So you think you understand Quantum Mechanics

Thanks for doing this! (I don't think these are properly called "meditations", though: "Research shows that you're much more likely to remember useful info if you try to solve the problem yourself before reading the solution. Succeed or fail, the important thing is to have tried first." I think in this case, the primary point isn't to remember the correct answer once you post it, but to see how far off our own were, to correct our confidence in our understanding of QM.)

Okay, I do not place much confidence in any of the following, and to do it properly I'd probably have to spend far more time on this than I can spare, but I guess it's still useful to find out how wrong I'll be...

My mental image is the following stark simplification: I think of the state at any point in time as two probability distributions in 2D space (2D = the S-G apparatus viewed from the side, with the atom moving left-to-right and being deflected up/down), one probability distribution for the position of the atom if it's aligned along the field, one for if it's aligned against the field. I'm imagining each distribution to be concentrated around a single point at any point in time, but not necessarily the same point for both distributions. By "probability distribution" I do not mean that I'm actually doing measurements that have a certain probability to come out one way or another (though I guess I could), I just mean that I only look at the squared amplitude of the wave function, forgetting about the phase, which I'm guessing is not relevant to the problem (one place where I might be wrong). Before the atom enters the S-G apparatus, the two distributions are the same. I'm guessing that what the magnetic field does is, it makes the center of one distribution move up and the center of the other move down; again, this is a place where I might be going wrong, but my reasoning is that this is what seems to be required to get the right behavior if I put in an atom that is definitely aligned along/against the field.

M1: My intuitive way of thinking about it is that the atom aligns itself when the centers of the two distributions start moving apart, i.e., in the apparatus. But this is not what you're talking about. For the MWI "world" to split, we need entanglement with something big, which happens when the atom hits the screen. Therefore, in the Copenhagen picture, hitting the screen is when the measurement happens.

M2: I assume the probability distributions are very concentrated, and the holes are big enough to let practically all of the "probability mass" through, so that the shape of the distribution looks the same after passing the holes, and continue to move in a straight line (unlike in the double-slit experiment, where after passing the slit, the particle seems to move in all directions from the slit). Then, the reverse magnetic field will make the two blobs come together again, and the atoms are "unaligned" (= aligned in whatever direction each of them was when it entered the apparatus ... no wait, I guess atoms aren't spin-1/2 and I can't think of their state as being given by an alignment in a particular direction -- I think...; but anyway: I think they are again in whatever spin state they were before entering the apparatus).

M3: "Two equally bright blobs corresponding to aligned and anti-aligned atoms respectively", because my two blobs of probability will just move through the holes undisturbed and then hit the second screen just as if the first hadn't been there.

M4: Alright, this makes me question whether the mental model I've been using can be correct, because I've assumed that no entanglement with the apparatus happens, but I guess for the action of the apparatus on the atom there probably has to be an equal and opposite reaction of the atom on the apparatus, of some form... and I'm not sure how to think of that in the context of quantum mechanics. I can't do this one, and wonder whether my answers to the others are wrong because of this.

I hope you'll post a solution set at some point?

Alright, I'll take a crack at this. I haven't read the comments, so likely (I hope?) there's a lot of duplicate information here.

1: When does the atom align itself?

I'm not 100% sure what you mean by this, so let me know if I misinterpreted the question. Consider a single electron. Think of the wavefunction as the product of the spin wavefunction (which is representable as some linear combination of spin up and spin down), and the position-space wavefunction (which is probably a pretty tight gaussian wavepacket). This goes propagating along happily until it reaches the SG aperatus. Up until now, the spin wavefunction didn't affect the time evolution of the position wavefunction at all: this is no longer true.

Quantum mechanics is linear: If you calculate what happens to one half of the wavefunction, and what happens to the other half, and add them together, you get the whole thing. So, you start with:

=%0AA\left%20%7C%20\text{packet%20moving%20forward}\right%3E%20\left%7C\uparrow\right%3E+%0AB\left%20%7C%20\text{packet%20moving%20forward}\right%3E\left%7C\downarrow\right%3E)

A and B are some complex coefficients whose squares add to one. Now, the first term is pure spin up. As such, it will move up, while the second term will move down. This means your new wavefunction will be:

Now we get into interpretational differences. I believe in the Copenhagen interpretation, it goes something like this:

The atom hits the screen: this is a position measurement. The wavefunction collapses, and one of the position eigenstates is chosen. It's either going to be an eigenstate of a position in the top section of the screen, or of a position in the bottom section of the screen. Now, there are multiple eigenstates for the same position: spin up or spin down, because positon measurements don't care about spin. However, if you wind up with a top section position eignenstate, it must be spin up, as your pre-measurent wavefunction didn't have any component that was both in the top section and spin down. Likewise, if you measure the position of the atom as in the bottom section, you must be spin down.

