A Spherical Tensor Problem

Since last I wrote about it, my continued sojourn through Sakurai has brought me back to spherical tensors, a topic I didn’t well understand when last I saw it. The problem in question is Sakurai 3.21. We will get to this problem shortly…

I’ve been thinking about how best to include math on this blog. The fact of the matter is that it isn’t easy to do very fast. It looks awful if I photograph pages from my notebook, but it takes forever if I use a word processor to make it nice and neat and presentable. I’ve tried a stylus in OneNote before, but I don’t very much like the feeling compared to working on paper.

After my tirade the other day about the Smith siblings, I’ve been thinking again about everything I wanted this blog to be. It isn’t hard to find superficial level explanations of most of physics, but I also don’t want this to read like a textbook. If Willow Smith hosts ‘underground quantum mechanics teachings,’ I actually honestly envisioned this effort on my part as a sort of underground teaching –regardless of the nonexistent audience. What better way to put it. I didn’t want to put in pure math, at least not quite; I wanted to present here what happens in my head while I’m working with the math. How exactly do you do that?

Here’s an image of the mythological notebook where all my practicing and playing takes place:

4-16-16 Notebook image

I’ve never been neat and pretty while working with problems, but all that scratching doesn’t look like scratching to me while I’m working with it. It’s almost indescribable. I could shovel metaphors on top of it or take pictures of beautiful things and call that other thing ‘what I see.’ But there isn’t anything like it. If you’ve spent time on it yourself, maybe you know. It’s addictive. It’s conceptual tourism in the purest form, standing on the edge of the Grand Canyon looking out, then climbing down inside, feeling the crags of stone on my fingertips as I pass down toward where the river flows. It’s tourism in a way, going to a place that isn’t a place, not necessarily pushing back the frontiers since people have been there before, but climbing to the top of a mountain that nobody ever just visits in daily life. You can’t simply read it and you don’t just walk there.

The pages pictured above are of my efforts to derive a formula from Schwinger’s harmonic oscillator representation to produce the rotation matrices for any value of angular momentum. Writing the words will mean nothing to practically anybody who reads this. But what do I do to make it genuine? How do you create a travelogue for a landscape of mathematical ideas?

For the moment, at least, I hope you will forgive me. I’m going to use images of my notebook in all its messy glory.

Where we started in this post was mentioning Spherical Tensors. I hit this topic again while considering Sakurai problem 3.21. ‘Tensor’ is admittedly a very cool word. In Brandon Sanderson’s “Steelheart,” Tensors are a special tool that lets people use magic power to scissor through solid material.

For all the coolness of the word, what are Tensors really?

In the most general sense, a tensor is a sort of container. Here is a very simple tensor:

Ai

This construct holds things. Computer programmers call them Arrays sometimes, but here it’s just a very simple container. The subscript ‘i’ could stand for anything. If you make ‘i’ be 1,2 or 3, this tensor can contain three things. I could make it be a vector in 3 dimensions, describing something as simple as position.

In the problem I’m going to present, you have to think twice about what ‘tensor’ means in order to drag out the idea of a ‘spherical’ tensor.

Here is Sakurai 3.21 as written in my notebook:

Sakurai 3.21Omitting the |j,m> ket at the bottom, Sakurai 3.21 is innocuous enough. You’re just asked to evaluate two sums between parts a.) and b.). No problem right? Just count some stuff and you’re done! Trick is, what the hell are you trying to count?

Contrary to my using the symbol ‘Ai’ above to sneak in the meaning of ‘love,’ the dj here do not play in dance clubs, even if they are spinning like a turntable! These ‘d’s are symbols for a rotation operation which can transform a state as if rotating it by an angle (here angle β). Each ‘d’ transforms a state with a particular z-axis angular momentum, labeled by ‘m’, to a second state with a different label, where the angle between the two states is a rotation of β around the y-axis. Get all that? You’ve got a spinning object and you want to alter the axis of the spin by an angle β. Literally you’re spinning a spin! That’s a headache, I know.

Within quantum mechanics, you can know only certain things about the rotation of an object, but not really know others. This is captured in the label ‘j’. ‘j’ describes the total angular momentum contained in an object; literally how much it’s spinning. This is distinct from ‘m’ which describes the rotation around a particular axis. Together, ‘m’ and ‘j’ encapsulate all of the knowable rotational qualities of our quantum mechanical object, where you can know it’s rotating a certain amount and that some of that rotation is around a particular axis. The rest of the rotation is in some unknowable combination not along the axis of choice. This whole set of statements is good for both an object spinning and for an object revolving around another object, like a planet in orbit.

The weird trick that quantum mechanics plays is that only a certain number of rotational state are allowed for a particular state of total angular momentum; the more total angular momentum you have, the larger the library of rotational states you can select from. In the sum in the problem, you’re including all the possible states of z-axis angular momentum allowable by the particular total angular momentum. Simultaneous rotation around x and y-axis is knowable only to an extent depending on the magnitude of rotation about the z-axis (so says the Heisenberg Uncertainty Principle, in this case–but the problem doesn’t require that…).

Here is an example of how you ‘rotate a state’ in quantum mechanics. I expect that only readers familiar with the math will truly be able to follow, but it’s a straightforward application of an operator to carry out an operation at a symbolic level:

Rotating a state 4-20-16

All this shows is that a rotation operator ‘R’ works on one state to produce another. By the end of the derivation, operator R has been converted into a ‘dj’ like what I mentioned above. Each dj is a function of m and m” in a set of elements which can be written as a 2-dimensional matrix… dj is literally mapping the probability amplitude at m onto m”, which can be considered how you route one element of a 2-dimensional matrix into another based upon the operation of rotating the state. In this case, the example starts out without a representation, but ultimately shifts over to representing in a space of ‘j’ and ‘m.’ The final state can be regarded as a superposition of all the states in the set, as defined by the sum. In all of this, dj can be regarded as a tensor with three indices, j, m and m”, making it a 3-dimensional entity which  contains a variable number of elements depending on each level of j: dj is only the face-plate of that tensor, coughing up whatever is stored at the element indexed by a particular j, m and m”.

