Sunday, December 25, 2011

Visitor from Jordan

One of the great things about Google Blogger is you get statistics telling you how many people read your posts; you also get info on what country the hits come from, and what website they got here from. I noticed someone was visiting from Jordan via Google Translate, so I clicked on the link and was amazed to see my blog translated into Arabic. Here's a small piece of what it looks like:

Gorgeous stuff: I just hope the translation isn't too much of a hack job. Google translate is an amazing program, but I pity the fool that tries to make too much sense of my stuff in the original English let alone through a machine translation. Good luck, and while you're here, I hope you'll check out a suggestion I made a couple of months ago:"Why not write Hebrew with Arabic Script?"

Friday, December 23, 2011

The Quantization of Spin Revisited

I started this series on the Stern Gerlach experiment two weeks ago because I wanted to make a certain point about just where and how the wave function supposedly "collapses". Stern Gerlach is a nice discussion topic because people say the particle jumps into one of two possible spin states...but just where does it jump? When it passes through the magnetic field, or when it hits the screen?

Then I got sidetracked by a fascinating Master's Thesis by a guy named Jared Stenson, who argues that we should really look at what the silver atoms do when they pass through a pure quadrupole field, instead of the traditional Stern Gerlach. Stenson claims, with some justification, that the beam of atoms then spreads into a donut. This is very different from the traditional Stern Gerlach where the beam simply splits in two.

Stenson did his analysis for an unpolarized beam, and last week I refined his analysis for the case of the polarized beam. I got a beautiful pattern of spin distributed about the donut. For an initial spin-up beam configuarion, this is the pattern at the detector screen:

It's very different from the pattern we get for the traditional Stern Gerlach setup...OR IS IT????

Stenson points out that the original experiment was done, not with a pencil-shaped beam, but with a fan-shaped source. There never were two disrete dots representing "spin-up" and "spin-down". There were in fact two bands, representing the splitting of the fan-shaped beam. No, that's not eve right: the two bands came together at the ends, making an elongated ellipse. And isn't an elongated ellipse just another name for an elongated circle?

What if Stern and Gerlach had used a pencil beam source instead of a fan-shaped beam? Would they have gotten Stenson's donut? Well, let's see: Stenson calculates his donut for the case of the pure quadrupole field, and Stern-Gerlach has a sharp magnetic pole in proximity with a wide flat pole. If we calculate the Stern-Gerlach field, to the second order approximation we get a constant field plus...a pure quadrupole.

And what is the effect of the constant field on the beam of silver atoms? AB-SO-LUTE-LY nothing! Ok, to be fair, if the spin is lined up cross-wise to the field, there should be precession; but because the field is constant, any precession is also constant across the beam profile; so when it leaves the field, there is no shifting of phases which would tend to make the beam curve or bend. It just keeps going straight.

Any diffraction or bending of the beam can only come from the quadrupole component; and that's what we've calculated. The silver atoms don't suddenly jump into one of two possible states at all...they spread themselves around the axis of the beam in a symmetric pattern, showing all possible valuse of spin orientation. In the picture, I've shown the result for a polarized beam, where the circular distribution is skewed in the direction of the spin polarization. For the unpolarized beam, you get the uniform circular donut pattern.

Yes, there is a way you can set up the experiment so the spin appears to jump into one of two possible values: instead of using a pencil-shaped beam, you spread the beam out into a fan-shape. Then you get diffraction effects and interference so that the deposition pattern on the screen spreads out into an elongated ellipse, which, ignoring the endpoints, looks like two discreet bands. But that's not the Stern-Gerlach experiment the way the whole world talks about it. The real story is right here on this blog, and I just wonder if you can find this analysis ANYWHERE else.

Tuesday, December 20, 2011

Spinor Algebra by the Seat of your Pants

I spent the last week putting together all the analytical machinery for this beautiful problem of the polarized beam in a quadrupole field, and today I’m going to solve it. Ironically, I’m not going to use any of my intiricate machinery after all. I’m going to solve the problem by guessing the answer, and then showing that it’s the only answer that makes sense. It’s another way of doing things.
As you will recall this is a modified Stern Gerlach experiment proposed by a grad student at Brigham Young by the name of Stenson. He wants us to analyze the splitting of the beam not in the traditional Stern Gerlach filter, but in a pure quadropole magnetic field. It’s a really good problem, and Stenson gives us the answer for the case of the unpolarized beam. Now I’m going to give the answer for a polarized beam.

Stenson starts off by observing that the solution for the ordinary case is almost universally described as the beam splitting into two paths, so you get two dots of silver on the screen. He points out that this is wrong: that in the original experiment, the beam was fan-shaped, and the deposit was in the shape of an ellipse. He goes on to doubt whether you would even get the famous two-dot result if you used a pencil-shaped beam, and I agree with him.
He then presented the solution for the pencil-shaped beam going through a quadrupole field, and he shows it as a donut pattern. This makes physical sense. But he only does the case of the randomly polarized beam. What about the polarized beam?

