Sorry, not in this post, but upcoming. I’m scheduled to do another episode of Bloggingheads.tv with David Albert, and we’ve decided to spend the whole hour talking about quantum mechanics. Start with the basics, try to explain this crazy theory and some of its outlandish consequences in ways that anyone can understand, and then dig into some of the mysteries of measurement, superposition, and reality.
So — what do you want to know? What are the really interesting questions about QM that we should be talking about?
One thing I don’t think we science-explainers get as clear as we could is the idea of the Wave Function of the Universe. It sounds scary and/or pretentious — an older colleague of mine at MIT once said “I’m too young to talk about the wave function of the universe.” But it’s a crucial fact of quantum mechanics (arguably the crucial fact) that, unlike in classical mechanics, when you consider two electrons you don’t just have a separate state for each electron. You have a single wave function that describes the two-electron system. And that’s true for any number of particles — when you consider a bigger system, you don’t “add more wavefunctions,” you beef up your single wave function so that it describes more particles. There is only ever one wave function, and you can call it “of the universe” if you like. Deep, man.
Here is another thing: in quantum mechanics, you can “add two states together,” or “take their average.” (Hilbert space is a vector space with an inner product.) In classical mechanics, you can’t. (Phase space is not a vector space at all.) How big a deal is that? Is there some nice way we can explain what that means in terms your grandmother could understand, even if your grandmother is not a physicist or a mathematician?
(See also Dave Bacon’s discussion of teaching quantum mechanics as a particular version of probability theory. There are many different ways of answering the question “What is quantum mechanics?”)
Comments
165 responses to “Everything You Ever Wanted to Know About Quantum Mechanics, But Were Afraid to Ask”
Sean, your
compares quantum mechanics with classical particle physics, which is a straw man. Compare quantum field theory with “probability densities over classical field states” (aka random fields) — which, given that all our best theories are field theories, and given that probability is ever-present in Physics, seems less likely to be a straw man — we find that superposition is present in both. Indeed, a random field can be formulated as a commutative algebra of operators, making differences between quantum and classical only of the algebraic structure and interpretation of their measurement algebras. Bell inequalities become an essentially negligible issue (I argue that the conspiracy assumption is natural for a random field, whereas I agree with the conventional argument that it is unreasonable for classical particle models).
With apologies for grinding my axe, for “Bell inequalities for random fields”, see cond-mat/0403692, J. Phys. A: Math. Gen. 39 (2006) 7441-7455; for an algebraic formulation of interacting random fields, “Lie fields revisited”, see Arxiv:0704.3420, J. Math. Phys. 48, 122302(2007). These will not convince you of much, I have not yet developed my approach enough for it to be much more than speculative, but you might perhaps take away that superposition is not a good separator between classical and quantum. If you understand and start to implement in your thinking that classical particle physics is a straw man you will be ahead of the curve. Asking how much of this can be put into a discussion directed at the layman is an awkward question of course.
Here’s one for today…
Interesting discussion everyone. A question for the experts. I recently came across Ballentine’s quantum mechanics book and he makes a fairly convincing case to me that this “collapse of the wave function” business is not needed in the measurement problem if one uses the statistical point of view for the wave function. How generally accepted is this viewpoint ?
Thinking more about the question of “deriving” the schrod eqn, maybe an analogy would be to the “derivation” of maxwell’s eqns in, e.g., Schwartz’s Principles of Electrodynamics, where he takes coulomb’s law, invariance of charge under lorentz boosts, and lorentz invariance, and argues not only that a magnetic field is required, but gets the full maxwell equations (if i recall correctly). So he is starting with three very well-established experimental facts and shows that consistency requires this extra structure (loosely speaking).
(I’m not sure whether his argument can really be viewed as an air-tight logical deduction, but it certainly provides motivation and insight).
It would be nice if there were a similar set of well-established experimental facts that could similarly lead you to schrod eqn. My guess is that there *isn’t* such an argument, because it seems like the experimental facts would have to involve the wavefunction, but we dont’ really *have* any experimental facts about wavefunctions. The fact that the wavefunction – the very quantity whose evolution schrod eqn describes – is not an observable gives it a very different flavor than the electromagnetic field.