Now on to MWI. We need to describe the state of the screen now, so I'm going to add another term to the wavefunction. Before the silver atom hits the screen:

)

Now, similarly to how the position of the atom got entangled with the spin of the particle, the state of the screen is going to become entangled with both of them, resulting in:

Now here comes the "world splitting" bit. When you look at the screen, or otherwise become causally entangled with the state of the screen in any way (that is to say, when your wavefunction depends on the state of the screen in some way). Before this happens, you have:

)

And afterwards, you have:

Now, remember that QM is linear. As such, you can treat each term completely separately. The first term looks like a world where the screen is marked near the top, and the atom is purely spin up. The second term looks like a world where the screen is marked near the top, and the atom is purely spin down. As soon as you become entangled with the state of the screen, the spin no longer seems to be in superposition at all, but is simply up or down, depending on if we're discussing the experiences of You_A or You_B. I would say that the world splits when you interact with the screen.

I may continue, but this level of detail is excruciating and I'm a bit burned out from it atm.

For those interested in the proper QM calculation of what is observed in the classic S-G experiment, I warmly recommend this old paper (PDF), especially Appendix 1. It shows why it is possible to treat the wave function of a spin-half atom as a superposition of spin-up and spin-down trajectories, why the spin precession in magnetic field can be ignored, and how to account for the velocity variation and beam width in calculating the deflection distribution.

1: Whatever the original alignment of the atom was, the N and S components experience a different lateral force, so the amplitude blobs corresponding to those two cases will be shoved apart sideways. I guess you could say they begin aligning the moment the blobs become distinguishable - and that'll depend on a variety of other factors like how you're measuring the deflection. This is the case whether you are working in MWI or Copenhagen.

2: back to unaligned, if you can completely reverse the forces in all cases and equalize the path-lengths - though the way the S-G apparatus really works, that's not going to happen. So you're going to end up with a complicated mishmash of altered phases, and thus reorientations.

3: The atoms heading through the hole in the first screen would end up with two single-slit interference patterns - though since atoms are really heavy I wouldn't count on this being a very noticeable interference pattern. It's two single-slit patterns because the left and right contributions are not feeding into the same quantum state (different spin), so they can't interfere.

4: By measuring the deflection, you've provided an earlier mechanism for distinguishing the blobs, so it would be at that earlier time.

Some of these "quantum" concepts that you mention don't require actual quantum physics in the first place. "Quantum dice" is just a short-hand for "truly random decision making method", and a quantum multiverse is for decision-theoretical purposes equivalent to having an infinite ensemble of classical universes with small differences between them, plus self-location uncertainty.

That's my impression anyway.

You're doing a world of good with this. People take their uneducated guesses far, far too seriously (QM and morality etc).

I object to “the following questions (usually called meditations in the idiosyncratic language used on this forum).” The term “meditation” was introduced relatively recently, not as an alternative to “question”, but a question serving a certain role, which these questions do not. I find this both a misuse of jargon and unnecessarily bizarre in this context. I request that you use what you think is ordinary language, instead.

M2: Apparently it's not possible to recover the original spin, so I guess you just end up with a random spin.

Stern-Gerlach Interferometer with Realistic Magnetic Field:

Abstract: In this report, we investigate the spin dynamics of a neutron beam in a Stern-Gerlach experiment. In contrast to the simple constant gradient magnetic field assumed in most literatures which violates Maxwell Equations, we work with a model of magnetic field which satises Maxwell Equations. The spin dynamics is investigated by solving the Schrodinger equation using Fast Fourier Transform method. The spin coherence of the neutron beam is found to exhibit the humpty-dumpty behaviour[1, 2, 3] even though there is no fluctuation to the magnetic field nor any environmental noise. The main cause of the spin decoherence is identified as the inhomogeneity of the magnetic field in the x and y components. In addition, the nonlinear terms in the z component of the magnetic field, zn, n > 1 also contribute to the loss of the spin coherence. Although only magnetic field model considered is very specific, the cause of the loss of spin coherence identified using this model is a common features of any realistic magnetic to be used in a SGA experiment. Therefore the humpty-dumpty nature of the spin coherence exist even without any fluctuation to the magnetic field and is inherent to the SGA experiment itself.

And see also this earlier paper, Is spin coherence like Humpty-Dumpty? I. Simplified treatment (free full text), which gives a more understandable qualitative argument:

Why should it be so difficult to maintain spin coherence? The essential features of the following qualitative argument are due to Heisenberg.

I won't try to quote the actual argument, which is on page 3 of the PDF, but it seems that the main problem isn't entanglement between the particle and the apparatus.

Thanks! Great post.

I was expecting more math-looking stuff in physics problems. Is it implicit?

shminux,

I think this test demonstrates how little I understand quantum mechanics, but the test also serves as an impetus for me to learn physics at a more rigorous level. So thank you for that!

Do you have any textbook recommendations for teaching myself quantum mechanics? Ideally, I need an introductory textbook suitable for someone already studying mathematics at an undergraduate level.

(I also welcome others to answer my question and debate each others' recommendations. Answers in this style get extra Paragon points.)

In other words, in the MWI picture, when does the world split into two, one with the atom aligned and one with the atom anti-aligned?