Of course now it’s complicated because the polarization can be in any direction with respect to the quadrupole. But then we get back to the old idea of basis states: if you can analyze it for the two orthogonal basis state polarizations, namely spin-up and spin-down, you should get all other cases as superpositions of these.
Furthermore, there is a four-fold symmetry to the field. If you solve the spin-up case, then rotate 90 degrees, you are once again aligned in the same way (except the relative spins are opposite) so your solution should have the same form. Ditto for 180 and 270 degrees. Four-fold symmetry.

Finally, whatever solution you find, when you turn it upside down and add your two solutions together, you have to get the uniform donut pattern shown by Stenson. Things have to add up.
Putting all these conditions together, lets take a wild guess at the solution. That’s what I did, and I came up with the theory that the intensity of the pattern has to vary sinusoidally about the periphery of the ring. I’ve drawn it here:

The nice thing about this solution is that it approximates the traditional Stern-Gerlach to the extent that the spin-up particle is drawn (more or less) towards the upwards-pointing field; and at the same time,to the extent that the field points sideways relative to the particle, the trajectory splits left and right. It’s a bit like you had two traditional S-G’s one after another.
Furthermore, the solution is nice because if you then take a spin-down particle, you should get the opposite pattern. If you combine these two patterns as you would in the case of a random beam, you just add the intensities together, and the result is a uniform density distribution about the ring. So we’ve replicated Stenson’s solution.

But there is one rather nasty problem with this distribution. It’s a density function, which is of course the square of the wave function, and you really haven’t solved the problem until you’ve determined the wave function itself. The obvious thing is to take the square root of the density function, but then you get cosine of the half-angle. The problem is when you come around full circle, you’ve reveresed polarity. The two branches of the function don’t line up.
What I figured out is you can add a phase to the cosine so that it comes full circle: you just multiply it by an exponential, which preserves the amplitude but reverses the sign after one full revolution. The function looks like this:

It turns out this function can be rewritten in terms of sums instead of products, and it is actually quite simple:
There’s a factor of ½ which I haven’t bothered to keep track of, but no matter. You can see that the amplitude of the function peaks at the top of the circle, goes to zero as you move around the circle clockwise, and comes back up to the peak when you complete the circle.
I have to admit I thought I had nailed it with this, and then I noticed that it doesn’t quite do everything I think it ought to do. The beam enters the magnetic field, and it is spread out into a donut shape: but you look at the silver atoms as they land, they are still all pointing upwards, and I don’t think that can be right. I the atoms to be trying to line themselves up with the quadrupole field as they pass through it, so they land on the screen pointing all different ways. Incredibly, I can make them do this without too much trouble. We just have to remember two things: first, the formula for breaking up exponentials into sines and cosines; and secondly, that the function psi is really a spinor function and has to have two components. Let’s just try assigning the real part to the up component and the imaginary part to the down component and see if it works:
A careful inspection shows this function preserves all the properties we wanted: it peaks at top dead center, goes to zero at the bottom, and the intensity (square of the amplitude) varies as cosine-squared. And oh yes, the endopoints match up when you come full circle.
What’s more, we get an interesting distribution of spin orientations as we go around the circle. I’ve sketched them out here:

If you remember what the quadrupole field looked like (I posted the sketch about a week ago!) you can see that whatever the orientation of the local field, the silver atoms try to line themselves up parallel to minimize their energy. For example when theta is 90 degrees, for instane, the spinor evaluates (1, i) which is pointing sideways. Which is what the quadrupole field is doing for theta = 90. So it’s a physically satisfying picture.

The big challenge is yet to come: first, we have to write out the wave function for a beam that enters the magnets with spin down.  And then we have to examine combinations of those two wave functions, representing beams that enter the field with arbitrary spin orientations, e.g. spin sideways, and we have to see if the function does things that make sense. In particular, keeping in mind the four-fold symmetry of the field, we have to see if a beam entering with sideways spin polarization does the same thing as the form we’ve already calculated, only rotated by 90 degrees.

Hold on to your pants.

Monday, December 19, 2011

Stern Gerlach Corkscrew Diffraction Pattern

It’s taken a while but I’ve laid out all the machinery for analyzing the case of the silver atom entering the magnetic field tilted sideways. Here is how it goes.
1. The atoms precess in the magnetic field. The ones higher up precess faster.
2.  Because of the differential rates of precession, when they leave the magnetic field the spin orientations form aa corkscrew pattern across the cross-section of the beam.

3. Now break the corkscrew into up and down components and analyze the output as a diffraction pattern.
For purposes of calcualtion we can take the precession to be zero in the center of the beam. From basic spinor algebra, we know that the spin-up component will vary as the cosine of the distance from the center; likewise, the spin-down component will vary as the sine. Here is a map of the spin components across the cross-section:

Of course, I’ve taken my “up/down” orientation to be aligned with the spin of the atoms. So what I’m calling “spin-up”, the original orientation of the atoms, is actually spin-sideways to the magnetic field. It’s easy to analzye if I take my spin basis to line up with the magnet…I did that last week. The interesting thing is how it works if I take the alternate spin basis, and that’s what we’re about to see.

From everything we’ve gone over in the last week, it should now be obvious that the beam splits in two. The spin-up component splits in two, and the spin-down component splits in two. From the incoming beam, we get two outgoing beams.
What can we say about the spins of the two outgoing beams? They appear to be made up of equal portions of spin-up and spin down. But look carefully at the relative phases of the two components. They are out by 90 degrees, and they are out in opposite directions! In one beam the spin-down is 90 degrees behind the spin-up, and in the other beam it is 90 degrees ahead.  

I’m not going to go back and derive all of spinor algebra from scratch, but if you know anything about spinors you can see that these two beams have their spins alignede in opposite directions. In fact, one is spin-up relative to the Stern-Gerlach magnet and the other is spin-down. It’s the same result we got when we analzyed it in the spin-basis parallel to the Stern-Gerlach field, but now you can actually see how it works with precesssion and recombination.
The next problem on my agenda is the one proposed five years ago by Brigham Young grad student Jared Rees Stenson: the splitting of the beam in a pure quadrupole field. I wonder how that’s going to look.

Friday, December 16, 2011

How to Avoid Math by Using Physics

When I left off yesterday, I was talking about how you can analyze reflection from a mirror by looking at waves of current in the metal surface. These current waves flow in response to the incoming electric field; but if you ignore the driving field, they just look like waves of current flowing in the metal. It’s an odd fact that these waves seem to skim over the surface of the metal at speeds much greater than the speed of light; but there’s nothing physically mysterious about this. It’s just a geometrical consequence of the glancing angle with which the driving waves impinge upon the surface.
There are of course physical consequences to these current waves: they must in turn create electromagnetic radiation. It’s a true consequence of electromagnetic theory that a source travelling faster than the speed of light must radiate, and the simple mirror is a perfect example of this phenomenon. It is a bit of work to actually calculate it from scratch, but the result of the calculation can be easily guessed from physical considerations. There are two waves generated, shown in my sketch by the solid and the dashed arrows. The dashed arrow represents the reflected wave, and the solid arrow represents the wave which perfectly cancels the incoming wave. That’s why you can’t shine a beam of light through a mirror. The currents flowing in the metal surface generate a wave which is exactly equal in magnitude to the incoming wave, but opposite in phase.

Actually, when I drew this picture, it wasn’t meant to represent the case of the mirror. I was really doing the  analysis of a particle trajectory bending as it passes through an external field. The particle could have easily been an electron passing through a charged capacitor, but again that’s wasn’t the idea. The diagram was actually drawn to show a beam of silver atoms passing through an inhomogenous magnetic field: the Stern Gerlach experiment. But either way, the drawing is the same! The “particles” are represented by a wave function, and the phase of the wave function changes more rapidly where the energy is higher. The advancing phase thereby imparts a curvature to the path. That’s how it works.

Unfortunately it’s not always that easy to follow the wave trajectory. It’s easy in this case because we can identify surfaces of constant phase and magnitude, and it’s easy to see that the power in the wave must flow perpendicular to these surfaces. But it’s not always easy to identify such surfaces; indeed, they don’t always exist. So we need to develop other techniques. Which brings us to the present case.
I’ve already shown what the waves do if you throw a mirror into the picture. But the funny thing is they do pretty much the same thing without the mirror! Oh, it’s not really the same at all once you get right down to it. But the case of a mirror gives us really good insight into the mathematical form of the actual solution.

The main idea is that every point on a wave acts as a source for spherically outgoing waves. Every point. The overall sum of all these infinitely many spherical sources gives you the resulting wave. In the case of the ordinary plane wave propagating in the forward direction, all those little spherical wavelets cancel out everywher except in the normal direction of propagation. The backwards-going wavelets cancel out. The sideways-going wavelets cancel out. Only the forward-going waves are constructively reinforced.
In the case of the Stern-Gerlach experiment, instead of looking for surfaces of constant phase, we can just throw an arbitrary imaginary surface into the picture and ask what the wave looks like as it passes through that surface. Since in this case it is just slightly oblique to the alignment of the phasefronts, it works just like the mirror. The Schroedinger wave seems to be skimming along this surface at a speed much greater than the actual particle velocity, or even the velocity of light. It’s just like the wavefronts on the surface of the mirror, and there’s nothing unusual about it.

The difference is that in the mirror situation, these were actual currents flowing in response to the driving field. In this case there are no currents flowing because there is no surface for the to flow within: it’s just the mathematical value of the wavefunction that we’re considering. But remember how we said we would analyze it: we treat every point on the wave function as an independent source of spherically outgoing waves, and calculate the sum of all those sources.
It sounds like a monstrous mathematical task, and in fact it is. It’s almost impossible to do it. But fortunately we don’t have to. Nature has already shown us the solution in the case of the mirror. The solution consists of two outgoing waves shown by the dashed arrow and the solid arrow.

The difference is that in the case of the mirror, the dashed arrow represented the actual solution, namely the reflected wave; and the solid arrow represented the imaginary wave that serves to cancel out the incoming power. But the beam of silver atoms is a different story.
You still get the dashed wave and you still get the solid wave. But you get much more than that. I drew an arbitray plane in the middle of the picture. I could have drawn a similar plain one micrometer to the left, or one micrometer to the right. Each one of those planes would show similar solutions to the wave equation, just displaced by the corresponding distance.

