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…
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.”
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….
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).
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.
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:
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…
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…
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…
(edit 5-22-18: The image right here contains a bit of wrong math, see the end of the post for a more comprehensive and correct version. I made a mistake and I won’t try to hide it: see if you can find it!;-)
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…
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.
There is a quirk in this derivation for the interaction picture that continues to bother me. I didn’t really see it at first, but it bothers me having thought some time about it. The full Hamiltonian is defined to be some basic part plus some separable time-dependent potential. In the derivative that produces the evolution from the time-dependent potential, there is a basic assumption that this time-dependent potential does not contain time explicitly, meaning that no time derivative is taken on the potential. This seems like a self-contradiction to me: the potential is defined as time dependent, but must be the same form as the basic part of the Hamiltonian and not contain explicit time dependence in order for the derivation to work as shown above. I’m still thinking about it.
Here is a better version of the derivative that gives the time dependent Schrodinger form involving only the potential within the interaction picture: