’m very disappointed. A physicist who doesn’t know Einstein’s box is a travesty. I’m not the greatest physicist that ever lived, but I work at it and I know what I’m talking about… where I have gaps, I do my best to admit it.

edit: 11-18-17

(Statement redacted. It was an unfairly insulting comment)

edit: 11-20-17

As this is still nagging at me, one further thought. What I consider the last statement of the conversation before it simply became obvious trolling, the fellow accused me of not including “a variable speed for light” in my calculation. In Einstein’s calculation, the speed of light is given the constant “c.” This is a constant which comes with a caveat; “c” is the speed of light when it is not passing through anything material, the speed of light in a vacuum. This distinction is important because light can travel at lower speeds when it’s passing through a material. This situation is well-handled by E&M and is considered a “solved effect” by Relativity, whose postulates include the explicit notion that E&M simply be true everywhere. The constant “c” is the maximum possible speed that light can travel, but it will travel at lower speeds in a medium with an index of refraction greater than 1, where permittivity and permeability might have values other than their vacuum values, which has the wonderful result of making lenses possible in glasses and microscopes. In my lavatory derivation above, a little screw up on my part is that I didn’t clobber the reader over the head with the constancy of the value “c,” I said “a box in zero gravity” and I said light travels at “c,” but I didn’t say “this is definitely all in a vacuum” which I probably should have. If index of refraction is “n”… the velocity of light in a medium with that refractive index is v = c/n. There are other ways to encode refractive index which allow for more sophisticated optical behaviors, but everything in that line is completely out of the pail for the argument in question, and drawing attention to it is simply chaff intended to shift the focus of the argument.

Light can travel at speeds lower than “c,” but “c” itself is so far found to be invariant. Moreover, the fact that light can travel at speeds other than “c” does not change the Einstein’s box derivation, which is set in explicit conditions where light would travel at “c.” Somebody who doesn’t know this isn’t a physicist (11-20-17: I’ll moderate this it’s unfair and was too angry.)

Also, as an aside, I mention above that Special Relativity can produce E = mc^2. Thinking about it, but not running through the calculations, I think this is actually backward; E=mc^2 is sort of needed first before it shows up in Special Rel. Einstein made some amazing leaps.

Edit: 11-20-17

As an added extra, here is a derivation of E=pc from the stress-energy and electromagnetic power continuity equations. These were written a few years ago, but I had the good sense to scan them:

EMC31 001

EMC32 001

EMC33 001

The E=pc derivation begins on the second page above. The first page is the end of the continuity equation derivation. I’ll neglect that. No relativity here, just pure E&M. There are a couple pieces in here that I don’t remember so well and I need to think about to decide if they’re correct. The first page is included to show clearly the relation between force and the stress-energy tensor divergence.

Edit: 11-21-17

I’ve spent some time thinking about the form Narad put forward in the comments.

Epc approximate

First of all, we have to be really sure of what is meant by “p” on the left side of this equation. My first reading of it was as “momentum,” but I’m realizing that it isn’t, and this may be leading to some misunderstandings about what is meant by E=pc. The thing in the middle is average poynting vector divided by speed of light… Poynting vector has units of Watts/meter^2 and speed of light has units of meters/sec, which works out to Newtons/m^2, or force per area, which is pressure, not momentum. The thing on the right is actually in units of energy… permittivity times peak E-field^2 over 2, which is just a form of electromagnetic energy, in units of Joules. For a literal reading of the equation above, unit analysis put me at momentum = pressure = energy, which is not right (apple can’t equal orange can’t equal pear). If I take “p” as pressure rather than momentum, the left side makes sense, but the right side still doesn’t quite work.

It’s a nice try, all the elements are there. It has energy and momentum can be massaged out of it. I think the route being taken here is to try to use the form of a plane wave to figure out the momentum based on the pressure and specifically for a plane wave form of the poynting vector, or else the peak E-field intensity wouldn’t be needed.

The approach in the E=pc derivation I posted above is really different. My starting point is with a classical structure called the Electromagnetic stress-energy tensor and with a second structure which is conservation of power given energy flux. (Wikipedia actually kind of pissed me off about this: they want to masturbate over the four-dimensional relativistic version, but wouldn’t provide me a clean on-line citation for the classical version shown above; the form given here is the same as it appears in Jackson E&M) The first equation is a consequence of the Lorentz force law (F = qE+qvxB) where the system has electromagnetic waves, but is sealed so that there is no net force… the equation says that the change in Poynting vector per unit time is equal to the divergence of the electromagnetic stress-energy tensor, all of which is in units of force or change of momentum with time. The second equation is a consequence of Power=current*voltage, believe it or not, and just says that the change in energy density in the system is equal to the divergence of the Poynting vector, all in units of power. These structures make no real initial assumptions about the form that the electromagnetic fields are taking, they speak only of change of momentum per time and change of energy density given energy flux and are derived directly from application of Maxwell’s laws.

The first step is to take the stress-energy continuity relation and to hold it as change in Poynting vector with time is equal to change in momentum density with time by direct application of Newtonian force. You end up with an expression that says that Poynting vector is equal to momentum density times speed of light squared.

The second step is to throw this Poynting vector relation into the power equation so that you get a relation that says that the momentum flux out of a volume of space is equal to the change of energy density with time. This gives you a “momentum current vector” equation, which is analogous to the relationship between electrical current “I” and current vector “J.”

I next establish a momentum current, basically just a beam of light with no specific frequency or field configuration. You could write this as white light in a Fourier composition. A set of very simple manipulations gets you to a relation that directly says that energy density is equal to momentum density times speed of light. Integrate out the density and you get E=pc directly. Please note, this set-up is explicitly agnostic on the idea of photons since it depends on a mixture of frequencies to produce a constant envelope of plane waves with constant momentum density distributed everywhere and therefore does not require quantum mechanics to work. I can’t claim this work is Einstein’s because I didn’t follow anyone to make it… this is me using Jacksonian E&M technique to prove E=pc for myself, all using classical physics.

With E=pc in hand by these means, the classical derivation of E=mc^2 is pretty much a shoe-in. Again, I used no quantum and no relativity. If I could do this, the geniuses at the turn of the century got it faster;-)

Edit: 11-22-17

I must’ve done something wrong with the Latex, it doesn’t seem to want to render in the body of my post; I’m still looking into whether I need to get the plugin…

Further, I figured out what was wrong with the unit analysis I did above… the right side of that equation is energy density (J/m^3) rather than energy (J)… and since J =N*m, J/m^3 is N/m^2…. the equation above is all in units of light pressure. To get to E = pc in approximate form in the plane wave, you just need to sub in the relation for momentum density per Poynting vector S = pc^2, then cancel the density by integrating over volume.

One additional thing about the Einstein’s box derivation that is important; it works in a classical framework. What I’ve provided above, then, is E=mc^2 as a classical equation, which is really torturing the point that it was “proven.” I’ve been thinking about whether or not I was doing this right since the whole discussion started and the derivation is only consistent from the standpoint that there are no effects included taking into account the potential relativistic characteristics of the box as it moves. I’m sorry about that, Narad. The derivation above would be insufficient from a modern physics standpoint because the box would undergo length contractions and dilations as it moves. To be perfectly honest, this nagged at me a tiny bit as I wrote the derivation, but maybe not as much as it should have… I drew the box as strictly “before” and “after” so that I ended up looking at the system only when it is located in the inertial frame of reference. That would call into question the nature of the boost pushing it into motion. I was assuming that the completely undisclosed relativistics located between the end-points were sufficient to conspire that the end-points be right! And, that’s an open end since length contraction would place the wall of the box in a different location depending on the frame… throwing off the whole calculation.

(For the people at home, here is something very important about how I designed to write this blog. I leave my edits visible so that the progression of my thinking is clear… one of the hardest, most human aspects of working in sciences is facing the fact that nobody is always right about everything. I think that being a good scientist is not about being right all the time, but about changing your mind when it’s important to do so. And, it’s about admitting when someone else was right, sometimes very publicly! Are you smart if you’re unwilling to abandon a sinking ship? I think not. Smart is being able to turn the steering wheel and to grow when its necessary to do so –especially when it effects your pride. I think this is the difference between arguing loudly and arguing productively.)

Here is the derivation converting the light pressure equation Narad offered into E=pc…

Epc approximate 2

Hopefully that ties up all the loose ends! (Don’t be surprised to see me back here playing with a relativistic E=mc^2 proof at some point.)

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Interaction Picture

It’s not always about the cat. Here, I will show how to hop from the time dependent Schrodinger equation to the Interaction picture form.

This post is intended to help recover a tiny fraction of the since-destroyed post I originally entitled “NMR and Spin Flipping part II.” I have every intention to reconstruct that post when I have time, but I decided to do it in fragments because the original loss was 5,000 words. I don’t have time to bust my head against that whole mess for the moment, but I can do it in bits, I think.

One section of that post which stands pretty well as a separate entity from the NMR theme was the fraction of work where I spent time deriving a version of the time dependent Schrodinger equation in the interaction picture.

I thought I would go ahead and expand this a little bit and talk generally about some of the basic structural features of non-relativistic quantum mechanics. Likely, this will mostly not be very mathematical, except for the derivation at the end. I’ll warn you when the real derivation is about to start if you are math averse…

Everybody has heard about Schrodinger’s cat. Poor cat is dragged out and flogged semi-dead, semi-alive pretty much any time anybody wants to speak as if they know something about “quantum physics.” The cat might be the one great mascot of quantum in popular culture. The kitty drags with it a name that you no doubt have heard: Erwin Schrodinger, the guy who first coined the anecdote of feline torture as an abstraction to describe some features of quantum mechanics on a level that laymen can embrace, if not totally understand. This name is immediately synonymous with the spine of quantum mechanics as the Schrodinger equation. This equation is not so simple as E = mc^2 or F = ma, but it is a popular equation…

Schrodinger equation

I’ve included it here in its full-on psi-baiting time-dependent form with Planck’s constant uncompressed from ħ.

You hardly ever see it written this way anymore.

All this equation says is that the sum of kinetic and potential energy is total energy, which is tied implicitly to the evolution of the system with time. This equation is popular enough that I found it scrawled on a wall along with some Special Relativity inside the game “Portal 2” once. Admittedly, the game designers used ħ instead of h for Planck’s constant. It may not look that way, but the statement of this equation is no more complicated than F = ma or E = mc^2. It just says “conservation of energy” and that’s pretty much it.