The result of adding all these plane source solutions is easy to guess: in the backwards direction, shown by the dashed arrows, the sources are cumulatively out of phase so they end up completely cancelling out. A freely travelling wave does not spontaneoulsy generate a backward-moving reflection.
But the forward-travelling directions reinforce each other. The wave moves forward, and its direction is shown by the solid arrow. We can be lazy and pretend that we’ve solved the for the resultant wave just by looking at the effect of the sources in our single, arbitrarily selected plane, but in honest truth that only gives the barest whisper of the shape of the solution. The true solution is identical in form except it’s reinforced by the cumulative effect of every other imaginary plane.

What is the point of all this? Simply that for normally propagating waves, we can identify the general form of the complete trajectory by evaluating the wave function over a single, planar surface within the path of the wave. Ordinarily this is not a tremendous benefit, because we still have to do a monstrous amount of calculus to integrate over all the sources. But sometimes, as in the present case, we can identify the form of the wave with a physical situation where we already know the answer. That’s what we’re going to do when we take up the next topic: the case of the sideways-polarized beam in the Stern Gerlach experiment.

Thursday, December 15, 2011

Stern-Gerlach without the Quantum Leap

The Stern-Gerlach experiment is often put forward as a showcase example of wave function collapse. The silver atom enters the magnetic field with a random spin, and suddenly it has to make a choice: spin up or spin down? Depending on the answer, it takes one of two paths to the screen where it is detected.
This widely believed description is in fact a deliberate, almost criminal mystification of the process. Last week I said I would show how to understand the Stern Gerlach experiment as the normal time-evolution of the Schroedinger wave function, and today I’m going to get started.
The simplest case to deal with is when you have a beam of atoms prepared in the spin-up state: that is, their spins are already aligned with the magnet. We want to show that the beam curves upwards. Of course, it does so classically: the tiny atomic magnet is simply drawn to the stronger pole. The problem with this analyis is that if the magnet is tilted, it should still be drawn in the same direction, only not so strongly. That’s not what happens.

We need a way to analyze it which not only gives the right curvature in the case of aligned spin, but also gives us the bifurcation of paths for the case of arbitrary spin. We have to analyze the wave function of the beam. And it is especially helpful to let the beam have some finite width, as we will see in the picture below:

You can see that like any wave in quantum mechanics, the phase of the wave precesses faster when it has a higher energy…in this case, the portion of the beam near the bottom. (EDIT: This is really wrong. I didn't realize until over a year after I posted this that I didn't have the right explanation for the bending of an electron wave in a potential field. I finally worked it out and you can read the correct explanation HERE. But even so, my picture is still correct!) So the wavefronts tend to get ahead of the rest of the beam. That has the effect of tilting the imaginary suface which represents a constant phase. You can see that the effect of this is to bend the beam upwards. You would draw exactly the same diagram to show the path of an electron bending around a charged object.
Similarly, if the silver atoms were aligned the opposite way, the phase would advance faster at the top of the beam, and it would bend downwards.
But what if the spin is pointing sideways? The truly magical thing about spin is that you can represent a sideways spin as the composition of up and down spins, just as in electromagnetics you can represent a 45 degree polarization as the composition of vertical and horizontal. You just break up the spin into its up and down components, and analyze them separately. When you put the results together, you see that the beam path divides in two when it passes through the magnets. That’s what happens.
However, I don’t find this mathematical answer to be completely satisfying. We know, for example, that if an atom enters the field with its magnetic axis tilted sideways, then classically it ought to precess. That is, the direction of spin should rotate within the plane perpendicular to the external field. Does it also do this quantum mechanically? And if so, how can we reconcile this with the path dividing in two when the beam leaves the magnet?
It turns out we can indeed analyze it this way, and although the picture looks quite different, it turns out to have the same result. It’s a matter of your choice of “basis states”, and it’s an exercise well worth carrying through in order to develop your intuitions of how these things work. But before we go there I want to return to the picture we already drew and put a slightly different perspective on it.
I’ve drawn the same picture but I’ve added an imaginary surface representing the place where the magnetic field ends. I’ve drawn it supposedly parallel to the orientation of the original phase fronts; so the waves are tilted slightly with respect to this plane as they leave the magnetic field.

If you carefully trace the wavefronts in that imaginary plane, you’ll see that they’re something like I’ve shown with the orange line. It’s not quite right because these are actually exponential waves, not sine waves, so the amplitude never goes to zero. But it’s kind of traditional to draw them as sine waves anyhow because that’s the best we can do.
The thing to notice is that if we “unfreeze” the static picture and imagine what those wavefronts are doing in real time, they are actually zipping along from south to north at quite a good pace, quite possibly even faster than the speed of light. There is nothing unusual about this: it’s a very normal circumstance and it happens all the time in physics. Remember these aren’t real physical waves carrying energy: they’re just mathematical points along an imaginary surface.
In fact, it’s exactly the same picture you’d draw if you were trying to explain reflection from a mirror. The light waves come in from the left and strike the mirror at a glancing angle. The fields set up oscillating charges in the surface of the mirror. If we trace the charge oscillations, we find that they are just the waves shown by the orange line in the picture: they move from south to north and the “wave” definitely travels faster than the speed of light. It’s an obvious consequence of the glancing angle.
It’s this rushing wave of charge oscillations in the mirror that generates the reflection. There’s something exotic sounding called Cerenkov radiation and it has to do with the waves generated by a source travelling faster than light. It sounds like stuff that you get in cyclotrons or whatever, but in fact it’s a very ordinary thing. The current waves in the mirror travel faster than light, and as a result they generate wave systems peeling off at an angle, represented by the blue arrows. The dashed arrow is the reflected wave.
What about the solid arrow? That’s the wave that cancels out the incident wave! Otherwise the incoming wave would just pass right through the mirror and keep on going. The waves of charge in the mirror do two jobs: they cancel out the incoming wave, and they generate the reflected wave.
This is the exactly the same methodology we are going to use when we analyze the Stern Gerlach experiment. I’m going to save that for another day.