In problem 3.21, what you’re counting up is a series of objects that transform other objects as paired with whatever z-axis angular momentum they represent within the total angular momentum contained by the system. This collection of objects is closed, meaning that you can only transform among the objects in the set. If there were no weighting factor in the sum, the sum of these squared objects actually goes to ‘1’… the ‘d’ symbols become probability amplitudes when they’re squared and, for a closed set, you must have 100% probability of staying within that set. The headache in evaluating this sum, then, is dealing with the weighting factor, which is different for each element in the sum, particularly for whatever state they are ultimately supposed to rotate to.

My initial idea looking at this problem was that if I can calculate each ‘d,’ then I can just work the sum directly. Just square each ‘d’ and multiply it by the weighting factor and voila! There was no thought in my head about spherical tensors, despite the overwhelming weight of that hint following part b.)

Naively, this approach could work. You just need some way of calculating a generalized ‘d.’ This can be done using Schwinger’s simple harmonic oscillator model. All you need to do is rotate the double harmonic oscillator and then pick out the factor that appears in place of ‘d’ in the appropriate sum –an example of which can be seen in the rotation transformation above. Not hard, right?

A month ago, I would have agreed with you. I had spent only a little bit of time learning how the Schwinger model works and I thought, “Well, solve ‘d’ using the Schwinger method and then boom, we’re golden.” It didn’t seem too bad, except that days eventually converted themselves into weeks before I had a good enough understanding of the method to be able to crank out a ‘d.’ You can see one of my pages of work on this near the top of this post… there were factorials and sums everywhere. By the time I had it completely figured out –which I really don’t regret, by the way– I had actually pretty much forgotten why I went to all that trouble in the first place. My thesis here is that, yes, you can solve for each and every ‘d’ you may ever want using Schwinger’s method. On the other hand, when I came back to look at Sakurai 3.21, I realized that if I were to try to horsewhip a version of ‘d’ from the Schwinger method into that sum, I was probably never going to solve the problem. The formula to derive each ‘d’ is itself a big sum with a large number of working parts, the square of which would turn into a _really_ large number of moving parts. I know I’m not a computer and trying to go that way is begging for a trouble.

It was a bit of a letdown when I realized that I was on the wrong track. As a lesson, that happens to everyone: almost nobody gets it first shot. This should be an abstract lesson to many people: what you think is a truth isn’t always a truth, or necessarily the simplest path to a truth. I still expect that if you were a horrific glutton for punishment, you could work the problem the way I started out trying, but you would get old in the attempt.

I spent some introspective time reading Chapter 3 of Sakurai, looking at simple methods for obtaining the necessary ‘d’ matrix elements. Most of these can’t be used in the context of problem 3.21 because they are too specific. With half-integer j or j of 1, you can directly calculate rotation matrices, except that this is not a general solution. I had a feeling that you could suck the weighting factor of ‘m’ back into the square of the ‘d’ and use an eigenvalue equation to change the ‘m’ into the Jz operator, but I wasn’t completely sure what to do with it if I did. About a week ago, I started to look a bit more closely at the section outlining operator transformations using spherical tensor formalism. I had a feeling I could make something work in these new ideas, especially following that heavy-handed hint in part b.)

The spherical tensor formalism is very much like the Heisenberg picture; it enables one to rotate an operator using the same sorts of machineries that one might use to rotate a state. This, it turns out, is the necessary logical leap required by the problem. To be honest, I didn’t actually understand this while I was reading the math and trying to work through it. I only really understood very recently. Rotating the state is not the same as rotating operators. The math posted above is the rotation of a state.

As it turns out, with an operator written in a cartesian form, different parts will rotate differently from one another; you can’t just apply one rotation to the whole thing and expect the same operator back.

This becomes challenging because the angular momentum operators are usually written in a cartesian form and because operator transformations in quantum mechanics are usually handled as unitary transformations. Constructing a unitary transformation requires careful analysis of what can rotate and remain intact.

Here is a derivation which shows rotation converted into a unitary operation:

Rotation as a unitary transform 4-21-16

In this case, the rotation matrix ‘d’ has been replaced by a more general form. The script ‘D’ is generally used to represent a transformation involving all three Euler angles, whereas the original ‘d’ was a rotation only around the y-axis. In principle, this transformation can work for any reorientation. In this derivation, you start with a spherical harmonic and show, if you create a representation of something else with that spherical harmonic, that you can rotate that other object within the Ylm. In this derivation, the object being rotated is just a vector used to indicate direction, called ‘n’. The spherical harmonics have this incredible quality in that they are ready-made to describe spherical, angle-space objects and that they rotate naturally without distortion… if you want to rotate anything, writing it as an object which transforms like a spherical harmonic is definitely the best way to go.

In the last line of that derivation, the spherical harmonic containing the direction vector has been replaced with a construct labeled simply as ‘T’. T is a spherical tensor. This object contains whatever you put into it and resides in the description space of the spherical harmonics. It rotates like a spherical harmonic.

The last line of algebra contains another ramification that I think is interesting. In this math, for this particular case, the unitary transform of D*Object*D reduces to a simple linear transform D*Object.

This brings me roughly full circle: I’m back at spherical tensors.

A spherical tensor is a multi-dimensional object which sits in a space which uses the spherical harmonics as a descriptive basis set. Each index of the spherical tensor transforms like the Ylm that resides at that index location. In some ways, this looks very like a state function in spherical harmonic space, but it’s different since the object being represented is an operator native to that space rather than a state function. Operators and state functions must be treated differently in quantum mechanics because they are different. A state function is a nascent form of a probability distribution while an operator is an entity that can be used to manipulate that distribution in eigenvalue equations.