Schrodinger’s equation is the source of wave mechanics, where Psi “ψ” is the notorious quantum mechanical wave function. If you care nothing more about Quantum mechanics, I could say that you’ve seen it all and we could stop here.

The structure of basic quantum mechanics has a great deal to it. Schrodinger’s equation tells you how dynamics happens in quantum physics. It says that the way the wave equation changes in time is tied to some characteristics related to the momentum of the object in question and to where it’s located. Structurally, this is the foundation of all non-relativistic quantum mechanics (I say “non-relativistic” because the more complete form of the Schrodinger equation competent to special relativistic energy is the Klein-Gordon Equation, which I will not touch anywhere in this post.) Pretty much all of quantum mechanics is about manipulating this basic relation in some manner or another in order to get what you want to know out of it. Here, the connection between position and momentum as well as between energy and time hides the famous “uncertainty relations,” all built directly into the Schrodinger equation and implicit to its solutions.

One thing you may not immediately know about Schrodinger’s equation is that it’s actually a member of a family of similar equations. In this case, the equation written above tells about the motion of an object in some volume of space, where the space in question in literally only one dimensional, along an effective line. Another Schrodinger equation (as the one written in this post) expands space into three dimensions. Still other Schrodinger equation-like forms are needed to understand how an object tumbles or rotates, or even how it might turn itself inside out or how it might play hopscotch on a crystalline lattice or bend and twist in a magnetic field. There are many different ways that the functional form above might be repurposed to express some permutation of the same set of general ideas.

This tremendous diversity is accomplished by a mathematical structure called “operator formalism.” Operators are small parcels of mathematical operation that transform the entity of the wave function in particular ways. An operator is sort of like a box of gears that hides what’s going on. You might fold down the gull-wing door in the equation above and hide the gears in an operator called the “Hamiltonian.”

Schrodinger equation 2

This just shuffles everything you don’t care about at a given time under the rug and lets you work overarching operations on the outside. Operators can encode most everything you might want. There are a ton of rules that go into the manipulation of operators, which I won’t spend time on here because it distracts from where I’m headed. A hundred types of Schrodinger equation can be written by swapping out the inside of the Hamiltonian.

An additional simplification of operators comes from what’s called “representation formalism.” The first Schrodinger equation I wrote above is within a representation of position. Knowing about the structure of the representation places many requirements which help to define the form of the Hamiltonian. I could as easily have written the same Schrodinger equation in a representation of momentum, where the position variable becomes some strange differential equation… momentum is in that equation above, but you would never know it to look at because it’s in a form related to velocity, which is connected back to position, so that position and time are the only variables relevant to the representation. By backing out of a representation, into a representation free, “abstract form,” operators lose their bells and whistles while wave functions are converted to a structure called a “ket.”

Ket is short for “Bra-Ket,” which is a representation free notation developed by Paul Dirac, another quantum luminary working in Schrodinger’s time. A “bra” is related to a “ket” by an operation called a “conjugate transform,” but you need only know that it’s a way to talk about the wave equation when you are not saying how the wave equation is represented. If you’ve dealt with kets, you’ve probably been in a quantum mechanics class… “wave function” has a place in popular culture, “ket” does not.

Most quantum mechanics is performed with operators and kets. The operators act on kets to transform them.

One place where this general structure becomes slightly upset is when you start talking about time. And, of course time is needed if you’re going to talk about how things in the real world interact or behave. The variable of time is very special in quantum mechanics because of how it enters into Schrodinger’s equation… this may not be apparent from what I’ve written above, but time is treated as its own thing. Schrodinger’s equation can be rewritten to form what’s called a time displacement operation.

You might take a breath, derivation begins here….

hamiltonian time dependence

This is just a way to completely twist around Schrodinger’s time dependent equation into a ket form where the ket now has its time dependence expressed by a time displacement modulated by the Hamiltonian. I’ve even broken up the Hamiltonian into static and time dependent parts (as this will be important to the Interaction Picture, down below). The time displacement operation just acts on the ket to push it forward in time. The thing inside the exponential is a form of quantum phase.

This ket is an example of a “state ket.” It is the abstract representation of a generalized wave function that solves Schrodinger’s time dependent equation. A second form of ket, called an “eigen ket,” emerges from a series of special solutions to the Schrodinger equation that have no time dependence. An eigen ket (I often write “eigenket”) remains the same at all times and is considered a “stationary solution” to the Schrodinger equation. “Eigen solutions” tend to be very special solutions in many other forms of physics: the notes on your flute or piano are eigen solutions, or stationary wave solutions, for the oscillatory physics in that particular instrument. In quantum mechanics, eigen modes are exceptionally useful because any general time dependent solution to the Schrodinger equation can be fabricated out of a linear sum of eigenkets. This math is connected intimately to Fourier series. The collection of all possible eigenket solutions to a particular Schrodinger equation forms a complete description of a given representation of that Schrodinger equation, which is called a Hilbert space. You can write any general solution for one particular Schrodinger equation using the Hilbert space of that equation. A particular eigenket solves the Hamiltonian of a Schrodinger equation with a constant, called an eigenvalue, which is the same as saying that an eigenket doesn’t change with time (producing Schrodinger’s time-independent wave equation).

eigen function equation 2

This is just the eigenvalue equation for the stationary part of the Hamiltonian written above, which could be expanded into Schrodinger’s time independent equation.

Deep breath now, this dives into Interaction Picture quickly.

How quantum mechanics treats time can be reduced in its extrema to two paradigms which are called “Pictures.” The first picture is called the “Schrodinger Picture,” while the second is called the “Heisenberg Picture” for Werner Heisenberg. Heisenberg and Schrodinger developed the basics of non-relativistic quantum mechanics in parallel from two separate directions; Schrodinger gave us wave mechanics while Heisenberg gave us operator formalism. They are essentially the same thing and work extremely well when used together. Schrodinger and Heisenberg pictures are connected to each other from the time displacement operator. In Schrodinger picture, the time displacement operation acts on the state ket, causing the state to evolve forward in time. In Heisenberg picture, the time displacement is shifted onto the operators and the eigenkets, while the state ket remains constant in time. Schrodinger picture is like sitting on a curbside and watching a car drive past, while Heisenberg picture is like sitting inside the car and watching the world drive past. Both pictures agree that the car is traveling the same speed, but they are looking at the situation from different vantage points. The Schrodinger time dependent equation is balanced by the Heisenberg equation of motion.

Where time dependence starts to become really interesting is if the Hamiltonian is not completely constant. As I wrote above, you might have a part of the Hamiltonian which contains some dependence on time. One way in which quantum mechanics addresses this is by a construction called the “Interaction Picture”… Sakurai also calls it the “Dirac Picture.” The interaction picture is sort of like driving along in your car and wondering at the car you’re passing; the world outside appears to be moving, as is the car you’re looking at, if only at different speeds and maybe in different directions.

I’ve likened this notion to switching frames of reference, but I caution you from pushing that analogy too far. The transformation between one picture and the next is by quantum mechanical phase, not by some sort transformation of frame of reference. Switching pictures is simply changing where time dependent phase is accumulated. As the Schrodinger picture places all this phase in the ket, Heisenberg picture places it all on the operator. Interaction picture splits the difference: the stationary phase is stuck to the operator while the time dependent phase is accumulated by the ket. In all three pictures, the same observables result (rather, the same expectation values) but the phases are broken up. Here is how the phases can be split inside a state ket.

State function in schrodinger picture 3

I’ve written the state ket as a sum of eigenkets |n>. The time dependence from a time varying potential “V” is hidden in the eigenket coefficient while the stationary phase remains behind. The “n” index of the sum allows you to step through the entire Hilbert space of eigenkets without writing any but the one. Often, the coefficient Cn(t) is what we’re ultimately interested in, so it helps to remember that it has the following form when represented in bra-ket notation:

expanding the ket 5

I’ve skipped ahead a little by writing that ket in the Interaction picture (these images were created for the NMR post that died, so they’re not quite in sequence now), but the effect is consistent. The usage of “1” here just a way to move into a Hilbert space representation of eigenkets… with probability normalized eigenkets, “spanning the space” means that you can construct a linear projection operator that is the same as identity. The 1 = sum is all that says. This is just a way to write the coefficient above in a bra-ket form.

The actual transformation to the Interaction picture is accomplished by canceling out the stationary phase…

Transformation to interaction picture 4

By multiplying through with the conjugate of the stationary phase, only the time varying phase in the coefficient remains. This extra phase will then show up on operators translated into the interaction picture…

Potential in interaction picture 7

This takes the potential as it appears in the Schrodinger picture and converts it to a form consistent with the Interaction picture.

You can then start passing these relationships through the time dependent Schrodinger equation. One must only keep in mind that every derivative of time must be accounted for and that there are several…

Time dependent Schrodinger in interaction pic 5

This little bit of algebra creates a new form for the time dependent Schrodinger equation where the time dependence is only due to the time varying potential “V”. You can then basically just drop into a representation and use all the equalities I’ve justified above…

Time dep in Interaction pic diff eq 8

The last result here has eliminated all the ket notation and created a version of the time dependent Schrodinger equation where the differential equation is for the coefficients describing how much of each eigenket shows up in the state ket. The dot over the coefficient is a shorthand to mean “time derivative.”

This form of the time dependent Schrodinger equation gives an interesting story. The interaction represented by the time dependent potential “V” scrambles eigenket m into eigenket n. As you might have guessed, this is one in the huge family of different equations related to the Schrodinger equation and this particular version has an apt use in describing interactions. Background quantum mechanical phase accumulated only by the forward passage of time is ignored in order to look at phase accumulated by an interaction.

I will ultimately use this to talk a bit more about the two state problem and NMR, as from the post that died. Much of this particular derivation appears in the Sakurai Quantum Mechanics text.

Magnets, how do they work? (part 2)

I will talk about the origin of the magnetic dipole construct here. Consider a loop of wire…

You may have noticed that I posted an entry entitled “NMR and Spin flipping (part 2)” which has since disappeared. It turns out that wordpress doesn’t synch so well between its mobile app and its main page: I had an incomplete version of the NMR post on a mobile phone which I accidentally pushed to publish and over-wrote the completed post that I had finished several days before. Thank you wordpress for not synching properly! The incomplete version had none of the intended content. As I don’t feel like reconstructing a 5,000 word post right now, I thought I would scale back a bit and bite off a tiny chunk of the big subject of how magnets work. In part, I figure I can use some of what is derived here in the next version of the NMR post, which I intend to rewrite.