Wednesday, December 14, 2011

Physics Retreat

It seems I'm organizing a kind of physics retreat at the Maskwa Wilderness Lodge between Christmas and New Year. If you don't know about Maskwa, the setting is a very cool but primitive lodge not far from Pine Falls, Manitoba, and I'm going to be there for three days from the 28th thru 30th. The theoretical cost for the 3 days is $250 (but that's flexible); we might also be getting some people on a drop-in basis. 

There will be wilderness activities, home-made music, wood-stove cooking, and of course some physics. The idea is that we will be adults who might have taken as much as high school or first-year physics but who still find it interesting. There are fun topics at almost any level, but I usually start with some very cool stuff about orbits and trajectories. It’s pretty much inevitable that sooner or later we end up talking about Schroedinger’s Cat, but I draw the line at String Theory or the Higgs Boson. Anyone tries to bring up that stuff and we’ll make him clean the outhouses. 

We can handle a maximum of 12 participants, so email me, marty(at) if you’re in the area and interested. We’re trying to avoid the classic sausage-fest scenario, but either way it should be fun.

Tuesday, December 13, 2011

Shout out to the Gaither Vocal Band

I know this is supposed to be about physics, but it's my blog and I can blog about anything I want to. And as my friend Neil pointed out, I've already "polluted" it (yes, that was his word) with my musings about Israel and the Jews. I guess we're lucky I decided to name this site in honor of physics before I decided to started writing about the Jews, because otherwise the name of the blog might have had unfortunate connotations.

Anyhow, I was browsing youtube two days ago and I landed on a video that's so beautiful I can't stop listening to it: it's The Gaither Vocal Band singing "There Is A River". You know, when gospel music is at its best, it puts every other form of music to shame. When I'm listening to this stuff, I can't imagine choosing to listen to any other kind of music. Absolutely phenomenal.

Now I've got to get back to work on solving the Stern Gerlach wave function for the case of the polarized electron beam in a quadrupole field.

EDIT: The link above doesn't work any more for me. It may be blocked in certain countries. This one seems just fine...

Monday, December 12, 2011

Stern Gerlach with a Quadrupole Field

After I wrote my last article, I happened across an article with a different perspective on the Stern Gerlach experiment. It was actually a Master’s Thesis from one Jared Rees Stenson, a student at Brigham Young University, written in 2005.

This Jared makes an interesting point. Everyone talks about the Stern Gerlach experiment as though you shoot a thin beam between two magnets, and it divides into two: so you get two splotches of silver on the glass plate. I pointed out in my last article that you really need to consider the thickness of the beam, as though it were the size of a pencil. Jared makes a slightly different point: he says that Stern and Gerlach used a collimated slit  for their source, so the silver beam was really more fan-shaped than ray-like. And that the pattern on the glass plate was more of an ellipse than anything else:

Jared then goes on to doubt that if you actually had a pencil-thin beam, you would in fact get the two perfect dots that people like to talk about. And I think he may be right. He points out that classically, you really can’t create a magnetic field which gets stronger from top to bottom without also having it fan out. It’s basically Gauss’s Law applied to magnetostatics, and nobody really accounts for this when they analyze the “ideal” Stern-Gerlach system.

But what I like most about his thesis was where he suggested a different way of setting up the magnetic field, so as to eliminate the DC component. He arranges four wires to create a perfect quadrupole field, like so:

He then asks the question: if you shoot a pencil beam of silver atoms through this magnetic field, what pattern do you get on the glass plate? If you think about it, it's a funny question.

It turns out Jared is some kind of monster mathematician, and he does a bunch of stuff that I don't really follow to end up with the nice result that the pencil beam becomes a donut on the glass plate.

Now that sounds right to me if you start out with a random, unpolarized beam. But it occurs to me: what if you have a polarized that's already been selected for spin in one direction by a conventional Stern-Gerlach filter? What pattern do you get in that case?

It's a question that I have to ask myself almost as a matter of put-up-or-shut-up: in my last post, I said I could analyze the conventional S-G experiment by standard wave theory, and if I'm so smart, why can't I do it for this one too? It looks like a fun question to work on and I have some idea how it ought to come out. So give me a day or two and let's see what I come up with...