This may seem a non-sequitur, but I’ve just introduced you to a form of trans-dimensional travel. I’ve just shown you the gap for moving between a space involving the dimensions of length, width and depth into a space which replaces those descriptive commodities with angles. A being living in spherical harmonic space is a being constructed directly out of turns and rotations, containing nothing that we can directly witness as a physical volume. You will never find something so patently crazy in the best science fiction! Quantum mechanics is replete with real expressions for moving from one space to another.

The next great challenge of Sakurai 3.21 is learning how to convert a cartesian operator construct into a spherical one. You can put whatever you want into a spherical tensor, but this means figuring out how to transfer the meaning of the cartesian expression into the spherical expression. As far as I currently understand it, the operator can’t be directly applied while residing within the spherical tensor form –I screwed this problem up a number of times before I understood that. To make the problem work, you have to convert from cartesian objects into the spherical object, perform the rotation, then convert backward into the cartesian object in order to come up with the final expression. The spherical tensor forms of the operators end up being linear combinations of the cartesian forms.

Here is the template for using spherical harmonics to guide conversion of cartesian operators into spherical tensor components:

Conversion to spherical tensor 4-21-16

In this case, I’m converting the momentum operators into a spherical tensor. This requires only the rank 1 spherical harmonics. The spherical tensor of rank one is a three dimensional object with indices 1,0 and -1, which relate to the cartesian components of the momentum vector Jz, Jx and Jy as shown. For position, cosine = z/radius and the x and y conversions follow from that, given the relations above. Angular momentum needs no spatial component because of normalization in length, so z-axis angular momentum just converts directly into cosine.

As you can see, all the tensor does here is store things. In this case, the geometry of conversion between the spaces stores these things in such a way that they can be rotated with no effort.

Since I’ve slogged through the grist of the ideas needed to solve Sakurai 3.21, I can turn now to how I solved it. For all the rotation stuff that I’ve been talking about, there is one important, very easy technique for rotating spherical harmonics which is relevant to this particular problem. If you are rotating an m=0 state, of which there is only one in every rank of total angular momentum, the dj element is a spherical harmonic. No crazy Schwinger formulas, just bang, use the spherical harmonic. Further, both sections of problem 3.21 involve converting m into Jz and Jz converts to the m=0 element of the spherical tensor with nothing but a normalization (to see this, look at the conversion rules that I included above). This means that the unitary transform of Jz can be mediated either by rotating from any state into the m=0 state, or rotating m=0 toward any state, which lets the dj be a spherical harmonic in either direction.

Now, since part a.) is easy, here’s the solution to problem 3.21 part b.)

Sakurai 3.21 b1

I apologize here that the clarity of the images is not the best; the website downgraded the resolution. I included a restatement of problem 3.21 part b.) in the first line here and then began by expanding the absolute value and pulling the eigenvalue of m back into the expression so that I could recast it as operator Jz using an eigenvalue equation to give me Jz^2. Jz must then be manipulated to produce the spherical tensor, the process expanded below.

Sakurai 3.21 b2

Where I say “three meaningful terms,” I’m looking ahead to an outcome further along in the problem in order to avoid writing 6 extra terms from the multiplication that I don’t ultimately need. I do write my math exhaustively, but in this particular case, I know that any term that isn’t J0*J0, J1*J-1 or J-1*J1 will cancel out after the J+ and J- ladder operators have had their way. For anyone versed, J1 is directly the ladder operator J+ and J-1 is J-. If the m value doesn’t end up back where it started, with J+J- or J-J+ combinations, when you take the resulting expectation value, anything like <m|m+1> is zero. Knowing this a page in advance, I simply omitted writing all that math. I then worked out the two unique coefficients that show up in the sum of only three elements…

Sakurai 3.21 b3

In the middle of this last page, I converted the operators Jx and Jy into a combination of J^2 and Jz. The ladder operators composed of Jx and Jy served to strain out 2/3 of the mathematical extra and I more or less omitted writing all of that from the middle of the second page. After you’re back in the cartesian form, once you’ve made the rotation, which occurs once the sum has been expanded, there is no need to stay in terms of Jx and Jy because the system can’t be simultaneously expressed as eigen functions of Jx, Jy and Jz. You can have simultaneous eigen functions of only total angular momentum and one axis, typically chosen to be the z-axis. By converting to J^2 and Jz only, I get the option to use eigen values instead of operators, which is almost always where you want to end up in a quantum problem. This is why I started writing |m> as |j,m>… most of the time in this problem I only care about tracking the m values, but I understand from the very beginning of the problem that I have a j value hiding in there that I can use on choice.

One thing that eases your burden considerably in this problem is understanding how j compartmentalizes m values. As I mentioned before, each rank of j contains a small collection of m value eigenfunctions which only transform amongst themselves. Even though the problem is asking for a solution that is general to every j, by using transformations of the angular momentum operator, which is a rank 1 operator, I only needed the j=1 spherical harmonics to represent it. This allows me to work in a small space which can be general across all values of j. This is part of what makes the Schwinger approach to this problem so unwieldy; by trying to represent d for every j, I basically swelled the number of terms I was working with to infinity. You can work with situations like this, but it just gets too big too quickly in this case –I’m just not that smart.

It’s also possible to work omitting the normalization coefficients needed in the spherical harmonics, but do this with caution. It can be hard to tell which part of the coefficient is dedicated to flattening multiplicity and which is canceling out of the solid angle. In cases where terms are getting mixed, I hold onto normalization so that I know down the line whether or not all my 2s and -1 are going to turn out. I always screw things like this up, so I do my best to give myself whatever tools I can for figuring out where I’ve messed up arithmetic. I found an answer to this problem on-line which leaves cartesian indices on the transformations through the problem and completely omits the normalization… technically, this sort of solution is wrong and bypasses the mechanics. You can’t transform a cartesian tensor like a spherical tensor; getting yourself screwed up by missing the proper indices misses the math. How the guy hammered out the right answer from doing it so poorly makes no sense to me.