So, this will be the continuation of my series about magnets.

Reading through the initial magnets post, you will see that I did a rather spectacular amount of math, some of it unquestionably uncalled for. But, hey, the basic point of a blog is excess. One of the windfalls of all that math can yield an important theoretic construct which turns out to be one of the most major contributors of the explanation of how magnets work.

What this has to do with a loop of wire, I’ll come back to…

When an exact answer is not available to a physics question, one of the go-to strategies used by physicists is series approximation. Often, the low orders of a series tend to contribute to solutions more strongly than the high orders, meaning that the first couple terms in an expansion can be good approximations. One such expansion is used in magnetism.

Recall the relation between the magnetic field and the magnetic vector potential:

BS to amp 2

This expression is useful because the crazy vector junk is moved outside the integral. The magnetic potential is easier to work with than the magnetic field as a result. The expansion of interest is usually directed at the vector potential and is called the “multipole expansion.” There are many ways to run the multipole expansion, but maybe the easiest (for me) is to come back to our old friends the spherical harmonics Ylm.

 

1 multipole expansion

In the vector potential of the magnetic field, that r-r’ factor in the denominator is really hard to work with. By itself, it is usually too complicated to integrate over. The multipole expansion lets us replace it with something that can be calculated. In this expansion, r is the location where we’re looking for the field while r’ is where the current which sources the field is located. The expansion is converting the difference in these (the propagator which pushes influence from the location of the current to the location of the field) into an infinite series of terms: in the sum, r< is whichever of the two distances is lesser, while r> is which of these two is greater. If you’re looking at a location inside the current distribution, r’ is bigger than r… but if you’re looking at a location outside of the current distribution, r is bigger than r’. The Ylms appear because space has a spherical polar geometry.

The substitution changes the form of the vector potential:

2 expanded vector potential

 

The vector potential is now a sum of an infinite number of terms inside the integral. You still can’t just compute that because this sequence converges to 1/r-r’, which you can’t calculate by itself anyway. What you can do is introduce a cut-off. This is literally where the multipole terms all come from: instead of calculating the entire series all at once, you only calculate one term (or one level of terms, as the case may be). If you take l=0, you get the monopole term, if you take l=1, you get the dipole term, and so on and so forth for higher orders of l.

Since I’m interested in magnetic dipoles right now, this is the crux: I’ve simply called the l=1 term “the dipole” by definition. Further, I care only about locations where I’m looking for the magnetic field well outside of the dipole, since I’m not going to look directly inside of the bar magnet to start with, so that r>r’. For the dipole, l=1 and I only care about m=-1,0 and 1 of the Ylms. This collapses the sum to just three terms.

3 dipole term of multipole

If you’ve spent any time messing around with either E&M or quantum, you may remember those three Ylms off the top of your head. They’re basically just sines and cosines.

4 ylms

I will note, this whole expansion can be done in terms of Legendre polynomials too, but I remember the Ylms better. For some expedience, I will focus on the Ylm part of the integral in order to help bring it into a more manageable form before moving on.

5 compressing ylmsThere’s a lot of trig in here, but the final form is actually very much more manageable than where I started. I’ve highlighted the pattern in red and green. If you squint really really hard at this, you’ll realize that it’s a dot product of the cartesian form of the hatted unit vector r. So, it’s just a dot product of cartesian unit vectors…

6 simplification of spherical polar terms

This dials down to just a dot product of two unit vectors pointing in the directions toward either where the current is located or where the field is. I’ve installed it in the vector potential in the last line. I note explicitly that both of these are functions of the spherical polar angles since this will be important when I start working integrals.

If all things were equal, I could start doing calculus right now. Unfortunately, I don’t know the form of the current vector. That could be any distribution of currents imaginable and not all of them have pure dipole contributions. Working the problem as is, the set-up will respond to the dipole moment of whatever J-current I choose to install. You could imagine a case with a non-zero current where this particular integral goes to zero –if I did a line of current going in some constant direction, that would probably kill this integral. But, I do know of one current distribution in particular that has a very high dipole contribution… you might recognize this as post hoc reasoning, but I’m doing this to try to focus our attention on how one particular term in the multipole expansion behaves. The current distribution which is most interesting here is a loop of wire with a electric current circling it.

7 dipole current

I’ve sketched out the current vector here as well as a set of axes showing the relationship between the spherical polar and cartesian coordinates where the unit vectors are all labeled. This vector current is just a current ‘I’ constrained to the X-Y plane, maintaining a loop around the origin at a radius of R. The current runs in a direction phi, which is tangential to the loop in a counterclockwise sense, and presumably has a positive current definition. The delta functions do the constraining to the X-Y plane. The factor of sine and radius in the denominator is a correction for use of the delta function in a spherical polar measure. The factor 2 is included to avoid a double-counting problem with a loop which shows up more explicitly, for example, in Jackson E&M, where the definition of the magnetic dipole moment is directly written with respect to the current vector. You’ll be happy to know that pretty much none of my work here actually follows Jackson, though the set-up is based strongly on the methods used in Jackson (I hated how Jackson set up his delta functions because I found them opaque as hell! But, that’s Jackson for you…)

The measure of integration is the typical spherical polar measure. You may remember my defining this in my post on the radial solution of the hydrogen atom. I’ll just quote it here. If you’ve done any vector calculus, it should be familiar anyway.

8 integration measure

I can then put these all together in the vector potential, collect the terms and begin solving it.

9 beginning integration

In the third line, I pulled everything out front that I don’t need inside the integral. The radial portion of the integral collapses on the delta function. The angular portion is somewhat harder because it involves a couple unit vectors that vary with the angles; one of the unit vectors, the unprimed r, could actually be pulled outside the integral, but I left it in to help display a useful construct that will help me simplify the integral again. I will again focus on the vector portion inside this integral:

10 unit vector adjustment

This use of the BAC-CAB rule allows me to change the unit vectors around into a cross product and flip the direction slightly. In the next step, by converting the theta unit vector into a cartesian form, the integral becomes trivial.

11 solving integral

This solves the integral. Use of the delta function guts the theta coordinate and no remaining dependence exists for phi. After the hatted unit vectors are decoupled from the integration coordinates, the cross product gets pulled out front in an uncomplicated form. You can then collect and cancel in what remains:

12 vector potential of magnetic dipole moment

Here, I’ve collected a particular quantity dependent on electric current running around in a loop which I have called a “magnetic dipole moment.” I conspired pretty strongly to get all the variable terms to pop out in a form that people will find familiar. A magnetic dipole is simply a loop, which can be of arbitrary shape, it turns out. This current loop is always right-hand defined, as above, to be “current x area” pointed in a direction normal to the area. This object could simply be a wire loop. At this point, you should be having images of stereotypical electromagnets which are many wire loops wrapped around some solid core. This electrical current configuration is very special because of the magnetic field that it tends to produce.

As an aside, I’ve seen dipole moment derived in a much more simplistic fashion than presented here, but my purpose was to be a bit more complete without actually duplicating Jackson… which I’ve mostly avoided, believe it or not… and to produce the form which can generate the whole dipole magnetic field, which can’t be done in the E&M 102 variety derivation. The simple derivation tends to operate on the axis of the magnetic dipole only, and does not calculate the shape of the field elsewhere in space. To get the whole field, you need to be a bit more sophisticated.

Magnetic field is produced by taking the curl of the vector potential, as I wrote far above. The fastest way I’ve found to take this curl is using the spherical polar definition of the curl, found here. You can derive this form of the curl in a manner very similar to what I did in my hydrogen atom radial equation post, but I’m going to hold off deriving it here: I’m somewhat short on time and I had hoped that this post wouldn’t get too very long.

13 setting up curl

My starting point here is to figure out how much of the curl I actually need. If you massage the terms inside the vector potential, you rapidly discover that only one of the three vector components is present, thus simplifying the curl. And, of course, to get to the magnetic field from here, I just need to take a curl…

14 getting magnetic field

The last thing I end up with here is an accepted form for the dipolar magnetic field:

15 magnetic dipole field

This is an exact solution for the magnetic field from a current dipole. This particular solution is dependent on the assumption that the location where you’re examining the field is large compared to the size of the loop; for real physical dipoles of appreciable size, there can be other non-zero terms in the multipole expansion, meaning that the field will be predominantly what’s written here with some small deviations.

Admittedly, this mathematical equation doesn’t have a very intuitive form. Why in the world do I care about deriving this particular equation? To understand, we need some choice pictures…

edit 10-25-17: It seemed kind of ridiculous that I worked through all that math to find the dipole field and then stole other people’s diagrams of it. For completeness, here’s a vector plot of mine in Mathematica of the field equation written above:

magnetic dipole.jpg

Another magnetic dipole picture with the location of the dipole explicitly drawn in:

440px-vfpt_dipole_magnetic3-svg

This image, where the dipole is rotated by 90 degrees from how I plotted it, is taken from wikipedia.

My interest in this field becomes more obvious when compared side-by-side with the magnetic field produced by a bar magnet…

This image is taken from how-things-work-science-projects.com. The field produced by a bar magnet is very similar in shape to the field produced by the loop of wire. Going further out, here is a diagram of the magnetic field produced by the Earth:

f823ef888c061dd0ca7f663b103fc4866ff9a197

Notice some similarity? You’ll notice the Earth’s field lines are assigned to point oppositely from my diagram above, but that has to do with how compass needles orient rather than from any actual fundamental difference in the field!

Physical ferromagnets tend frequently to have dipolar magnetic fields. As such, the quantity of the magnetic dipole moment has huge physical importance. Granted, the field of the Earth isn’t perfectly dipolar, but it has an overwhelming dipole contribution. Other planets also have fields that are dipolar in shape.

Understanding how magnets work, compass needles, bar magnets and most sorts of permanent magnets, requires dipolar behavior as the underlying structure. Even the NMR post that got ruined was about a quantum mechanical phenomenon which revolves around magnetic dipoles.