Jason Dick, thanks for the helpful attempt. However, I still don’t think you get the deep objection, which is that even that one resulting wave “that we observe” still has no reason to suddenly shrink into a tiny space, it should still be an extended wave anyway, etc. You and others are still taking the observation regime for granted and can’t seem to “get above it”, you are IMHO like fish who can’t appreciate what their being in water does.
In any case, the multiple transmission thought experiment I have been describing above doesn’t rely on any particular interpretation of decoherence, etc., it is based on the logical implications of what we already know about those particular interactions between photons and wave plates.
weichi, I think you’re missing andy s.‘s point a bit. It’s all fine and good to say the Schrodinger eqn. “just is”, and we may not have some sort of derivation from first principles, but nevertheless Schrodinger himself must have got it from somewhere, right?
That is to say, Schrodinger didn’t simply write down random symbols until he found an equation that predicted the spectrum of the Hydrogen atom, did he?
Most intro-level text books do seem to just state Schrodinger’s equation without any explanation of where it comes from. I’ll go check my graduate QM book and see if it does any better.
Well, according to Merzbacher’s Quantum Mechanics textbook, Schrodinger wanted an equation that agreed with classical mechanics in the classical limit.
Taking Schrodinger’s equation and using psi = exp(i*S/hbar), we are led to the Hamilton-Jacobi equation (which is one formulation of classical mechanics, equivalent to Newton’s laws, Lagrangian mechanics, etc.) Except, this equation has one extra term proportionate to hbar.
In other words, it agrees with classical mechanics in the limit where hbar is negligibly small.
That’s a reverse derivation — what Schrodinger really did was probably used his knowledge of classical mechanics to constrain the form of the quantum mechanical equation, and then played around with it a bit until he got one that also reproduced known experimental data. At least, that’s my best guess without actually looking up Schrodinger’s original papers (or rather, translations of them to English).
To clarify the above post, I don’t mean that the Hamilton-Jacobi equation has an extra term compared to classical mechanics. I mean that the Schrodinger equation leads to the Hamilton-Jacobi equation plus an extra term. Hamilton-Jacobi by itself is exactly equivalent to classical mechanics.
I heard the story that Schrodinger got his idea from attending a lecture given by De Broglie on his thesis work
on wave-particle duality. Schrodinger and Kramers were sitting next to each other and at the end of the talk, Kramers said to Schrodinger that if matter had wave proporties, then there must be a wave equation. Supposingly this comment started Schrodinger thinking about the topic though of course it does not answer andy s.’s question.
Jason,
Thank you for responding to my handwaving.
Given my mental dyslexia I think QM makes more sense if this statement is reversed, that matter is a property of energy. We assume there is some underlaying monolithic property, but by all evidence so far, reality seems to be a function of relative interactions of opposing forces, with matter as a fairly stable manifestation of this. The search for this underlaying property leads to smaller and smaller points of observation, while the macrocosmic reality continues to balance itself between polarities. What if they don’t find the Higgs? Are they just going to keep looking? String theory is seemingly about the strings, but it’s only their fluctuation that really matters. What if they are simply vortices?
TimG,
Well, I did provide a pointer to Schrod’s original paper, so I don’t think I *totally* missed his point 😉 Agreed that it would be nice if textbooks explained why schrod choose particular eqn.
Probably not historical, but I think the clearest way to look at Schroedinger’s equation is that energy is the generator of time-translation in exactly the same way that momentum is the generator of space-translation. Noether’s theorem, etc.
Agreeing with Aaron: once we represent the results of experiments by operators acting on a Hilbert space, time evolution gives a one-parameter group, so by Stone’s theorem there is a self-adjoint operator, which in the case of time evolution we call the Hamiltonian, that generates the group. It’s convenient to represent statistical measurement by operators, and to represent the impossibility of joint probability distributions, for certain combinations of measurements, by using noncommuting operators.
More on “deriving” Schrodinger’s equation:
Consider a plane wave: psi = e^[i (k x – w t)]
This has
k psi = -i d/dx psi
and
w psi = i d/dt psi
From the de Broglie relations we have p = hbar k and E = hbar w
For this reason, we define:
p = -i hbar d/dx
and
E = i hbar d/dt
Now take the classical equation E = p^2 / 2m + V
and replace with our expressions for E and p in terms of differential operators.
This (acting on psi) is the Schrodinger equation.