This problem took a considerable amount of work and thought. It may not show in the writing, but I had been thinking about it for weeks. One great difference between doing this for class and doing it on my own is that there is no time limit on completing it except for the admission of defeat. I never made that admission and I gradually became more and more clear on what to do in the problem. I had been thinking about it on and off so hard that it was losing me sleep and leaving me foggy headed on other daily tasks. It takes real work. Eventually, there was a morning while I was in the shower where I just saw it. Clear as day, the solution unfolded to me. I do my best thinking in the morning while taking my shower. Under some circumstances, the stress of this process can be soul-breaking. It can also be profoundly illuminating. Seeing through it can be addictive… but you must not give up when the going gets tough.

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How to learn Quantum Mechanics: not like this!

This particular topic never fails to get my hackles up. At breakfast this morning, I stumbled over an article about Willow Smith. This article linked to a profile on the same person.

Up front, Jaden and Willow Smith have both made their feelings about school well known. They are home schooled and actively decry spending time in the classroom.

The Smith kids are a master course on how to game the media. Every article about them comes off crooning over how smart and otherworldly they are. Jaden Smith opens his mouth and utters endless solipsisms with offhand references to ‘mathematics’ and ‘theoretical physicists’ and the tabloids bend over in awe about his perceived intellectual might. Never mind that for all his talk about ‘math’ and ‘physics’ he has not once pulled out any associated skill in a place where it’s clear that he doing anything but reading Deepak Chopra –which is a really bad way to learn anything about Quantum Mechanics.

When I think of Willow Smith, I will always think of this:

I went to school for one year. It was the best experience but the worst experience. The best experience because I was, like, “Oh, now I know why kids are so depressed.” But it was the worst experience because I was depressed.

Followed closely by this:

maybe even attending the Massachusetts Institute of Technology (MIT) one day. It’s why she hopped on a flight to tour the campus in Cambridge directly after Paris Fashion Week (where she excited a flashing mob of paparazzi with karate kicks at Chanel’s #frontrowonly show—NBD!). “It was nice to be able to talk to female students and professors about science and logic because that’s just such a man’s world,” she notes.

This pair of quotes is separated by a couple years. The first outlines why she won’t stay in school while the second is suggesting that she’s thinking about jumping into the iconic school of STEM education. Does she want to be in school or not? She’s been in school for only a year total in her life and doesn’t like the process, but she wants to jump into MIT??? If you can’t handle the classroom in a normal highschool, you are not only flunking out of a University, but talking about a place like MIT just for the name recognition is a complete joke. To get into MIT and make it meaningful, the bare minimum is not merely a GED, it’s excellent scores on ACT and SAT. You can’t just talk your way through that. Maybe money is enough to open all the doors, but what do you do when the professor has shoved a test in front of you and demanded that you either get an ‘A’ on it or fail? You can’t cry about gender discrimination if you are not capable of handling the task put in front of you… and it sure as hell won’t be a photo shoot with you dressed head to foot in swanky leather.

In the GRE, before heading into graduate school as a physicist, I managed nearly an 800 on my math score. That is not kidding you. To get that number required a meaningful amount of bashing my skull against the wall –I think that most people can probably do it, but it’s not trivial preparing to be able to. There were no solipsisms that I was uttering. I did better than the average on my Physics GREs. The time spent being able to do this would never have allowed for a trip to Paris fashion week; it required months of diligent, focused work! It’s like sharpening a razor blade or learning the form for an Olympic level high jump or learning how to shoot a sniper rifle and being able to hit the bull’s eye every time at two thousand yards. You know how marines get that good? They sit on the firing range all day with their faces glued into the sighting scope, shooting round after round after round. Just buying the gun isn’t enough. You have to work at it really hard and there’s not much time to do other stuff. Getting there leaves you pissing the intellectual equivalent of blood for your efforts –literally dinged and light headed as if you just stood a round against Connor McGregor and got the crap kicked out of you! My first two years in graduate school several times had me in tears. I was dreaming in mathematics and waking up in the middle of the night with the equations swirling around me. You’ve not lived until you’re deriving vector expressions for E-field polarization in light from Maxwell’s equations in your sleep. Breathing the topic, eating it, changes you. I regretted it at the time, but that place where demands are being made that are just beyond where you think you can go really does force you to break your own limits. The reality of physics very rapidly squeezes out anything resembling Osho or Prana energy.

Now, it’s with me forever and not just words. The feeling and the vision are still there along with the addiction of cracking problems apart and seeing where they implode into order.

In truth, there are people in this world who are ‘that’ smart. There is a small chance the Smith kids are among them. I do believe them to be bright, but how bright? Not like you can actually tell something like that from a tabloid interview. I’ve met and interacted with a few of these people, though I wouldn’t qualify myself among them. These people are truly scary because no matter how hard you work, they do better than you on the same test without having seemingly put in the effort. On the other hand, when they talk, it’s clear that they know what they’re talking about… with the Smith kids, it’s endless alliteration and metaphor without any substantial verification of the background. They use the words, but shove them into metaphors as if they have no concept of what the words actually mean. I was completely capable of that myself at that age! I talked endlessly about hypercubes and event horizons when I was seventeen, but ask me now if I truly understood it.

This particular article about Willow Smith drove me to write something on this blog because of this:

Willow sews clothing, hosts underground teachings in quantum mechanics, and is studying how to produce songs from mathematical equations

And in the other article, this:

1. Willow Smith likes to sew.

2. Willow Smith is a self-proclaimed “STEM freak.”

3. Willow Smith chatted with female professors at MIT about science and logic, and might attend school there.

4. Willow Smith’s eyeliner philosophy is “all about emulating the colors you feel inside,” which is probably the best eyeliner philosophy, tbh.

5. This whole interaction between Willow Smith and Siri: “‘I see myself as a — €”hold on, let me ask Siri.’ The teenager whips out her iPhone and speaks into it, drawing stares from tables nearby. ‘Siri, define artisan.’ Everybody’s favorite robotic voice serves up a satisfying definition: Artisan is a worker in a skilled trade, especially one that involves making things by hand. ‘Ah, OK, yes!’ Willow allows. ‘I call myself an artisan.'”

6. Willow Smith is studying how to produce songs from mathematical equations.

7. Willow Smith “hosts underground teachings in quantum mechanics.”

A big portion of what’s written here is about some combination of science, physics and math… having never actually demonstrated a competence at any of these in a public framework! What does it mean ‘hosts underground teachings in quantum mechanics?’ Is she sitting around with a group of her friends learning how to derive Klebsch-Gordon recurrence formulas?  For all that the article applauds her for using the words, there’s zero demonstration of the substance. You can look back at my post about Quantum University to know what I think of that.

I’m all for STEM education in women. I’ve dealt with female physicists who pulled higher scores than me in my quantum classes. I have a deep respect for women in STEM and I know flat out that they are capable of wickedly cool things. Take Emmy Noether, for instance. General Relativity would not be the same as a field without her. The difference here is that Emmy Noether actually did it. She didn’t wear around the aura of doing it, she didn’t abstractly pay lip service to ‘quantum mechanics,’ she produced real mathematical topology results never having once wasted her time singing about it in a recording booth, going on photo shoots for whoever or beaming about Prana energy, Ayurveda or Osho. After they get to the level I’m at, most of these women stop worrying so much about how they dress or what makeup they wear because they simply don’t have the time to spend the effort. I respect them because of the bias they wade against in many fields and it blows me away when they dress just as comfortably as the guys and pull through tests neck-in-neck. They can certainly do what they want with their clothing, but with many of them, it’s clear where they put their budget of energy. Most stop troweling on the eyeliner or lipstick unless they feel like it. Give me a clean-faced, smiling woman in flat, comfortable shoes with a three second pony-tail talking competently about Fourier transforms over this paper-tiger with her photo shoots and nonsensical posing any day. Willow Smith is casting the image of what popular culture wants smart to be… not what it actually is!

This girl could be an incredible benefit to her generation: she could make it cool for girls to do well in engineering type disciplines (if she were actually doing that, of course, which is debatable). She and her brother could make it cool for kids to be science geeks like I was. However, in order to actually accomplish in these fields, you can’t just wear the aura. If you are divided in your path and can’t get behind the idea of sitting in a classroom learning how to tell when you’re wrong, and really being pushed into a territory where almost no one is just ‘comfortable,’ you aren’t making it. Don’t delude yourself, it’s never just the words! Anybody can sound smart.

Much ado is made about them ‘teaching themselves’ in the book of life and how much better that is than the classroom. The problem with self study education (autodidacticism at the extreme), is that it’s hard to learn how to error-check your own thoughts. There are not that many true autodidacts because of that. People don’t easily step outside of themselves to know whether the information they’ve picked up is correct or not. Adding to a library means sometimes knowing when to discard a book. For instance, reading Deepak Chopra is not going to take you anywhere in learning quantum mechanics, but you can’t necessarily know this without some external direction since Chopra actively works to make what he says look like a legitimate truth deserving of as much consideration as what the topic of quantum mechanics actually contains. If you’re spending too much time pondering such chaff, you go nowhere with the things that have substance which take huge amounts of time to understand, like real Quantum Mechanics. And, if you accept what Chopra says as a valid interpretation because of some eastern mystic mentality, you have sabotaged yourself from knowing what real quantum can give you. For something like this type of physics, which contains no close conceptual parallels in our workaday world, without some guidance toward where the field lies, there’s not any reason why anyone can teach themselves to sort out what’s truth from what isn’t.

I think that the path to deep understanding of something like quantum physics is not along the path she’s walking… if she were actually walking it, it wouldn’t be ‘learning how to make songs about mathematical equations’ it would be ‘learning math.’ You can’t go deep or far if you’re divided. Your purpose can’t be distantly around, it has to be directly through. For instance, you can’t really learn quantum if you’re wasting time pretending that Ayurveda contains some kind of equivalent truth. That would have you spending time on the Ayurveda and not on the math. Quantum is a handful all by itself, let me tell you! I wish very much for her to prove me wrong, to prove to me that she has some substance to go with the borrowed legitimacy from using the words ‘quantum’ and ‘math’ to describe half of what she does. If I were invited to one of her ‘underground quantum mechanics teachings’ believe me, I could knock the topic straight and make it real… but I doubt anybody like me is ever invited. Likely, it’s all Ayurveda and Chopra and 2% real quantum.

Prove me wrong, girl! In five years, when you’re 20, after you’ve scraped off the makeup and put away the toys, gotten the GED and dominated the aptitude tests, then spent enough real effort on learning higher math to know what an equation is, we’ll chat again about what you like at MIT.

Math is fun, but you’ve got to really do it!

“Breakthrough Starshot” or Are You Serious, Mr. Hawking?

His heart’s in the right place, I will admit that much.

The “Breakthrough Starshot” project was bound to be proposed by someone eventually. Postage stamp-sized space probes riding a beam of light to Alpha Centuri, then beaming messages back to us of what they find there, all with Stephen Hawking’s stamp of approval. Explore another star! Nifty idea, truly, but can we really do it?