Sunday, December 11, 2011

Spatial Quantization and the Measurement Postulate

One of the most baffling aspects of quantum mechanics is the notion that spin must be spatially quantized: that an electron can have its spin axis pointing up, or down, but nothing in between. This goes back to the Stern-Gerlach experiment: a beam of silver atoms with random spins is passed through a magnetic field, and instead of being spread out smoothly like you would expect “classically”, the beam splits in two. Half the atoms are “spin-up”, and half are “spin-down”.
Copenhagen explains this as an example of the Measurement Postulate. We have a silver atom intially in a random state: that is, in a superposition of up and down states. The Stern Gerlach apparatus is designed to clearly identify atoms in either of those two pure states. Therefore, when the atom enters the apparatus, it makes a decision: spin up, or spin down. The Born Postulate tells us that the probability of this decision is given by the amplitudes of the respective states.
Copenhagen is a bit sketchy on the question of just when and where the silver atom makes that decision. Some people think it happens when the atom passes between the magnets. Others defer the moment of truth to the point of impact on the screen. Perhaps I’m being unfair when I say “Copenhagen” is sketchy on this point; it is probably more accurate to say that the followers of the Copenhagen Interpretation are not, on the whole, especially clear on what they are supposed to believe about this question.
It is hard to believe that with all the nonsense written about the quantum leap and the collapse of the wave function, that nowhere will you find the straightforward explanation of the Stern Gerlach experiment that I am going to give you here, based on the the simple premise of matter waves as originally conceived by De Broglie and put into mathematical form by Schroedinger. From this perspective there is no issue of spatial quantization or quantum leaps. Everything happens through a natural time evolution of the wave function.

The critical step in demystifying the physics is to start by realizing that the beam of silver atoms is not a geometrical ray, but rather a spread-out beam which we can think of as more like a pencil than a thread. When we treat it as a wave function, it is obvious that the portion of the beam closer to the pointy magnet – that is, in the strongest part of the field - must have a different phase velocity than the portion farther away. And anyone who has analyzed wavefronts passing through different media, such as glass with a variable index of refraction, knows what this means: the beam must curve. The baffling aspect of this is, of course: why does the curvature of the path follow exactly two trajectories? Why is it not infinitely variable between the two extremes of spin alignment? Specifically, why does an atom whose spin is aligned perpedicular to the field axis not pass through undisturbed?
All of these questions are answered when we understand that for electrons, any arbitrary spin can be represented as the superposition of two spins, which we call “up” and “down”. The method is straightforward application of wave mechanics: we represent an arbitrary spin as the superposition of our two eigenstates, and analyze each eigenstate separately according to the straightforward method of wavefronts and phase velocities. It is crucial to represent the beam as a pencil and not a geometrical ray, because only then do we see clearly how the beams curve. Each of the two cases has its own characteristic path. After analyzing them separately, we then combine them and calculate the superposition of those two paths. The result gives us the trajectory of the beam, and we will show that the beam is simply split in two.

Where exactly is the deep mystery in all this? Where does an atom, initially in a superposition of states, decide to arbitrarily make that quantum leap into one or the other final state? I will have more to say on this in a future post, but first I want to point out that the exact same thing happens with light, and nobody goes around saying that a photon which is polarized at 45 degrees suddenly decides that it must be either vertically or horizontally polarized. Which is exactly what they do say for the silver atom.
Consider the well-known case of iceland spar, the prototypical “birefringent crystal”. A beam of light shone through the crystal is split in two. The vertically polarized component is refracted to a different extent than the horizontally polarized component. The mechanism whereby this happens is well understood. The crystal structure is not cubically symmetric, so it is easier to polarize in one direction than the other. Light passing through this crystal will travel at different speeds whether it is polarized along the easy axis or the stiffer axis. Ordinary light, which is polarized randomly, will naturally divide into two paths.

This is exactly the same thing that happens to silver atoms in the Stern Gerlach experiment, but no one points to iceland spar and says that it proves there is a spatial quantization of polarization: that light can be either vertically or horizontally polarized, but nothing in between.  That is just typical of the nonsense that is spouted everywhere you turn with regard to quantum mechanics.

Thursday, December 8, 2011

Quantum Leap or Superposition?

Earlier this year a guy named Andrew asked an interesting question on  Are these two quantum systems distinguishable?
The idea was this: suppose you had a machine that could randomly generate atoms in either the ground state or the first excited state. Then someone tried to sell you a cheap knock-off of this machine, except it generated atoms in a random superposition of these two states. Question: could you actually tell which machine you bought, the cheap knock-off or the Real McCoy?
It turns out that some of the people who frequent this site are pretty good at handling things like density matrices, and according to them these two machines were actually indistinguishable! The different output states described by these two machines ultimately reduce to the same density matrix, so expermentally you can’t tell them apart.

