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)onforeignsoil.com 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.

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...

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 beam...one 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...

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.

Quantum Leap or Superposition?

Earlier this year a guy named Andrew asked an interesting question on stackexchange.com:  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?

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?

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!”