Hawking’s intentions are good. From a long-term perspective, there’s a 100% chance that the Earth will endure another extinction-level event. It has happened repeatedly in the history of our world. All the species of life living here and now are dead. Eventually. If we’re here when it happens, we’re dead too. There’s no chance it won’t happen. It may happen tomorrow, next week, next year, maybe next century or even a million years from now. If anything is living here a billion years from now, our sun will have changed its radiant output enough to render this place uninhabitable anyway. Our little oasis in the void won’t last forever. It will happen, even if the probability of it happening in the life span of anybody living now is very very small.

Admittedly, we’re biasing the statistics a little at the moment. There are enough people spread across the face of this planet that everything bad that ever happens will have a human audience caught up in it. And, worse, news of every bad event can now circle the whole planet in minutes by Twitter. It’s also an active discussion that there are enough humans on this planet that our activities are fouling the environment to the point where we may not be able to live here any longer. The notion is scary especially since we really don’t know enough one way or another to tell what can or should be done, if anything. It’s like being diagnosed with cancer, but not having been told yet how serious the disease is or what form the treatment will take.

By truth or distortion, it’s no mistake that we increase the odds of our long term survival by not leaving all the eggs in one basket. This is Stephen Hawking’s thought: we increase our chances of surviving a thousand years from now if we have people living elsewhere… and not merely on Mars, but as far elsewhere as we can get. Hawking is in the unenviable position of being aware that extinction events may originate not here on Earth, or even anywhere in our solar system, but elsewhere nearby in the stars. From his perspective, the further we spread ourselves, the greater the chances we survive that big event that no one can see coming.

Someone eventually has to propose a trip elsewhere beyond the edge of our solar system and it has to happen in such a way that information can return to Earth quickly enough for us to do something about it. Hawking and a couple billionaires proposing it are not surprising to me.

Where I got tripped up is that the proposal was essentially a big piece of Alistair Reynolds Sci-fi. God love him, it’s an interesting read, but not a very plausible one.

The Yahoo article calls it an “Audacious” plan. Postage stamp-sized, laser-beam riding autonomous mini-robots that will get there in 20 years. Seems like they hit all the sci-fi buzz words: nano-beamriding-laser-AIs. Moore’s Law gone crazy. I’m sure it will sell well to the public and they can wave off the ‘audacity’ by pushing the technology requirements into the future. We will develop what’s needed.

Or can we?

If you stop to think about what they’re proposing, they’re talking about essentially being able to detect emissions from a cellphone at a range of 4 light years. That’s essentially all the more power this device can carry with it… that or less depending on how small you build it. I do like lasers, but they aren’t all-powerful. There are no sensors available to detect such a thing. We are not capable of detecting the albedo light reflected off the surface of a planet at a distance of 4 light years and they’re talking about detecting radiant output from an object not able to carry as much battery power as the typical cellphone…

I had no details on the solar sail they’re planning to propel this probe with, but I did some back-of-the-envelop calculations on the power requirements for propelling a ‘stamp-sized’ object with a laser. The concept itself is not really a fictional one: light contains momentum and can be said to have a pressure –I would recommend studying Poynting’s vector for anyone curious. Knowledge that light carries momentum was actually a prerequisite for Einstein’s E=mc^2 postulate (what, you’ve never heard of E=pc?) You shine light on a surface and that surface rebounds from the light it absorbs or reflects. The whole concept underlies the idea of solar sailing. Using a laser as your light source is not a big leap: lasers are more columnated than sunlight and can deliver a great deal of intensity to a tiny spot. Some calculations about the system are easy to make.

I estimate that a postage stamp sized probe needs to be hit by something like a 600 megawatt/m^2 intensity laser for the duration of a year in order to achieve the speeds they’re talking about. That’s the average amount of light power a 1 gram mass space probe must absorb to be accelerated to 20% of light speed in the course of one year. Laser lights are extraordinarily intense, but this rating becomes even steeper when you start to realize that even laser light must fall off as 1/distance^2 for the spherical radius of the laser wave fronts, however well columnated that light is. For a high finesse laser cavity, where the mirrors are curved, that radius actually ends up being shorter –corrective lenses can help, but this is still imperfect. For all existent lasers, the spot size disperses to meters across at a distance of thousands of kilometers, at maximum. How far does a little laser pointer reach? For radiation delivered for a year, coping with the inverse square fall-off of light intensity as the object moves away from us ever more rapidly, we’re talking about a laser that is probably as intense as terawatts/m^2 (in order to continue to deliver the average 600 megawatts/m^2 to our postage stamp sized craft when it is a distance of 1/10th of a Light year away from us). The power capacity required to drive that is terawatt*hrs, which is the energy necessary to drive a small (or big) country. The US generates something like 25 terawatt*hrs of energy. It would certainly take a billionaire to buy up 1/20th of the power generation capacity of the US for a  year.

This is not even considering that precarious problem of how to know where to aim the laser. If it gets some other force input (space isn’t completely empty) it might veer out of the laser path –it doesn’t have to move far, just meters, or even centimeters where you aren’t expecting it to. And, it won’t even travel straight with respect to gravitational interactions it encounters from the planets in our solar system as it leaves. We can’t detect objects that are a centimeter square on planetary scales, let alone extra-solar scales. You could possibly track it by back reflection from the laser propulsion beam, but if it falls out of the beam, then what? They talk about Stealth aircraft having radar cross sections the size of a pigeon… this has a cross section the size of a postage stamp. Maybe the best way to go would be lidar using the propulsion laser, just scanning the beam around until you find back scatter again, but that would depend on the probe maintaining its orientation. The whole problem gets harder and harder as a function of a square of the distance with the rate at which light intensity falls off… the solar system is light hours across where the propulsion system would need to be workable out into light months of distance, or about a fifth of a light year given how far and fast they’re talking about. Feedback of how to move the steering laser beam if the probe disappears when it’s light minutes away means that you can’t find it again for at least minutes and then your knowledge is always minutes old. What do you do when it’s light days or even light weeks away?