I actually proposed a way to build this machine: you take a vial of hot plasma composed of a fifty-fifty mixture of atomic nitrogen and carbon-14. You understand that in a nuclear sense, carbon-14 is really just an excited state of nitrogen, because that is what it decays to in 7000 years of so. I said just open a little door and let one atom out at a time. You might assume that there is a 50-50 chance that you get a carbon atom and an equal chance that you get a nitrogen atom. But according to those guys with their density matrices, I have just as much right to declare that each atom is in a 50-50 superposition of carbon and nitrogen…and experimentally, no one can prove me wrong!
In my original post I added a couple of stipulations: first, that the atoms might be in any number of superpositions, e.g. 80-20 or whatever, so long as they averaged out to 50-50. Secondly, that there was a real issue with the assumption that any machine was capable of letting out exactly one atom at a time. Maxwell’s demon and all that, but I think it goes even deeper.
No matter. That’s not why I brought up the question today. The reason is that in his original post, this Andrew fellow promised that pending the answer to this question, he would have a follow-up question. I was eagerly awaiting the follow-up and it never came. Somehow Andrew just disappeared.
The reason I was waiting for the follow-up is that I think I know where Andrew was going with this. It’s something I have been arguing for years and constantly getting shot down for. It’s about whether the universe is really as described by Copenhagen, with it’s quantum leaps and collapse of the wave function, or whether Schroedinger was on the right track when he looked for the natural time-evolution of the wave function. This is the question:
In the Copenhagen interpretation, we say that a gas consists of atoms in the ground state and a variety of excited states. The probability of finding an atom in an excited state is inversely and expontially proportional to the energy of that state. From time to time an atom jumps from one energy level to another, emitting or absorbing a photon. The probability of such transitions is calculated according to something called Fermi’s Golden Rule.

Following Schroedinger, I have an alternate description of the universe. I say that the same gas consists of atoms in a superposition of states. When you look at an atom in a superposition of eigenstates, you find that the charge distribution is not stable: it oscillates at frequencies corresponding to the difference in energy levels between the eigenstates. Because you have an oscillating charge distribution, it emits and absorbs radiation like a tiny antenna. The amount of radiation emitted and absorbed is calculated according to Maxwell’s Equations.
The question I ask, which is the question I believe Andrew meant to ask, is the following: is there any way to experimentally distinguish my model of the universe, my “cheap knock-off”, from the Copenhagen Model, the “real McCoy” according to everybody who is anybody. I’m saying there isn’t. Anybody care to disagree?

Monday, December 5, 2011

Entanglement and the Crossed Polarizers

I said last week that there was something wrong with the whole narrative concerning entanglement, and today I’m going to explain it. It has to do with the central role of Bell’s theorem in the ongoing debate. What I realized only a year ago is that Bell’s theorem hardly matters. The barn door was already open and the horse long gone before Bell came up with the business of the 22.5 degrees.

Don’t get me wrong. What Bell did was extremely clever, and he showed how to close an important philosphical loophole in the argument of local realism. But what people don’t seem to realize is that for all practical purposes, local realism was already in a shambles before anyone thought of varying the angle of the crossed polarizers.
I wrote last week about how Einstein pointed out the philosophical problem with two particles shooting off from each other with opposite momenta. According to the theory, the actual trajectory of either particle was completely indeterminate up to the surface of a sphere, until the moment when one of them was detected. At that moment, the wave function of the second particle collapsed simultaneoulsy: the measurement of one had affected the properties of the other.
This was the clear and unmistakeable implication of the theory, and it should have been extremely troubling. However there was a catch: no conceivable experiment could distinguish this philosophical nightmare from the more prosaic explanation that the particles had simply been endowed with their complementary momenta at the moment of separation; that the randomness in their detection was merely a lack of information on our part. To be sure, the theory was clear on the distinction: but that distinction remained, so it seems, only theoretical.

People seem to think that this happy state of affairs came crashing down at the moment Bell proposed his experiment with the crossed polarizers in 1964. That’s what I don’t understand. For me the disaster occurred in 1950 when Bohm proposed the experiment with spin states. The disaster didn’t depend on varying the angle of the polarizers: it should have been evident from the get-go.
We know how the experiment must look if two electrons are created with opposite spins: let one be in the positive z direction and the other in the negative z direction. If we set up two Stern-Gerlach detectors at opposite ends of the lab, we know what must happen. When one detects an “up” electron, the other must detect a “down” electron. There is nothing mysterious about this.

(Yes, I know the Stern Gerlach apparatus does not work on charged electrons but only on neutral atoms: but the theory is the same and for all I know, modern experimenters are able to adapt Stern Gerlach to charged species.)
Where the problem occurs is when we get a stream of electrons at each end of the lab, randomly up and down: what happens is each time detector A clicks “up”, detector B clicks simultaneously “down”. That’s a real problem.

But how is this different from the first situation, I hear you ask, where I said there was nothing mysterious? Sometimes the particle detected at A is up, so B must be down. And vice versa. Each detection stream appears random, but compare the streams and you get perfect anti-correlation. It’s all very ordinary, isn’t it?
No it isn’t! It is indeed possible to prepare pairs of electrons in complementary states, one up and one down (or at least it’s possible to write down an expression for the wave function!) and it’s very clear what must happen when we detect them: if we detect one of them at A with spin up, the other must be detected spin down at B…with a 75% probability! This is the result we get if the particles are endowed with opposite spins at the moment of creation: the detection streams are anticorrelated but not to the extent of 100%. It is only a 75% correlation.

Why is it only 75%? Because in practise, there is no source which prepares electrons in states of alignment only along the z axis: any real source must produce pairs of electrons anti-aligned along a random axis. So, for example, if they are aligned along the x axis, the z polarizer at A will detect its electron up or down with 50% probability; likewise, and completeley independently, the polarizer at B. You can see there is a 25% chance that both polarizers will detect up coincidences, and another 25% chance they will detect both down coincidences. You simply cannot get perfect anti-correlation with this kind of setup.
Unless, that is, you prepare the electrons in a very different type of state. It’s the quantum state we’ve alluded to already whereby the two electrons have opposite spins but there is no actual axis along which the spins are defined, until the moment of detection. That’s the mysterious entangled state that leads to all the philosophical headaches about local realism, and you don’t need to tilt your polarizers to 22.5 degrees to come face-to-face with it. It was, or should have been, obvious from the moment Bohm wrote his spin-modified version of EPR in 1950. There was no need to wait for Bell to come along in 1964 to set off all the excitement.