This is a large step forward. I haven’t explained much, but I will write another post later showing why it is that magnets, particularly dipoles, respond to magnetic fields, as well as what the source of magnetism is in ferromagnets (no off-switch on the current for God’s sake!) Stay tuned for part 3!

edit 11-5-17

Playing around with matplotlib, I constructed a streamplot of the magnetic field produced by three dipoles, all flattened into the same plane and oriented facing different directions in that plane. This is all just superpositions using the field determined above. Kind of pretty…

mag_dipole1

Chemical Orbitals from Eigenstates

A small puzzle I recently set for myself was finding out how the hydrogenic orbital eigenstates give rise to the S- P- D- and F- orbitals in chemistry (and where s, p, d and f came from).

The reason this puzzle is important to me is that many of my interests sort of straddle how to go from the angstrom scale to the nanometer scale. There is a cross-over where physics becomes chemistry, but chemists and physicists often look at things very differently. I was not directly trained as a P-chemist; I was trained separately as a Biochemist and a Physicist. Remarkably, the Venn diagrams describing the education for these pursuits only overlap slightly. When Biochemists and Molecular Biologists talk, the basic structures below that are frequently just assumed (the scale here is >1nm), while Physicists frequently tend to focus their efforts toward going more and more basic (the scale here is <1 Angstrom). This leads to a clear non-overlap in the scale where chemistry and P-chem are relevant (~1 angstrom). Quite ironically, the whole periodic table of the elements lies there. I have been through P-chem and I’ve gotten hit with it as a Chemist, but this is something of an inconvenient scale gap for me. So, a cat’s paw of mine has been understanding, and I mean really understanding, where quantum mechanics transitions to chemistry.

One place is understanding how to get from the eigenstates I know how to solve to the orbitals structuring the periodic table.

1200px-periodic_table_chart

This assemblage is pure quantum mechanics. You learn a huge amount about this in your quantum class. But, there are some fine details which can be left on the counter.

One of those details for me was the discrepancy between the hydrogenic wave functions and the orbitals on the periodic table. If you aren’t paying attention, you may not even know that the s-, p-, d- orbitals are not all directly the hydrogenic eigenstates (or perhaps you were paying a bit closer attention in class than I was and didn’t miss when this detail was brought up). The discrepancy is a very subtle one because often times when you start looking for images of the orbitals, the sources tend to freely mix superpositions of eigenstates with direct eigenstates without telling why the mixtures were chosen…

For example, here are the S, P and D orbitals for the periodic table:

ao

This image is from http://www.chemcomp.com. Focusing on the P row, how is it that these functions relate to the pure eigenstates? Recall the images that I posted previously of the P eigenstates:

P-orbital probabiltiy densityorbital21-1 squared2

In the image for the S, P and D orbitals, of the Px, Py and Pz orbitals, all three look like some variant of P210, which is the pure state on the left, rather than P21-1, which is the state on the right. In chemistry, you get the orbitals directly without really being told where they came from, while in physics, you get the eigenstates and are told somewhat abstractly that the s-, p-, d- orbitals are all superpositions of these eigenstates. I recall seeing a professor during an undergraduate quantum class briefly derive Px and Py, but I really didn’t understand why he selected the combinations he did! Rationally, it makes sense that Pz is identical to P210 and that Px and Py are superpositions that have the same probability distribution as Pz, but are rotated into the X-Y plane ninety degrees from one another. How do Px and Py arise from superpositions of P21-1 and P211? P21-1 and P211 have identical probability distributions despite having opposite angular momentum!

Admittedly, the intuitive rotations that produce Px and Py from Pz make sense at a qualitative level, but if you try to extend that qualitative understanding to the D-row, you’re going to fail. Four of the D orbitals look like rotations of one another, but one doesn’t. Why? And why are there four that look identical? I mean, there are only three spatial dimensions to fill, presumably. How do these five fit together three dimensionally?

Except for the Dz^2, none of the D-orbitals are pure eigenstates: they’re all superpositions. But what logic produces them? What is the common construction algorithm which unites the logic of the D-orbitals with that of the P-orbitals (which are all intuitive rotations).

I’ll actually hold back on the math in this case because it turns out that there is a simple revelation which can give you the jump.

As it turns out, all of chemistry is dependent on angular momentum. When I say all, I really do mean it. The stability of chemical structures is dependent on cases where angular momentum has tended in some way to cancel out. Chemical reactivity in organic chemistry arises from valence choices that form bonds between atoms in order to “complete an octet,” which is short-hand for saying that species combine with each other in such a way that enough electrons are present to fill in or empty out eight orbitals (roughly push the number of electrons orbiting one type of atom across the periodic table in its appropriate row to match the noble gases column). For example, in forming the salt crystal sodium chloride, sodium possesses only one electron in its valence shell while chlorine contains seven: if sodium gives up one electron, it goes to a state with no need to complete the octet (with the equivalent electronic completion of neon), while chlorine gaining an electron pushes it into a state that is electronically equal to argon, with eight electrons. From a physicist stand-point, this is called “angular momentum closure,” where the filled orbitals are sufficient to completely cancel out all angular momentum in that valence level. As another example, one highly reactive chemical structure you might have heard about is a “radical” or maybe a “free radical,” which is simply chemist shorthand for the situation a physicist would recognize contains an electron with uncancelled spin and orbital angular momentum. Radical driven chemical reactions are about passing around this angular momentum! Overall, reactions tend to be driven to occur by the need to cancel out angular momentum. Atomic stoichiometry of a molecular species always revolves around angular momentum closure –you may not see it in basic chemistry, but this determines how many of each atom can be connected, in most cases.

From the physics, what can be known about an orbital is essentially the total angular momentum present and what amount of that angular momentum is in a particular direction, namely along the Z-axis. Angular momentum lost in the X-Y plane is, by definition, not in either the X or Y direction, but in some superposition of both. Without preparing a packet of angular momentum, the distribution ends up having to be uniform, meaning that it is in no particular direction except not in the Z-direction. For the P-orbitals, the eigenstates are purely either all angular momentum in the Z-direction, or none in that direction. For the D-orbitals, the states (of which there are five) can be combinations, two with angular momentum all along Z, two with half in the X-Y plane and half along Z and one with all in the X-Y plane.

What I’ve learned is that, for chemically relevant orbitals, the general rule is “minimal definite angular momentum.” What I mean by this is that you want to minimize situations where the orbital angular momentum is in a particular direction. The orbits present on the periodic table are states which have canceled out angular momentum located along the Z-axis. This is somewhat obvious for the homology between P210 and Pz. P210 points all of its angular momentum perpendicular to the z-axis. It locates the electron on average somewhere along the Z-axis in a pair of lobes shaped like a peanut, but the orbital direction is undefined. You can’t tell how the electron goes around.

As it turns out, Px and Py can both be obtained by making simple superpositions of P21-1 and P211 that cancel out z-axis angular momentum… literally adding together these two states so that their angular momentum along the z-axis goes away. Px is the symmetric superposition while Py is the antisymmetric version. For the two states obtained by this method, if you look for the expectation value of the z-axis angular momentum, you’ll find it missing! It cancels to zero.

It’s as simple as that.

The D-orbitals all follow. D320 already has no angular momentum on the z-axis, so it is directly Dzz. You therefore find four additional combinations by simply adding states that cancel the z-axis angular momentum: D321 and D32-1 symmetric and antisymmetric combinations and then the symmetric and antisymmetric combinations of D322 and D32-2.

Notice, all I’m doing to make any of these states is by looking at the last index (the m-index) of the eignstates and making a linear combination where the first index plus the second gives zero. 1-1 =0, 2-2=0. That’s it. Admittedly, the symmetric combination sums these with a (+) sign and a 1/sqrt(2) weighting constant so that Px = (1/sqrt(2))(P21 + P21-1) is normalized and the antisymmetric combination sums with a (-) sign as in Py = (1/sqrt(2))(P211 – P21-1), but nothing more complicated than that! The D-orbitals can be generated in exactly the same manner. I found one easy reference on line that loosely corroborated this observation, but said it instead as that the periodic table orbitals are all written such that the wave functions have no complex parts… which is also kind of true, but somewhat misleading because you sometimes have to multiply by a complex phase to put it genuinely in the form of sines for the polar coordinate (and as the polar coordinate is integrated over 360 degrees, expectation values on this coordinate, as z-axis momentum would contain, cancel themselves out; sines and cosines integrated over a full period, or multiples of a full period, integrate to zero.)

Before I wrap up, I had a quick intent to touch on where S-, P-, D- and F- came from. “Why did they pick those damn letters?” I wondered one day. Why not A-, B-, C- and D-? The nomenclature emerged from how spectral lines appeared visually and groups were named: (S)harp, (P)rincipal, (D)iffuse and (F)undamental. (A second interesting bit of “why the hell???” nomenclature is the X-ray lines… you may hate this notation as much as me: K, L, M, N, O… “stupid machine uses the K-line… what does that mean?” These letters simply match the n quantum number –the energy level– as n=1,2,3,4,5… Carbon K-edge, for instance, is the amount of energy between the n=1 orbital level and the ionized continuum for a carbon atom.) The sharpness tends to reflect the complexity of the structure in these groups.

As a quick summary about structuring of the periodic table, S-, P-, D-, and F- group the vertical columns (while the horizontal rows are the associated relative energy, but not necessarily the n-number). The element is determined by the number of protons present in the nucleus, which creates the chemical character of the atom by requiring an equal number of electrons present to cancel out the total positive charge of the nucleus. Electrons, as fermions, are forced to occupy distinct orbital states, meaning that each electron has a distinct orbit from every other (fudging for the antisymmetry of the wave function containing them all). As electrons are added to cancel protons, they fall into the available orbitals depicted in the order on the periodic table going from left to right, which can be a little confusing because they don’t necessarily purely close one level of n before starting to fill S-orbitals of the next level of n; for example at n=3, l can equal 0, 1 and 2… but, the S-orbitals for n=4 will fill before D-orbitals for n=3 (which are found in row 4). This has purely to do with the S-orbitals having lower energy than P-orbitals which have lower energy than D-orbitals, but that the energy of an S-orbital for a higher n may have lower energy than the D-orbital for n-1, meaning that the levels fill by order of energy and not necessarily by order to angular momentum closure, even though angular momentum closure influences the chemistry. S-, P-, D-, and F- all have double degeneracy to contain up and down spin of each orbital, so that S- contains 2 instead of 1, P- contains 6 instead of 3, and D- from 10 instead of 5. If you start to count, you’ll see that this produces the numerics of the periodic table.

Periodic table is a fascinating construct: it contains a huge amount of quantum mechanical information which really doesn’t look much like quantum mechanics. And, everybody has seen the thing! An interesting test to see the depth of a conversation about periodic table is to ask those conversing if they understand why the word “periodic” is used in the name “Periodic table of the elements.” The choice of that word is pure quantum mechanics.

Powerball Probabilities

If you’ve read anything else in this blog, you’ll know I write frequently about my playing around with Quantum Mechanics. As a digression away from a natural system that is all about probabilities, an interesting little toy problem I decided to tackle is figuring out how the “win” probabilities are determined in the lottery game Powerball.

Powerball is actually quite intriguing to me. They have a website here which details by level all the winners across the whole country who have won a Powerball prize in any given drawing. You may have looked at this chart at some point while trying to figure out if your ticket won something useful. A part of what intrigues me about this chart is that it tells you in a given drawing exactly how much money was spent on Powerball and how many people bought tickets. How does it tell you this? Because probability is an incredibly reliable gauge of behavior with big samples sizes. And, Powerball quite willingly lays all the numbers out for you to do their book keeping for them by telling you exactly how many people won… particularly at the high-probability-to-win levels which push into the regime of Gaussian statistics. For big samples, like millions of people buying powerball tickets, where N=big, the errors on average values become relatively insignificant since they go as sqrt(N). And, the probabilities reveal what those average values are.

The game is doubly intriguing to me because of the psychological component that drives it. As the pot becomes big, people’s willingness to play becomes big even though the probabilities never change. It suddenly leaps into the national consciousness every time the size of the pot becomes big and people play more aggressively as if they had a greater chance of winning said money. It is true that somebody ultimately walks away with the big pot, but what’s the likelihood that somebody is you?

But, as a starter, what are the probabilities that you win anything when you buy a ticket? To understand this, it helps to know how the game is set up.

As everybody knows, powerball is one of these games where they draw a bunch of little balls printed with numbers out of a machine with a spinning basket and you, as the player, simply match the numbers on your ticket to the numbers on the balls. If your ticket matches all the numbers, you win big! And, as an incentive to make people feel like they’re getting something out of playing, the powerball company awards various combinations of matching numbers and adds in multipliers which increase the size of the award if you do get any sort of match. You might only match a number or two, but they reward you a couple bucks for your effort. If you really want, you can pick the numbers yourself, but most people simply grab random numbers spat out of a computer… not like I’m telling you anything you don’t already know at this point.

One of the interesting qualities of the game is that the probabilities of prizes are very easy to adjust. The whole apparatus stays the same; they just add or subtract balls from the basket. In powerball, as currently run, there are two baskets: the first basket contains 69 balls while the second contains 26. Five balls are drawn from the first basket while only one, the Powerball, is drawn from the second. There is actually an entire record available of how the game has been run in the past, how many balls were in either the first or second baskets and when balls were added or subtracted from each. As the game has crossed state lines and the number of players has grown, the number of balls has also steadily swelled. I think the choice in numbering has been pretty careful to make the smallest prize attainably easy to get while pushing the chances for the grand prize to grow enticingly larger and larger. Prizes are mainly regulated by the presence of the Powerball: if your ticket manages to match the Powerball and nothing else, you win a small prize, no matter what. Prizes get bigger as a larger number of the other five balls are matched on your ticket.

The probabilities at a low level work almost exactly as you would expect: if there are 26 balls in the powerball basket, at any given drawing, you have 1 chance in 26 of matching the powerball. This means that you have 1 chance in 26 of winning some prize as determined by the presence of the powerball. There are also prizes for runs of larger than three matching balls drawn from the main basket, which tends to push the probabilities of winning anything to a slightly higher frequency than 1 in 26.

For the number savvy this begins to reveal the economics of powerball: an assured win by these means requires you to spend, on average, $48. That’s 26 tickets where you are likely to have one that matches the powerball. Note, the prize for matching that number is $4. $44 dollars spent to net only $4 is a big overall loss. But, this 26 ticket buy-in is actually hiding the fact that you have a small chance of matching some sequence of other numbers and obtaining a bigger prize… and it would certainly not be an economic loss if you matched the powerball and then the 5 other balls, yielding you a profit in the hundreds of millions of dollars (and this is usually what people tell themselves as they spend $2 for each number).

The probability to win the matched powerball prize only, that is to match just the powerball number, is actually somewhat worse than 1 in 26. The probability is attenuated by the requirement that you hit no matches on any other of the five possible numbers drawn.

Finding the actual probability is as follows: (1/26)*(64/69)*(63/68)*(62/67)*(61/66)*(60/65). If you multiply that out and invert it, you get 1 hit in 38.32 tries. The first number is, of course, the chances of hitting the powerball, while the other five are the chance of hitting numbers that aren’t picked… most of these probabilities are naturally quite close to 1, so you are likely to hit them, but they are probabilities that count toward hitting the powerball only.

This number may not be that interesting to you, but lots of people play the game and that means that the likelihood of hitting just the powerball is close to Gaussian. This is useful to a physicist because it reveals something about the structure of the Powerball playing audience on any given week: that site I gave tells you how many people won with only the powerball, meaning that by multiplying that number by 38.32, you know how many tickets were purchased prior to the drawing in question. For example, as of the August 12 2017 drawing, 1,176,672 numbers won the powerball-only prize, meaning that very nearly 38.32*1,176,672 numbers were purchased: ~45,090,071 numbers +/- 6,715, including error (notice that the error here is well below 1%).

How many people are playing? If people mostly purchase maybe two or three numbers, around 15-20 million people played. Of course, I’m not accounting for the slavering masses who went whole hog and dropped $20 on numbers; if everybody did this, 4.5 million people played… truly, I can’t really know people’s purchasing habits for certain, but I can with certainty say that only a couple tens of millions of people played.

The number there reveals quite clearly the economics of the game for the period between the 8/12 drawing and the one a couple days prior: $90 million was spent on tickets! This is really quite easy arithmetic since it’s all in factors of 2 over the number of ticket numbers sold. If you look at the total prize pay-out, also on that page I provided, $19.4 million was won. This means that the Powerball company kept ~$70 million made over about three days, of which some got dumped into the grand prize and some went to whatever overhead they keep (I hear at least some of that extra is supposed to go into public works and maybe some also ends up in the Godfather’s pocket). Lucrative business.

If you look at the prize payouts for the game, most of the lower level prizes pay off between $4 and $7. You can’t get a prize that exceeds $100 until you match at least 4 balls. Note, here, that the probability of matching 4 balls (including the powerball) is about 1 in 14,494. This means, that to assure yourself a prize of $100, you have to spend ~$29,000. You might argue that in 14,494 tickets, you’ll win a couple smaller prizes ($4 prizes are 1 in 38, 1 in 91, and $7 prizes are 1 in 700 and 1 in 580) and maybe break even. Here’s the calculation for how much you’ll likely make for that buy-in: $4*(14,494*(1/38 + 1/91)) + $7*(14,494*(1/700 + 1/580))… I’ve rounded the probabilities a bit… =$2482.65. For $29,000 spent to assure a single $100 win, you are assured to win at most $2500 from lesser winnings for a total loss of $27,500. Notice, $4 on a $44 loss is about 10%, while $2500 on $27,500 is also about 10%… the payoff does not improve at attainable levels! Granted, there’s a chance at a couple hundred million, but the probability of the bigger prize is still pretty well against you.

Suppose you are a big spender and you managed to rake up $29,000 in cash to dump into tickets, how likely is it that you will win just the $1 million prize? That’s five matched balls excluding the powerball. The probability is 1 in 11,688,053. By pushing the numbers, your odds of this prize have become 14,500/11,688,053, or about 1 chance in 800. Your odds are substantially improved here, but 1 in 800 is still not a wonderful bet despite the fact that you assured yourself a fourth tier prize of $100! The grand prize is still a much harder bet with odds running at about 1 in 20,000, despite the amount you just dropped on it. Do you just happen to have $30,000 burning a hole in your pocket? Lucky you! Lots of people live on that salary for a year.

Most of this is simple arithmetic and I’ve been bandying about probabilities gleaned from the Powerball website. If you’re as curious about it as me, you might be wondering exactly how all those probabilities were calculated. I gave an example above of the mechanical calculation of the lowest level probability, but I also went and figured out a pair of formulae that calculate any of the powerball prize probabilities. It reminded me a bit of stat mech…

prob without powerball

prob with powerball

number for hits

I’ve colored the main equations and annotated the the parts to make them a little clearer. The final relation just shows how you can see the number of tries needed in order to hit one success, given a probability as calculated with the other two equations. The first equation differs from the second in that it refers to probabilities where you have matched numbers without managing to match the powerball, while the second is the complement, where you match numbers having hit the powerball. Between these two equations, you can calculate all the probabilities for the powerball prizes. Since probabilities were always hard for me, I’ll try to explain the parts of these equations. If you’re not familiar with the factorial operation, this is what is denoted by the exclamation point “!” and it denotes a product string counting up from one to the number of the factorial… for example 5! means 1x2x3x4x5. The special case 0! should be read as 1. The first part, in blue, is the probability relating to either hitting on missing the powerball, where K = 26, the number of balls in the powerball basket. The second part (purple) is the multiplicity and tells you how many ways that you can draw a certain number of matches (Y) to fill a number of open slots (X), while drawing a number of mismatches (Z) in the process, where X=Y+Z. In powerball, you draw five balls, so X=5 and Y is the number of matches (anywhere from 0 to 5), while Z is the number of misses. Multiplicity shows up in stat mech and is intimately related to entropy. The totals drawn (green) is perhaps mislabeled… here I’m referring to the number of possible choices in the main basket, N=69, and the number of those that will not be drawn M = N – X, or 64. I should probably have called it “Main basket balls” or something. The last two parts determine the probabilities related to the given number of hits (Y) (orange) and the given number of misses (Z) (red) and I have applied the product operator to spiffy up the notation. Product operator is another iterand much like the summation operator and means that you repeatedly multiply successive values, much like a factorial, but where the value you are multiplying is produced from a particular range and given a set form. In these, the small script m and n start at zero (my bad, this should be under the Pi) and iterate until they are just less than the number up top (Y – 1 or Z – 1 and not equal to). At the extreme cases of either all hits or all misses, the relevant product operator (either Miss or Hit respectively) must be set equal to one in order to not count it.

This is one of those rare situations where the American public does a probability experiment with the values all well recorded where it’s possible to see the outcomes. How hard is it to win the grand prize? Well, the odds are one in 292 million. Consider that the population of the United States is 323 million. That means that if everybody in the United States bought one powerball number, about one person would win.