Another pesky problem is what to do about dust. Like I said before, space isn’t completely empty. The fragility of the probe is somewhat inversely proportional to its size. The smaller the probe is, the less likely it is to hit dust, but the more likely a dust strike can do critical damage. A small, thin, unarmored thing with highly reduced electronics and electromechanics getting hit by a relatively larger piece of dust is serious business. How many of these probes do they plan to fly in order to hedge their bets and how many lasers are they using? The Pluto New Horizon’s mission carried a dust counting experiment and it is known that dust will gradually wear a space probe down given enough time. For a probe so small traveling at tremendous speeds, that time is less.

All of this is still just dealing with departure and says nothing about the complication of how to return data once the probe reaches its destination –which will be a rapid flyby since there won’t be any way to ever slow back down again. Like I said, do they really think they can detect a broadcast, even a laser broadcast, from a cellphone at a distance of 4 light years? The stars themselves are just points of light and many not even visible!

I shook my head, “Surely you must be joking, Professor Hawking!”

I think that the problem with the proposal is that they’re trying to have their cake and eat it too. They want to explore another star and do it in a time horizon that is significant to our civilization. 25 years is a long time, but not really that long. Exploring another star is not a technically insurmountable task, I don’t think, but it is demanding of both technical expertise and patience. Realistically, 25 years may be a bit impatient on the scale of our universe. It’s true that we have to perform this task someday and this proposal itself may well simply be trying to force that ‘someday’ to come sooner, but is this a form in which that task can work? Given our current capabilities, no: it’s betting on what we don’t know yet. Of course the proposal sounds like science fiction… it is science fiction still.

The great problem may be that we don’t know how close that existential threat is. Perhaps the achievable exploration mission demands that our civilization remain stable for a longer duration than perhaps it can. Are we better than the Roman Empire? With 25 years, there’s a chance that all the difficulties we’re currently facing haven’t exacted their existential toll yet… with two or three hundred years, a slightly more reasonable time goal for the exploration mission at hand, maybe our civilization will already have disappeared and won’t remain to learn what information the probe brings back. We live in days where pre-Renaissance barbarians are actually knocking on the door… will we withstand that?

I’m skeptical, but… go for it, guys, I wish you the best of luck.

Black Holes Don’t Just ‘Eat’…

While reading an article in Gizmodo the other day, I ended up taking umbrage with a particular comment. I’ve found in the past that it’s difficult to respond to the comments people make on these heavy traffic Gawker blogs in part because of how extraordinarily censored the comments section is and because of the pace at which people move on. In the past, I’ve come to the conclusion that the fastest response to a comment is not always the best considered and I would rather take time to think about what I want to say before I say it. So, by the time I’m ready to say anything about a particular piece, frequently the blog has already moved elsewhere, where no one will see what I have to say. A little frustrating since the comment response is usually not directed at the person who said it, who probably will never admit that they said anything wrong, but at the casual reader who will file it away in the back of their mind and not make the same mistake. The comment was this:

“This supermassive black hole, however, was found in a modestly-sized elliptical galaxy”

Sure, modestly-sized NOW. That’s like commenting on how few fish are in the aquarium exhibit with the great white shark.

Welcome to snark infested waters, the place where metaphors fly fast and furious and where a metaphor should sound like it came out of the mouth of one of Joss Wheden’s characters for it to float. Never mind if the person who said it stopped to think whether it made sense.

The article in question was about an unusual supermassive black hole found in a galaxy that is not considered big enough to support a black hole of that mass and how the discoverers were hypothesizing that the object they found is actually the product of a merging event where two black holes have combined to form a new one, not unlike the event the LIGO observed in order to confirm the existence of gravitational waves recently.

I can’t really say much about a majority of the commenters. You know how the comments sections in those blog articles go: the usual mix of arrogant, stupid, trolling, attention seeking and sometimes very astute and thoughtful. I fixated on the comment quoted above because it reflects a misconception that I’ve seen a few times recently, going back to the Ant-Man article that I wrote previously.

The popular image of a black hole is a huge vacuum cleaner out in space that busily sucks up everything that gets near it. That image is a distortion. The implication that a black hole is a shark which will ultimately eat up every star in the galaxy around it, as if it were eating smaller fish in an aquarium, is wrong. If I could cure this popular image, I would, but changing the world when people actively resist listening to what others are saying is difficult.

First of all, black holes do suck in matter. They do! Nothing of what I’m about to say contradicts that. The issue at hand actually comes down to the scale of what they do and eating matter is only a small part. Can you say that a black hole is going to devour all the stars around it? Actually, no. The conditions at which a black hole can eat something are limited by whether physics allows the object falling in to ever reach the ‘mouth’ –and if you know anything about orbital mechanics and conservation of energy, you’ll immediately realize that the story is not as simple as ‘falling in.’

As you approach a black hole, in order to ‘fall in,’ your ultimate trajectory must cross through the event horizon, the Schwarzchild radius where all in-falling trajectories can’t lead back out. That may not seem like much of a distinction, but you have to consider that the radius of the event horizon around the singularity at the heart of a black hole is actually quite small relative to the volume of space that a black hole effectively influences with its gravity. Pure gravitational potential energy looks like this:

u-graph01

In this image from Physicstutor, you can see the differences in potential energy between a position far away from the gravitating body and a position nearer to it –energy value is read off the vertical axis while position is read off the horizontal axis. In this curve, your potential energy does not change much when you are far away over large radii. This means that if you are far away, as you get much closer, you do not gain much kinetic energy for the change in potential energy that accompanies your change in position. On the other hand, if you are relatively close, when you become closer, your potential energy can change by much more, approaching some asymptote relative to the origin where the potential energy goes to negative infinity. In this particular curve, the difference in energy between sitting far away and sitting infinitesimally close is basically infinite energy and by moving from one position to the other, you convert that potential energy into kinetic energy. This is a small part of why it takes the moon a month to go around the earth, but the International Space Station only takes about 90 minutes.