Now, starting from plane wave solutions doesn’t make sense if V varies over space. But let’s suppose the Schrodinger equation is still correct. From this we can see:
(1) The equation has the right classical limit (as I mentioned above)
(2) It correctly predicts the spectrum of the hydrogen atom.
So, it looks like the equation is right even for the case where V depends on x.
TimG,
I like it. So we can use electron diffraction experiments to justify both 1) treating electron as a wave (so we have wavefunction) and 2) p = h-bar k deBroglie relation. For E = h-bar omega I guess you can refer to planck blackbody, and make a leap that it will hold for matter waves as well (just like deBroglie did!)
So treating these as your the experimental “facts”, your simple argument leads to Schrodinger (and also, I believe, Klein-Gordon if you take the relativistic case). It’s not a derivation from first principles, but better than pulling things out of thin air.
Yep, it gives the Klein-Gordon equation if you start from:
E^2 = p^2 c^2 + m^2 c^4
I’ve read somewhere that Schrodinger actually first discovered the Klein-Gordon equation, but rejected it because it gave incorrect predictions for the hydrogen atom. (Of course we now know the equation is appropriate for spinless particles, not electrons.)
1. Why can’t QM predict, i.e. have an analytic solution, the line spectra of He or any multielectron atom?
2. How does QM account for the continuous spectra of molecules?
ndy.s on Jul 7th, 2008 at 3:50 pm
Question 3: In a delayed choice two-slit experiment, …. How does it GOD DAMN KNOW HOW … ahem excuse me again.
—————–
It should be possible to use an Einstein lens as a great beam splitter. EM radiation emitted by a distant source, then lensed around a large elliptical galaxy (Abell cluster), will be have split paths through the Schwarzschild metric. For EM radiation in the radio band, bandpass filters with a very narrow frequency range can exploit the Heisenberg uncertainty principle to get around the path length difference take by either side. So if a photon strikes our radio telescope it has a quantum amplitude for traversing one path or the other. We may then be able to perform a Wheeler Delayed Choice experiment across hundreds of millions of light years.
How does the photon know which path to take? It doesn’t, in fact it knows nothing. Think of there being two types of relationships in the universe, one which involves invariant quantities such as distances, intervals, the momentum interval
$latex
m^2c^4~=~E^2~-~p^2c^2,
$
and the like and covariant elements such as curvatures. The other relationship involves quantum nonlocality, entanglements, superpositions and the like. This relationship system is independent of the geometric type of relationship. There happens to be a representation of quantum waves or states in spacetime, where
$latex
psi({vec r},~t)~=~langle{vec r}|psi(t)rangle,
$
here nonrelativistic, which obeys wave equations. In this way we can identify oscillator modes on each ADM relativity spatial slice of spacetime. Yet the wave function is nonlocally defined on each time slice (spatial surface) and that relationship is nonlocal. The Wheeler Delay Choice experiment indicates this nonlocality is not just across spatial distances, but temporal ones as well.
Lawrence B. Crowell
Albatross (#77) quoting from http://www.thepaincomics.com:
>
> Lately I find myself feeling less contemptuous of Creationists than sorry for them.
Although agnostic, I am contemptous of Creationists. Nowhere in the Bible does it detail or even indicate *how* the Universe was created or, explicitly, state that creation was ever completed. Nor, aside from the seven day business, where the word “day” was probably meant more as “epoch”, can a reliable timescale be deduced.
So dogmatic creationists are presumptuous and foolish to insist on their interpretation when the facts clearly indicate otherwise in countless ways. Maybe God if he chose to comment, far from approving of their blind faith, would give them the same bollocking he gave Job (Job Chap 38):
Celestial mechanician wrote:
Some differential equations can’t be solved analytically. Hydrogen works out nicely because of the spherical symmetry of the problem. The electron feels a potential due to the nucleus — and if you put the nucleus at the origin of your coordinate system and rotate around the origin, that potential doesn’t change.
In helium, an electron feels a potential due to the nucleus and due to the other electron, so the spherical symmetry is broken. You can’t put both the nucleus and the second electron at the point of rotation.