Only one.

Thanks to the power of the media, everybody has the opportunity to know that somebody won. Or not. That this person exists, nobody wants to doubt, but consider that the odds of winning are so scant that you not only won’t win, but you pretty likely will never meet anyone who did. Sort of surreal… everything is above board, you would think, but the rarity is so rare that there’s no assurance that it ever actually happens. You can suppose that maybe it does happen because people do win those dinky $4 prizes, but maybe this is just a red herring and nobody really actually wins! Those winner testimonials could be from actors!

Yeah, I’m not much of a conspiracy theorist, but it is true that a founding tenant of the idea of a ‘limit’ in math is that 99.99999% is effectively 100%. Going to the limit where the discrepancy is so small as to be infinitesimal is what calculus is all about. It is fair to say that it very nearly never happens! Everybody wants to be the one who beats the odds, which is why Powerball tickets are sold, but the extraordinarily vast majority never will win anything useful… I say “useful” because winning $4 or $7 is always a net loss. You have to win one of the top three prizes for it to be anywhere near worth anything, which you likely never will.

One final fairly interesting feature of the probability is that you can make some rough predictions about how frequently the grand prize is won based on how frequently the first prize is won. First prize is matching all five of the balls, but not the powerball. This frequency is about once per 12 million numbers, which is about 26 times more likely than all 5 plus the Powerball. In the report on winnings, a typical frequency is about 2 to 3 winners per drawing. About 1 time in 26 a person with all five manages to get the powerball too, so, with two drawings per week and about 2.5 first prize winners per drawing, that’s five winners per week… which implies that the grand prize should be won at a frequency of about once every five to six weeks –every month and a half or so. The average here will have a very large standard deviation because the number of winners is compact, meaning that the error is an appreciable portion of the measurement, which is why there is a great deal of variation in period between times when the grand prize is won. The incidence becomes much more Poissonian and stochastic, and allows some prizes to get quite big compared to others and causes their values to disperse across a fairly broad range. Uncertainty tends to dominate, making the game a bit more exciting.

While the grand prize is small, the number of people winning the first prize in a given week is small (maybe none or one), but this number grows in proportion to the size of the grand prize (maybe 5 or 6 or as high as 9). When the prize grows large enough to catch the public consciousness, the likelihood that somebody will win goes up simply because more people are playing it and this can be witnessed in the fluctuating frequency of the wins of lower level prizes. It breathes around the pulse of maybe 200 million dollars, lubbing at 40 million (maybe 0 to 1 person winning the first prize) and dubbing at 250 million (with 5 people or more winning the first prize).

Quite a story is told if you’re boring and as easily amused as me.

In my opinion, if you do feel inclined to play the game, be aware that when I say you probably won’t win, I mean that the numbers are so strongly against you that you do not appreciably improve your odds by throwing down $100 or even $1,000. The little $4 wins do happen, but they never pay and $1,000 spent will likely not get you more than $100 in total of winnings. It might as well be a voluntary tax. Cherish the dream your $2 buys, but do not stake your well-being on it. There’s nothing wrong with dreaming as long as you understand where to wake up.

(edit 8-24-17)

There was a grand prize winner last night (Wednesday 8-23-17). The outcomes are almost completely as should be expected: the winner is in Massachusetts… the majority of the country’s population is located in states on either the east or west coast, so this is unsurprising. There were 40 match 5 winners, so you would anticipate at least one to be a grand prize winner, which is exactly what happened (1 in 26 difference between 5 with powerball and 5 without). There were about 5.9 million powerball-only winners, so 38.32*5.9 is 226 million total powerball numbers sold in the run-up to last night’s drawing… with grand prize odds of 1 in 292 million, this is approaching parity. This means that more than $452 million was spent since Saturday on powerball lottery numbers (calculation excludes the extra dollar spent on multipliers). About five times as many ticket numbers were sold for this drawing as when I made my original analysis a week ago. With that many tickets sold, there was almost assuredly going to be a winner last night. This is not to say there shouldn’t have been a winner before this –probability is a fickle mistress– but the numbers are such that it was unlikely, but not impossible, for the prize to grow bigger. The last time the powerball was won was on 6-10-17, about two months and thirteen days ago… you can know that this is an unusually large jackpot because this period is longer than the usual period between wins (I had generously estimated 6 weeks based on the guess of 2 match 5 winners per drawing, but I think this might actually be a bit too high).

There was only one grand prize winning number out of 226 million tickets sold (not counting all the drawings that failed to yield a grand prize winner prior to this.) Think on that for a moment.

Parity symmetry in Quantum Mechanics

I haven’t written about my problem play for a while. Since last I wrote about rotational problems, I’ve gone through the entire Sakurai chapter 4, which is an introduction to symmetry. At the moment, I’m reading Chapter 5 while still thinking about some of the last few problems in Chapter 4.

I admit that I had a great deal of trouble getting motivated to attack the Chapter 4 problems. When I saw the first aspects of symmetry in class, I just did not particularly understand it. Coming back to it on my own was not much better. Abstract symmetry is not easy to understand.

In Sakurai chapter 4, the text delves into a few different symmetries that are important to quantum mechanics and pretty much all of them are difficult to see at first. As it turns out, some of these symmetries are very powerful tools. For example, use of the reflection symmetry operation in a chiral molecule (like the C-alpha carbon of proteins or the hydrated carbons of sugars) can reveal neighboring degenerate ground states which can be accessed by racemization, where an atomic substituent of the molecule tunnels through the plane of the molecule and reverses the chirality of the state at some infrequent rate. Another example is translation symmetry operation, where a lattice of identical attractive potentials serves to hide a near infinite number of identical states where a bound particle can hop from one minimum to the next and traverse the lattice… this behavior essentially a specific model describing the passage of electrons through a crystalline semiconductor.

One of the harder symmetries was time reversal symmetry. I shouldn’t say “one of the harder;” for me time reversal was the hardest to understand and I would be hesitant to say that I completely understand it yet. Time reversal operator causes time to translate backward, making momenta and angular momenta reverse. Time reversal is really hard because the operator is anti-unitary, meaning that the operation switches the sign on complex quantities that it operates on. Nevertheless, time reversal has some interesting outcomes. For instance, if a spinless particle is bound to a fixed center where the state in question is not degenerate (Only one state at the given energy), time reversal says that the state can have no average angular momentum (it can’t be rotating or orbiting). On the other hand, if the particle has spin, the bound state must be degenerate because the particle can’t have no angular momentum!

A quick digression here for the laymen: in quantum mechanics, the word “degenerate” is used to refer to situations where multiple states lie on top of one another and are indistinguishable. Degeneracy is very important in quantum mechanics because certain situations contain only enough information to know an incomplete picture of the model where more information is needed to distinguish alternative answers… coexisting alternatives subsist in superposition, meaning that a wave function is in a superposition of its degenerate alternative outcomes if there is no way to distinguish among them. This is part of how entanglement arises: you can generate entanglement by creating a situation where discrete parts of the system simultaneously occupy degenerate states encompassing the whole system. The discrete parts become entangled.

Symmetry is important because it provides a powerful tool by which to break apart degeneracy. A set of degenerate states can often be distinguished from one another by exploiting the symmetries present in the system. L- and R- enantiomers in a molecule are related by a reflection symmetry at a stereo center, meaning that there are two states of indistinguishable energy that are reflections of one another. People don’t often notice it, but chemists are masters of quantum mechanics even though they typically don’t know as much of the math: how you build molecules is totally governed by quantum mechanics and chemists must understand the qualitative results of the physical models. I’ve seen chemists speak competently of symmetry transformations in places where the physicists sometimes have problems.

Another place where symmetry is important is in the search for new physics. The way to discover new physical phenomena is to look for observational results that break the expected symmetries of a given mathematical model. The LHC was built to explore symmetries. Currently known models are said to hold CPT symmetry, referring to Charge, Parity and Time Reversal symmetry… I admit that I don’t understand all the implications of this, but simply put, if you make an observation that violates CPT, you have discovered physics not accounted for by current models.

I held back talking about Parity in all this because I wanted to speak of it in greater detail. Of the symmetries covered in Sakurai chapter 4, I feel that I made the greatest jump in understanding on Parity.

Parity is symmetry under space inversion.

What?

Just saying that sounds diabolical. Space inversion. It sounds like that situation in Harry Potter where somebody screws up trying to disapparate and manages to get splinched… like they space invert themselves and can’t undo it.

The parity operation carries all the cartesian variables in a function to their negative values.

parity operation

Here Phi just stands in for the parity operator. By performing the parity operation, all the variables in the function which denote spatial position are turned inside out and sent to their negative value. Things get splinched.

You might note here that applying parity twice gets you back to where you started, unsplinching the splinched. This shows that parity operator has the special property that it is it’s own inverse operation. You might understand how special this is by noting that we can’t all literally be our own brother, but the parity operator basically is.

parity2.jpg

Applying parity twice is like multiplying by 1… which is how you know parity is its own inverse. This also makes parity a unitary operator since it doesn’t effect absolute value of the function. Parity operation times inverse parity is one, so unitary.

parity3 or parity4

Here, the daggered superscript means “complex conjugate” which is an automatic requirement for the inverse operation if you’re a unitary operator. Hello linear algebra. Be assured I’m not about the break out the matrices, so have no fear. We will stay in a representation free zone. In this regard, parity operation is very much like a rotation: the inverse operation is the complex conjugate of the operation, never mind the details that the inverse operation is the operation.

Parity symmetry is “symmetry under the parity operation.” There are many states that are not symmetric under parity, but we would be interested in searching particularly for parity operation eigenstates, which are states that parity operator will transform to give back that state times some constant eigenvalue. As it turns out, the parity operator can only ever have two eigenvalues, which are +1 and -1. A parity eigenstate is a state that only changes its sign (or not) when acted on by the parity operator. The parity eigenvalue equations are therefore:

parity5a

All this says is that under space inversion, the parity eigenstates will either not be affected by the transformation, or will be negative of their original value. If the sign doesn’t change, the state is symmetric under space inversion (called even). But, if the sign does change, the state is antisymmetric under space inversion (called odd). As an example, in a space of one dimension (defined by ‘x’), the function sine is antisymmetric (odd) while the function cosine is symmetric (even).

Graph

In this image, taken from a graphing app on my smartphone, the white curve is plain old sine while the blue curve is the parity transformed sine. As mentioned, cosine does not change under parity.

As you may be aware, sines and cosines are energy eigenstates for the particle-in-the-box problem and so would constitute one example of legit parity eigenstates with physical significance.

Operators can also be transformed by parity. In order to see the significance, you just note that the definition of parity is that the position operation is reversed. So, a parity transformation of the position operator is this:

parity6

Kind of what should be expected. Position under parity turns negative.

As expressed, all of this is really academic. What’s the point?

Parity can give some insights that have deep significance. The deepest result that I understood is that matrix elements and expectation values will conserve with parity transformation. Matrix elements are a generalization of the expectation value where the bra and ket are not necessarily to the same eigenfunction. The proof of the statement here is one line:

parity7

At the end, the squiggles all denote parity transformed values, ‘m’ and ‘n’ are blanket eigenstates with arbitrary parity eigenvalues and V is some miscellaneous operator. First, the complex conjugation that turns a ket into a bra does not affect the parity eigenvalue equation, since parity is its own inverse operation and since the eigenvalues of 1 and -1 are not complex, so the bra above has just the same eigenvalue as if it were a ket. So, the matrix element does not change with the parity transformation –the combined parity transformation of all these parts are as if you just multiplied by identity a couple times, which should do nothing but return the original value.

What makes this important is that it sets a requirement on how many -1 eigenvalues can appear within the parity transformed matrix element (which is equal to the original matrix element): it can never be more than an even number (either zero or two). For the element to exist (that is, for it to have a non-zero value), if the initial and final states connected by the potential are both parity odd or parity even, the potential connecting them must be symmetric. Conversely, if the potential is parity odd, either the initial or final state must be odd, while the other is even. To sum up, a parity odd operator has non-zero matrix elements only when connecting states of differing parity while a parity even operator must connect states of the same parity. This restriction is observed simply by noting that the sign can’t change between a matrix element and the parity transformed matrix element.

Now, since an expectation value (average position, for example) is always a matrix element connecting an eigenket to itself, expectation values can only be non-zero for operators of even parity. For example, in a system defined across all space, average position ends up being zero because the position operator is odd, while both eigenbra and eigenket are of the same function, and therefore have the same parity. For average position to be non-zero, the wavefunction would need to be a superposition of eigenkets of opposite parity (and therefore not an eigenstate of parity at all!)

A tangible, far reaching result of this symmetry, related particularly to the position operator, is that no pure eigenstate can have an electric dipole moment. The dipole moment operator is built around the position operator, so a situation where position expectation value goes to zero will require dipole moment to be zero also. Any observed electric dipole moment must be from a mixture of states.

If you stop and think about that, that’s really pretty amazing. It tells you whether an observable is zero or not depending on which eigenkets are present and whether the operator for that observable can be inverted or not.

Hopefully I got that all correct. If anybody more sophisticated than me sees holes in my statement, please speak up!

Welcome to symmetry.

(For the few people who may have noticed, I still have it in mind to write more about the magnets puzzle, but I really haven’t had time recently. Magnets are difficult.)

Magnets, how do they work? (part 1)

Subtitle: Basic derivation of Ampere’s Law from the Biot-Savart equation.

Know your meme.

It’s been a while since this became a thing, but I think it’s actually a really good question. Truly, the original meme exploded from an unlikely source who wanted to relish in appreciating those things that seem magical without really appreciating how mind-bending and thought-expanding the explanation to this seemingly earnest question actually is.

As I got on in this writing, I realized that the scope of the topic is bigger than can be tackled in a single post. What is presented here will only be the first part (though I haven’t yet had a chance to write later parts!) The succeeding posts may end up being as mathematical as this, but perhaps less so. Moveover, as I got to writing, I realized that I haven’t posted a good bit of math here in a while: what good is the the mathematical poetry of physics if nobody sees it?

Magnets do not get less magical when you understand how they work: they get more compelling.

magnet-stem-cell-therapy

This image, taken from a website that sells quackery, highlights the intriguing properties of magnets. A solid object with apparently no moving parts has this manner of influencing the world around it. How can that not be magical? Lodestones have been magic forever and they do not get less magical with the explanation.

Truthfully, I’ve been thinking about the question of how they work for a couple days now. When I started out, I realized that I couldn’t just answer this out of hand, even though I would like to think that I’ve got a working understanding of magnetic fields –this is actually significant to me because the typical response to the Insane Clown Posse’s somewhat vacuous pondering is not really as simple as “Well, duh, magnetic fields you dope!” Someone really can explain how magnets work, but the explanation is really not trivial. That I got to a level in asking how they work where I said, “Well, um, I don’t really know this,” got my attention. How the details fit together gets deep in a hurry. What makes a bar magnet like the one in the picture above special? You don’t put batteries in it. You don’t flick a switch. It just works.

For most every person, that pattern above is the depth of how it works. How does it work? Well, it has a magnetic field. And, everybody has played with magnets at some point, so we sort of all know what they do, if not how they do it.

KONICA MINOLTA DIGITAL CAMERA

In this picture from penguin labs, these magnets are exerting sufficient force on one another that many of them apparently defy gravity. Here, the rod simply keeps the magnets confined so that they can’t change orientations with respect to one another and they exert sufficient repulsive force to climb up the rod as if they have no weight.

It’s definitely cool, no denying. There is definitely a quality to this that is magical and awe inspiring.

But, is it better knowing how they work, or just blindly appreciating them because it’s too hard to fill in the blank?

The central feature of how magnets work is quite effortlessly explained by the physics of Electromagnetism. Or, maybe it’s better to say that the details are laboriously and completely explained. People rebel against how hard it is to understand the details, but no true explanation is required to be easily explicable.

The forces which hold those little pieces of metal apart are relatively understandable.

Lorentz force

Here’s the Lorentz force law. It says that the force (F) on an object with a charge is equal to sum of the electric force on the object (qE) plus the magnetic force (qvB). Magnets interact solely by magnetic force, the second term.

2000px-lorentz_force-svg

In this picture from Wikipedia, if a charge (q) moving with speed (v) passes into a region containing this thing we call a “magnetic field,” it will tend to curve in its trajectory depending on whether the charge is negative or positive. We can ‘see’ this magnetic field thing in the image above with the bar magnet and iron filings. What is it, how is it produced?

The fundamental observation of magnetic fields is tied up into a phenomenological equation called the Biot-Savart law.

Biotsavart1

This equation is immediately intimidating. I’ve written it in all of it’s horrifying Jacksonian glory. You can read this equation like a sentence. It says that all the magnetic field (B) you can find at a location in space (r) is proportional to a sum of all the electric currents (J) at all possible locations where you can find any current (r’) and inversely proportional to the square of the distance between where you’re looking for the magnetic field and where all the electrical currents are –it may say ‘inverse cube’ in the equation, but it’s actually an inverse square since there’s a full power of length in the numerator. Yikes, what a sentence! Additionally, the equation says that the direction of the magnetic field is at right angles to both the direction that the current is traveling and the direction given by the line between where you’re looking for magnetic field and where the current is located. These directions are all wrapped up in the arrow scripts on every quantity in the equation and are determined by the cross-product as denoted by the ‘x’. The difference between the two ‘r’ vectors in the numerator creates a pure direction between the location of a particular current element and where you’re looking for magnetic field. The ‘d’ at the end is the differential volume that confines the electric currents and simply means that you’re adding up locations in 3D space. The scaling constants outside the integral sign are geometrical and control strength; the 4 and Pi relate to the dimensionality of the field source radiated out into a full solid angle (it covers a singularity in the field due to the location of the field source) and the ‘μ’ essentially tells how space broadcasts magnetic field… where the constant ‘μ’ is closely tied to the speed of light. This equation has the structure of a propagator: it takes an electric current located at r’ and propagates it into a field at r.

It may also be confusing to you that I’m calling current ‘J’ when nearly every basic physics class calls it ‘I’… well, get used to it. ‘Current vector’ is a subtle variation of current.

I looked for some diagrams to help depict Biot-Savart’s components, but I wasn’t satisfied with what Google coughed up. Here’s a rendering of my own with all the important vectors labeled.

biotsavart diagram

Now, I showed the crazy Biot-Savart equation, but I can tell you right now that it is a pain in the ass to work with. Very few people wake up in the morning and say “Boy oh boy, Biot-Savart for me today!” For most physics students this equation comes with a note of dread. Directly using it to analytically calculate magnetic fields is not easy. That cross product and all the crazy vectors pointing in every which direction make this equation a monster. There are some basic feature here which are common to many fields, particularly the inverse square, which you can find in the Newtonian gravity formula or Coulomb’s law for electrostatics, and the field being proportional to some source, in this case an electric current, where gravity has mass and electrostatics have charge.

Magnetic field becomes extraordinary because of that flipping (God damned, effing…) cross product, which means that it points in counter-intuitive directions. With electrostatics and gravity, the field is usually going toward or away from the source, while magnetism has the field seems to be going ‘around’ the source. Moreover, unlike electrostatics and gravity, the source isn’t exactly a something, like a charge or a mass, it’s dynamic… as in a change in state; electric charges are present in a current, but if you have those charges sitting stationary, even though they are still present, they can’t produce a magnetic field. Moreover, if you neutralize the charge, a magnetic field can still be present if those now invisible charges are moving to produce a current: current flowing in a copper wire is electric charges that are moving along the wire and this produces a magnetic field around the wire, but the presence of positive charges fixed to the metal atoms of the wire neutralizes the negative charges of the moving electrons, resulting in a state of otherwise net neutral charge. So, no electrostatic field, even though you have a magnetic field. It might surprise you to know that neutron stars have powerful magnetic fields, even though there are no electrons or protons present in order give any actual electric currents at all. The requirement for moving charges to produce a magnetic field is not inconsistent with the moving charge required to feel force from a magnetic field as well. Admittedly, there’s more to it than just ‘currents’ but I’ll get to that in another post.

With a little bit of algebraic shenanigans, Biot-Savart can be twisted around into a slightly more tractable form called Ampere’s Law, which is one of the four Maxwell’s equations that define electromagnetism. I had originally not intended to show this derivation, but I had a change of heart when I realized that I’d forgotten the details myself. So, I worked through them again just to see that I could. Keep in mind that this is really just a speed bump along the direction toward learning how magnets work.