With most planets and stars, the surface of that body prevents you from ever approaching a position infinitesimally close to the center of mass, meaning that you crash against the surface before you ever acquire ‘infinite’ kinetic energy. In addition, if your trajectory does not take you straight through that mass, you experience an increase in your kinetic energy which effectively changes your orbital speed with respect to the center of mass, which causes you to climb back up away from the body toward higher orbit. Taking this effect into account, the falling mass actually tends to experience a potential energy curve that looks more like this:

graphpotential_434_271

In this image taken from this website, you can see the potential energy curve that a satellite tends to experience. You get down to a particular altitude and you’re suddenly going ‘up-hill’ on the energy landscape and the kinetic energy you’ve gained tends to cause you to slide back out to a wider radius. Even if the pure gravitational potential energy curve offers a seductively huge amount of potential energy change, the effect of ‘going there’ will tend to push you back out for many trajectories. Orbits work because an object can get stuck in the bottom of that effective well: if you have an energy no greater than ‘a’ in the image above, you bob back and forth between ‘A’ and ‘A’ on that graph.

The same situation is true with a black hole. If you’re in a particular orbit, as you move closer to the black hole, your own kinetic energy will tend to rescue you and keep you from getting spaghettified and then squished. In fact, most galaxies live in this situation around their central black hole: the stars bob back and forth on some effective potential energy curve, never coming closer to the black hole than a particular radius. You can live happily next to a black hole like that without ever having to worry about ‘getting eaten.’ Some shark, huh, never quite being able to reach that morsel of food!

Fact is that since most trajectories within a black hole’s region of influence are like this and black holes are usually very small, only a remarkably few paths through the object’s influence ever actually cross the event horizon. As such, black holes are usually not denuding the galaxies they inhabit of surrounding stars.

In one case in our own solar system, we sent a probe to the planet Mercury. In our solar system, if the Earth is at the bottom of that effective potential well with respect to the sun, Mercury is way over to the left close to the origin on a completely different potential. For most of the kinetic energies that our rockets can actually impart, the planet Mercury cannot be directly reached. Launching a space probe to reach Mercury required us to figure out a way to dump enough kinetic energy during the trip that the probe could ever reach the planet Mercury with enough fuel to be able to establish an orbit. This was done by an orbital maneuver called a slingshot, which is performed by flying in close to a planetary mass and allowing that planet to redirect your flight path in such a way that your velocity vector, essentially your speed relative to the sun, is significantly altered. You basically use gravity to trade angular momentum with whatever planet you approach and the planet’s mass is so much bigger than the spacecraft that the planet is essentially unaltered by the interaction while the spacecraft gains or loses energy. The mission to Mercury required multiple slingshot assists with Venus and Mercury both (it was five or six IIRC) before the probe had bled off enough energy to be able to go into orbit at Mercury. This anecdote may seem a non sequitur, but it actually plays a deep role in the functioning of a black hole.

The one great trick that a black hole plays which changes this whole story is that they have no surface. There is essentially no ground to crash into as you get close! The potential energy changes that are available therefore become colossal. For the size of the object, it also possesses monumental mass, meaning that if it’s moving, it’s momentum is stupendous. It does have the momentum of at least a star, even if it doesn’t have the physical size. A gravity slingshot off a black hole can impart can so much kinetic energy that the flung object might be traveling at relativistic velocities.

18lr5ep9z65o1jpg

This image, pulled from Kinja, shows the effects of a black hole. Gases encountering the black hole are flung away by gravity slingshot with velocities approaching the speed of light. If a black hole of many stellar masses were to pass through our solar system, it could fling all the planets in every direction and rip the sun in half by tidal force alone and fling it in several directions all at once, eating very little in the process.

Once you encounter a black hole, there is a good chance this will be the outcome. In order to be eaten, if your path doesn’t just take you straight into the singularity, which it probably won’t, some other stuff usually has to happen first. Usually you have to dump kinetic energy somehow. In a situation with gas and dust and moving bodies, physical interactions between grains of dust or molecules of gas can allow energy to move between materials, slowing some, speeding others and generally scrambling up everybody’s flight paths. Those slowed can fall inward while those accelerated will tend to not want to. As charged particles fly about, constantly changing direction, they tend to emit light, also bleeding off energy… which means that everything around a black hole is glowing bright and that the accretion disk is emitting radiation up into the X-rays, all so that particles trapped there ever have a chance of falling in. Deeper in, the changes of gravitational potential may be appreciable across the length of something as small as a human being, meaning that the gravity force at your feet is greater than that at your head by something more than the tensile strength of the materials in between, tearing you apart. Most black holes act something like a cosmic blender and the reasons can be understood somewhat classically without even adding in the craziness of applied general relativity, or mistakenly reducing it to ‘sucking stuff in.’

Believe it or not, the picture I just painted is applicable depending on the mass of the black hole. The more mass a black hole has, the less it tends to pathologically torture the spacetime outside of its event horizon. If the black hole is massive enough, like maybe the supermassive black hole in the gizmodo article, a person can live to travel down to the event horizon without getting shredded. Around such a black hole, the spacetime is locally very euclidean, meaning that bigger and bigger objects can reside intact in stable orbits closer. So, small black holes are ‘sloppy eaters,’ while big ones actually attain something resembling table manners and would constitute better neighbors. How you go from being small to being big is perhaps a cloudy question since it isn’t clear if the process which typically makes stellar black holes is necessarily the same as the one which created the supermassive black holes that typically reside in the centers of galaxies.

Whatever you might think, the bottom line is simply this: a black hole in the core of a galaxy is probably not eating the stars in the sky around it. Galaxies do not shrink appreciably in size over time because of the black holes inside them per se. It simply doesn’t work like that!