I see that this list of responses has been growing rapidly since the time that I first looked just a day or so ago. I think this is good news on the one hand (as I believe that general interest in scientific thinking and ideas to be a essential thing. Of course, experts have contributed but it’s the general audience that matters here). However I think it also reflects a real problem with the understanding of quantum mechanics and the consequent frustration held by the general public (well, unfortunately not quite general since in many countries most are more concerned with finding food and avoiding wars and oppression, and elsewhere a lot of people can’t be bothered…however…)
Time! That seems to me to be the key. It is one of the most natural, obvious, and simple things, yet I think it is the biggest enigma remaining in the understanding of the world. In GR the whole thing seems to be fixed and “cast in stone”. The Wheeler-de-Witt eqn (just an example of one bit of QM thinking applied to GR) doesn’t even mention it at all. But, time seems to keep plodding on regardless. There is, as well, the entropy question that Sean has commented on before, and which is also related to time.
And what is the “quantum measurement problem” if it is not a question about time – at least as we humans experience it? The unfortunate cat is observed first to be alive, and later to be dead (either as the result of the malevolent Mr. Schroedinger or later from natural causes). But it all seems to depend on who you ask and what they may know from their particular involvement with the whole sad business at any particular time. Thinking about this I have some enthusiasm for (but no real understanding of) Carlo Rovelli’s ideas of relational QM – but who knows?
The fact remains that IMHO nobody has got a clue. ALL modern theories are (or aspire to be) quantum mechanical. They increase in their sophistication and mathematical complexity. But, right now, and for the foreseeable future, nobody knows what is going on – despite the fact that the thing seems to work. The vast majority of physicists/chemists/engineers, etc. who use it day-in-day-out just get on with the calculations, having long-since exhausted their enthusiasm for the associated philosophy after too many nights spent arguing with friends while doing their degrees.
So – “Quantum mechanics (and GR if deemed appropriate) and the question of time” would be my request for your discussion, and my also my”Clay” $1000000 prize” (if I had any mony) for your solution.
-James
James Robson on Jul 9th, 2008 at 5:24 pm
In GR the whole thing seems to be fixed and “cast in stone”. The Wheeler-de-Witt eqn (just an example of one bit of QM thinking applied to GR) doesn’t even mention it at all.
————
The absence of time in the WD equation stems from its classical roots in the ADM “space plus time” approach to general relativity. It leads to a Hamiltonian constraint NH = 0, where the Hamiltonian is due to the Gaussian second fundamental form, and N is the lapse function. There is also a momentum contraint as well N^iH_i = 0, where N^i is a shift function that tells how points are slid around from one spatial surface to the next. The lapse and shift functions are not “God given,” but depend upon how one fixes a coordinate condition on a problem. So time is not something which has any “cast in stone” quality to it.
A similar thing happens with quantum mechanics. The momentum and position have commutator relationships [x, p] = hbar/2, which under canonical quantization come from the Poisson bracket in classical Hamiltonian mechanics. QM also has an uncertainty relationship between energy and time, but there is no Poisson bracket for the Hamiltonian and time, and there is neither a time operator. The most you can work up is a sort of periodicity operator. So even in QM time has a curious property and does not really have the same status as position.
I could go on about this at considerable length, but I might find myself accused of “theory mongering.”
Lawrence B. Crowell
Please discuss the paradox that arises when one tries to attribute expansion of the universe to the energy of quantum vacuum fluctuations. The fact that a discrepancy of 10^120 is found when making this attribution should I think first suggest that the two ideas have in fact nothing to do with each other and that there is really no difficulty at all. Nevertheless, physicists have expended a great effort in trying to reconcile this argument. What am I missing here? Why should I insist that vacuum energy causes the universe to accelerate?
The vacuum energy that physicists burden themselves with as an attempted explanation of cosmological acceleration is calculated the same way as the Casimir force, which has been measured and verified. So the discrepancy when one applies this idea to the cosmological expansion can only be a misapplication, somehow, of the theory and not the deeply troubling deficiency or inconsistency of theory that is always presented. But obviously I’m missing one or more ideas here.
Would it be possible to perversely attribute the Casimir force to the measured cosmological constant and calculate an inverse discrepancy of 10^-120? Why would this problem be improperly motivated?
Thank you to one and all with opinions.
Can undergraduates really run through walls, or is that just a ploy to get them more interested in the topic of QM?
TimG,
I’m pretty rusty and never was interested in it much but isn’t it the case that all differential equations have numerical solutions to arbitrary precision. Then it is not even possible to write down the differential equations that describe multi-electron atoms, not only that they have no analytic solution?