For your viewing pleasure, the derivation of the Maxwell-Ampere law from the Biot-Savart equation.

In starting to set up for this, there are a couple fairly useful vector identities.

Useful identities 1

This trio contains several basic differential identities which can be very useful in this particular derivation. Here, the variables r are actually vectors in three dimensions. For those of you who don’t know these things, all it means is this:

vectors

These can be diagrammed like this:

vector example

This little diagram just treats the origin like the corner of a 3D box and each distance is a length along one of the three edges emanating from the corner.

I’ll try not to get too far afield with this quick vector tutorial, but it helps to understand that this is just a way to wrap up a 3D representation inside a simple symbol. The hatted symbols of x,y and z are all unit vectors that point in the relevant three dimensional directions where the un-hatted symbols just mean a variable distance along x or y or z. The prime (r’) means that the coordinate is used to tell where the electric current is located while the unprime (r) means that this is the coordinate for the magnetic field. The upside down triangle is an operator called ‘del’… you may know it from my hydrogen wave function post. What I’m doing here is quite similar to what I did over there before. For the uninitiated, here are gradient, divergence and curl:

gradivcurl

Gradient works on a scalar function to produce a vector, divergence works on a vector to produce a scalar function and curl works on a vector to produce a vector. I will assume that the reader can take derivatives and not go any further back than this. The operations on the right of the equal sign are wrapped up inside the symbols on the left.

One final useful bit of notation here is the length operation. Length operation just finds the length of a vector and is denoted by flat braces as an absolute value. Everywhere I’ve used it, I’ve been applying it to a vector obtained by finding the distance between where two different vectors point:

lengthoperation

As you can see, notation is all about compressing operations away until they are very compact. The equations I’ve used to this point all contain a great deal of math lying underneath what is written, but you can muddle through by the examples here.

Getting back to my identity trio:

Useful identities 1

The first identity here (I1) takes the vector object written on the left and produces a gradient from it… the thing in the quotient of that function is the length of the difference between those two vectors, which is simply a scalar number without a direction as shown in the length operation as written above.

The second identity (I2) here takes the divergence of the gradient and reveals that it’s the same thing as a Dirac delta (incredibly easy way to kill an integral!). I’ve not written the operation as divergence on a gradient, but instead wrapped it up in the ‘square’ on the del… you can know it’s a divergence of a gradient because the function inside the parenthesis is a scalar, meaning that the first operation has to be a gradient, which produces a vector, which automatically necessitates the second operation to be a divergence, since that only works on vectors to produce scalars.

The third identity (I3) shows that the gradient with respect to the unprimed vector coordinate system is actually equal to a negative sign times the primed coordinate system… which is a very easy way to switch from a derivative with respect to the first r and the same form of derivative with respect to the second r’.

To be clear, these identities are tailor-made to this problem (and similar electrodynamics problems) and you probably will never ever see them anywhere but the *cough cough* Jackson book. The first identity can be proven by working the gradient operation and taking derivatives. The second identity can be proven by using the vector divergence theorem in a spherical polar coordinate system and is the source of the 4*Pi that you see everywhere in electromagnetism. The third identity can also be proven by the same method as the first.

There are two additional helpful vector identities that I used which I produced in the process of working this derivation. I will create them here because, why not! If the math scares you, you’re on the wrong blog. To produce these identities, I used the component decomposition of the cross product and a useful Levi-Civita kroenecker delta identity –I’m really bad at remembering vector identities, so I put a great deal of effort into learning how to construct them myself: my Levi-Civita is ghetto, but it works well enough. For those of you who don’t know the ol’ Levi-Civita symbol, it’s a pretty nice tool for constructing things in a component-wise fashion: εijk . To make this work, you just have to remember it as I just wrote it… if any indices are equal, the symbol is zero, if they are all different, they are 1 or -1. If you take it as ijk, with the indices all different as I wrote, it equals 1 and becomes -1 if you reverse two of the indices: ijk=1, jik=-1, jki=1, kji=-1 and so on and so forth. Here are the useful Levi-Civita identities as they relate to cross product:

levicivita

Using these small tools, the first vector identity that I need is a curl of a curl. I derive it here:

vector id 1

Let’s see how this works. I’ve used colors to show the major substitutions and tried to draw arrows where they belong. If you follow the math, you’ll note that the Kroenecker deltas have the intriguing property of trading out indices in these sums. Kroenecker delta works on a finite sum the same way a Dirac delta works on an integral, which is nothing more than an infinite sum. Also, the index convention says that if you see duplicated indices, but without a sum on that index, you associate a sum with that index… this is how I located the divergences in that last step. This identity is a soft stopping point for the double curl: I could have used the derivative produce rule to expand it further, but that isn’t needed (if you want to see it get really complex, go ahead and try it! It’s do-able.) One will note that I have double del applied on a vector here… I said that it only applies on scalars above… in this form, it would only act on the scalar portion of each vector component, meaning that you would end up with a sum of three terms multiplied by unit vectors! Double del only ever acts on scalars, but you actually don’t need to know that in the derivation below.

This first vector identity I’ve produced I’ll call I4:

useful vector id 1

Here’s a second useful identity that I’ll need to develop:

useful vector id 2

This identity I’ll call I5:

vector id 2

*Pant Pant* I’ve collected all the identities I need to make this work. If you don’t immediately know something off the top of your head, you can develop the pieces you need. I will use I1, I2, I3, I4 and I5 together to derive the Maxwell-Ampere Law from Biot-Savart. Most of the following derivation comes from Jackson Electrodynamics, with a few small embellishments of my own.

first line amp devIn this first line of the derivation, I’ve rewritten Biot-Savart with the constants outside the integral and everything variable inside. Inside the integral, I’ve split the meat so that the different vector and scalar elements are clear. In what follows, it’s very important to remember that unprimed del operators are in a different space from the primed del operators: a value (like J) that is dependent on the primed position variable is essentially a constant with respect to the unprimed operator and will render a zero in a derivative by the unprimed del. Moreover, unprimed del can be moved into or out of the integral, which is with respect to the primed position coordinates. This observation is profoundly important to this derivation.

BS to amp 1

The usage of the first two identities here manages to extract the cross product from the midst of the function and puts it into a manipulable position where the del is unprimed while the integral is primed, letting me move it out of the integrand if I want.

BS to amp 2

This intermediate contains another very important magnetic quantity in the form of the vector potential (A) –“A” here not to be confused with the alphabetical placeholder I used while deriving my vector identities. I may come back to vector potential later, but this is simply an interesting stop-over for now. From here, we press on toward the Maxwell-Ampere law by acting in from the left with a curl onto the magnetic field…

BS to amp 3

The Dirac delta I end with in the final term allows me to collapse r’ into r at the expense of that last integral. At this point, I’ve actually produced the magnetostatic Ampere’s law if I feel like claiming that the current has no divergence, but I will talk about this later…

BS to amp 4

This substitution switches del from being unprimed to primed, putting it in the same terms as the current vector J. I use integration by parts next to switch which element of the first term the primed del is acting on.

BS to amp 5

Were I being really careful about how I depicted the integration by parts, there would be a unit vector dotted into the J in order to turn it into a scalar sum in that first term ahead of the integral… this is a little sloppy on my part, but nobody ever cares about that term anyway because it’s presupposed to vanish at the limits where it’s being evaluated. This is a physicist trick similar to pulling a rug over a mess on the floor –I’ve seen it performed in many contexts.

BS to amp 6

This substitution is not one of the mathematical identities I created above, this is purely physics. In this case, I’ve used conservation of charge to connect the divergence of the current vector to the change in charge density over time. If you don’t recognize the epic nature of this particular substitution, take my word for it… I’ve essentially inverted magnetostatics into electrodynamics, assuring that a ‘current’ is actually a form of moving charge.

BS to amp 75

In this line, I’ve switched the order of the derivatives again. Nothing in the integral is dependent on time except the charge density, so almost everything can pass through the derivative with respect to time. On the other hand, only the distance is dependent on the unprimed r, meaning that the unprimed del can pass inward through everything in the opposite direction.

BS to amp 8

At this point something amazing has emerged from the math. Pardon the pun; I’m feeling punchy. The quantity I’ve highlighted blue is a form of Coulomb’s law! If that name doesn’t tickle you at the base of your spine, what you’re looking at is the electrostatic version of the Biot-Savart law, which makes electric fields from electric charges. This is one of the reasons I like this derivation and why I decided to go ahead and detail the whole thing. This shows explicitly a connection between magnetism and electrostatics where such connection was not previously clear.

BS to amp 9

And thus ends the derivation. In this casting, the curl of the magnetic field is dependent both on the electric field and on currents. If there is no time varying electric field, that first term vanishes and you get the plain old magnetostatic Ampere’s law:

Ampere's law

This says simply that the curl of the magnetic field is equal to the current. There are some interesting qualities to this equation because of how the derivation leaves only a single positional dependence. As you can see, there is no separate position coordinate to describe magnetic field independently from its source. And, really, it isn’t describing the magnetic field as ‘generated’ by the current, but rather that a deformation to the linearity of the magnetic field is due to the presence of a current at that location… which is an interesting way to relate the two.

This relationship tends to cause magnetic lines to orbit around the current vector.

magcur

This image from hyperphysics sums up the whole situation –I realize I’ve been saying something similar from way up, but this equation is proof. If you have current passing along a wire, magnetic field will tend to wrap around the wire in a right handed sense. For all intents and purposes, this is all the Ampere’s law says, neglecting that you can manipulate the geometry of the situation to make the field do some interesting things. But, this is all.

Well, so what? I did a lot of math. What, if anything, have I gained from it? How does this help me along the path to understanding magnets?

The Ampere Law is useful in generating very simple magnetic field configurations that can be used in the Lorentz force law, ultimately showing a direct dynamical connection between moving currents and magnetic fields. I have it in mind to show a freshman level example of how this is done in the next part of this series. Given the length of this post, I will do more math in a different post.

This is a big step in the direction of learning how magnets work, but it should leave you feeling a little unsatisfied. How exactly do the forces work? In physics, it is widely known that magnetic fields do no work, so why is it that bar magnets can drag each other across the counter? That sure looks like work to me! And if electric currents are necessary to drive magnets, why is it that bar magnets and horseshoe magnets don’t require batteries? Where are the electric currents that animate a bar magnet and how is it that they seem to be unlimited or unpowered? These questions remain to be addressed.

Until the next post…