What I don’t understand about the whole history is why I don’t read anywhere about experiments to detect the perfect anti-correlations predicted by Bohm in 1950? Why does everyone only talk about the crossed-polarizer results motivated by Bell’s Theorem? To be sure, the later experimenters record their data at a variety of different angles, so the Bohm correlations are part of the record. But why does nobody ever talk about them as being significant?

Thursday, December 1, 2011

Double-dipping and the Talmud

Of the many memorable episodes of Seinfeld, who can forget the double-dipping scene in Episode 59, where George is confronted by his girlfriends brother at the reception: "That's like putting your whole mouth in the dip! From now on, when you take a chip, just take one dip and end it!"

I mentioned in a recent post that not many Jewish people of my generation are able to read our national literature in its original language. Because I am one of these chosen few, I sometimes come across things that I feel duty-bound to share with my less literate co-religionists. Todays article concerns one such topic.

We Jews are not so well-educated any more in our old traditions. I'm not talking about the ultra-orthodox in their enclaves, I mean the run-of-the-mill modern assimilated Jews who go to synagogue and send their children to Hebrew School. It's an education, granted, but it's far from being a Jewish education in the sense of our real "old-time religion".

In Imperial Russia, every eleven-year-old Jewish boy was immersed to the point of despair in the study of Jewish Law, specifically thosee mythological tomes referred to collectively as The Talmud. People sometimes think Madonna is studying the Talmud when she goes off to her Kaballah Center and meditates on the nature of God and consciousness and whatever. I'm sorry to say, but that's not the Talmud. The Talmud is concerned with the fine details of Jewish Law, as typified by the archtypical case of "the ox who gores the neighbor's cow". Our eleven-year-old Jewish child was expected to be able to explain all the different degrees of liability which applied to the owner of the offending beast, depending on such circumstances as the animal's previous history of violence. That's what the Talmud is all about.

What else does the Talmud contain? In addition to the rules of civil litigation there are of course dietary regulations, the rules of consecrating a marriage, etc. there is also an auxilliary portion of the Talmud known as The Agodah which has a rather different nature. The Agodah is a collection of folkloric tales largely devoted to stories about the great rabbi's who codified the Talmud in the aftermath on the destruction of the Temple by the Romans. In the course of my personal study of  Jewish culture, I was some years ago bequeathed a small volume of excerpt from the Agodah, translated into Yiddish from the original Aramaic. I was recently amazed to discover that this volume includes a cautionary tale on no less than the dangers of double-dipping. I'm not kidding.

For the benefit of my many readers in Germany, I'm going to include the relevant passage in full. Yiddish is of course written in Hebrew characters, so I've taken the liberty of latinizing it according to my own orthographic system which is designed to be German-friendly. Here is how it goes: (I've italicised those words which derive from the Semitic component of the language to make it less confusing. If you know both German and Hebrew, you should be able to read this passage without difficulty!)

"A mensch soll nischt trinken vun a kos un nâchdem geben an anderen zu trinken vun dem, weil dâs kenn sein schädlech zum gesund. Es is amâl gewe’en asa maysseh mit Rebi Akiva, wen er is gewe’en zugast bei einem. Der balebus (=ba’al habayit)  hât ihm derlangt a becher wein âber zuerst hât er alléin früher a sup getân vun dem becher. Hât Rebi Akiva gesâgt: “Trink dâs aus alléin”. Nehmt der balebus un giesst aus far Rebi Akiva a zwéiten kos un hât wieder früher versucht vun dem alléin. Sâgt Rebi Akiva noch amâl: "Trink dâs aus alléin.” Ben Azzai, welcher is derbei gesessen hât dann ausgerufen zum balebus: “Bis vannen westdu altz geben zu Rebi Akiva trinken vun dein maul?”

For those who are unable to follow the Yiddish, this is what happened: the venerable Rabbi Akiva was a guest at the home of some wealthy man, who offered the Rabbi a glass of wine from which he had previously taken a sip. Rabbi Akiva declined, the host then poured him a second glass, but again took a sip before extending it to the Rabbi. At this point Ben Azzai, who was also present, was unable to contain himself: “Why don’t you just let Rabbi Akiva drink straight from your mouth?”

Granted, this is a translation of a translation, and we must admit the possibility that the flavor of the original Aramaic might have been somewhat altered in passing first to Yiddish and then to English. But for me, I have no doubt. I’m pretty sure this is how the story has been understood for two thousand years, and I just can’t help  seeing in my minds eye Keiran Mulroney (who played Timmy in the Seinfeld episode) as Ben Azzai, his face contorted with rage and his voice seething with anger, coming to the defense of the too-polite Rabbi Akiva and laying on the line for the boorish balebus (played by Jason Alexander!):  “When you take a sip from the wine, it’s like you put your whole mouth in the wine!”