This one I'm going to do un-rot13ed because this misconception bothers me: The world was split all along according to the modern understanding of the MWI.

Wasn't sure if posting my not-very-considered answers was in the spirit of the exercise, but since you're hoping for more answers, here are mine. (And despite taking a QM class that discussed the Stern-Gerlach experiment, I'd somehow forgotten that S-G used an inhomogeneous magnetic field. So I might as well solicit some schoolin' from LW on this topic.)

Q1. If "When exactly does the atom align itself?" means "When does the atom start changing trajectory in response to the magnetic field?", I'd say it does that as soon as it enters the field. If it means "When can we say the atom has a single, unambiguous spin direction?", I'd say that's not until the atom hits the screen — before then it's in a superposition of spin up and spin down states.

Q2. As the atoms all travel unimpeded through the first S-G field, through the holes, and then through the reverse field, I guess they'd remain in a 50/50 superposition the whole way along. So they're just as "unaligned" or "aligned" as they were when they started. Presumably most of us would say the silver atoms were originally "unaligned" when they came flying out of the furnace, in which case we'd have to say they're still "unaligned".

Q3. I'd see two equally bright blobs, like those I'd have seen on the original screen, only further apart (because the two groups of atoms move further apart as they propagate, since the spin down particles have downward velocity relative to the spin up particles).

Q4. I'd observe either an atom continuously accelerating upwards or an atom continuously accelerating downwards. Once the atom enters the S-G apparatus and I see the accelerometer needle start moving, it's no longer in a superposition; it's either spin up or spin down, and I observe it accelerate accordingly.

When will the solutions be posted? If you post any more of these tests, could you mention that in the post itself, so that we know how much time we have to finish them, and also when to remind you to post the answers in case you forget?

It's not clear to me what you're envisioning for M4. I'm able to translate M2 and M3 into questions that I would have seen in my quantum classes, but M1 (and by extension M4) remind me more of electrodynamics. Is that intentional?

Similarly, the compass needle only pointing up or down doesn't fit with the standard description what underlies Stern-Gerlach, and I can't tell if discovering that subtlety is an intended effect of the meditations or not.

M1: The worlds split over a short period of time starting as soon as the atom enters the field. They don't split very far. In the Copenhagen interpretation no measurement occurs (EDIT: until they hit the screen) (you can tell because later on we'll "unmeasure" this alleged measurement, which would be impossible if the wavefunction had collapsed.). EDIT: The worlds don't split very far while the atoms are in the field (they only split in one direction). When they hit the screen they suddenly split a lot further. In Copenhagen this would constitute a measurement.

M2: The atoms become unaligned again. EDIT: You can't necessarily tell this apart from 50/50 aligned unaligned. If I could be bothered I'd calculate the density matrices. The situations are distinguishable iff the density matrices are different. My intuition says they're the same.

M3: Two blobs. No interference. EDIT: Because the spins are different.

M4: The detector answers soon after the atom enters the field. It will of course end up agreeing with the light formed by the atom hitting the screen. EDIT: In particular the acceleration would vary with time exactly as if the atom were classical.

D1: Gur ngbz vzzrqvngryl ragref va n fhcrecbfvgvba bs fcva-hc naq fcva-qbja, naq rnpu grez bs gur fhcrecbfvgvba trgf qvfcynprq, ohg (cneqba gur unaqjnil Pbcrauntral ynathntr) Angher qbrfa'g qrpvqr juvpu grez bs gur fhcrecbfvgvba lbh bofreir (va ZJV + qrpburerapr, gur fhcrecbfvgvba qbrfa'g qrpburer) hagvy vg uvgf gur fperra.

D2: Gurer'f ab jnl gb gryy. Gur qrafvgl zngevk (|hc><qverpgvba|), fb gurer'f ab jnl gb gryy na rafrzoyr jurer unys gur ngbzf unir fcva hc naq unys unir fcva qbja sebz bar jurer nyy gur fcvaf ner qvfgevohgrq ng enaqbz vfbgebcvpnyyl. Guvf obguref zr orpnhfr gur oryvrsf “gur ngbz vf rvgure fcva-hc be fcva-qbja jvgu cebonovyvgl 50% rnpu” naq “gur ngbz vf rvgure fcva-rnfg be fcva-jrfg jvgu cebonovyvgl 50% rnpu” srry vapbzcngvoyr ohg gurl yrnq gb gur rknpg fnzr nagvpvcngrq rkcrevraprf.

D3: Ubj jvqr ner gur ubyrf?

D4: Gur gjb ornzf qba'g vagresrer fvtavsvpnagyl ba gur fperra, fb vg qbrfa'g znggre gung lbh qb gung. BGBU vs lbh qvq gung jvgu n gjb-fyvg rkcrevzrag, lbh'q qrfgebl gur vagresrerapr cnggrea.

Wouldn't a meaningful test include some sort of math?

This experiment doesn't seem to require any non-classical behavior beyond quantized basic properties and a better understanding of electromagnetism.