Mind-bender: Particle-wave duality

[Surlethe]

We're all geeks. We all know that light is a particle and a wave, and subatomic particles are also both particles and waves. So how do you resolve the contradiction?
Spoiler

DID YOU THINK ABOUT IT?

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NO, SERIOUSLY, THINK ABOUT IT.

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I AM NOT KIDDING.




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Okay, here's the answer. There's no contradiction. To see why, think about what a "particle" is and what a "wave" is. A "particle" is just pure position. In other words, a "particle" is something that's completely localized. It's intuitively appealing to us because things in our realm of experience have positions, and can be represented as particles -- for instance, billiard balls, airplanes, or planets. A wave is a self-propagating "disturbance" in some medium. Lots of things in our realm of experience can be represented as waves, like wind, water waves, or earthquakes.

We try to make sense of physical observations by tying them to our experience. So when we think of planets orbiting the Sun, we think of points on a plane moving in a central force field. Here, a particle is a good description of what's going on. Or when we think of molecules or atoms, we think of them as particles. Collisions are elastic, they're all bouncing around, they're localized --- again, a good description.

Similarly, interference is something waves do. So when light interferes with itself, we say it's a wave.

Are they actually particles? Well, what does that even mean? Asking for the "real," "intrinsic" nature of things is meaningless (and Aristotelian and maybe if you want to you should be a devout Catholic) -- things "are" what the best description of their observed properties says they are.

So is light a wave or a particle? The answer is, neither, because the best description of light is not as a wave and is not as a particle. The best description of light is as a ... well, I don't know QED well enough to say for certain. But it's not a wave, and it's not a particle.

[Johonebesus]

That's not much of a mind bender, is it? I learned at a relatively young age that light is neither a particle nor wave, but something that's behavior can sometimes be described as one or the other.

In fact, didn't I say the same thing a couple of years ago in a thread about whether quantum mechanics truly described reality or was like the Ptolemaic model?

[Skgoa]

I learned at a relatively young age that light is neither a particle nor wave, but something that's behavior can sometimes be described as one or the other.

this.

Is it really that astonishing things aren't actually completely like the metaphores we sometimes use to describe them?

[Fingolfin_Noldor]

Particles behave like waves within certain limits. Otherwise, they can be described classically.

If you cool atoms to 10^(-6) Kelvin and lower, atoms start to adopt wave like behaviour, simple as that.

[Simon_Jester]

I think pretty much everyone who seriously follows the physics comes to the same conclusion, Surlethe. As my first year graduate quantum professor put it, there comes a time at which you just have to "shut up and calculate-" the equations are what they are, and if you want to visualize things accurately you're forced to adjust your intuition to the reality, not the other way round.

[Kanastrous]

What I ended up with after reading a few books is the idea that it's particles whose probabalistic distribution in space is described by a wave function. At least, if particles is what your experiment is set up to detect.

[Simon_Jester]

You could equally well view a "particle" as a tight knot of "wave" which happens to be concentrated in a small space- so that if you're concerned with areas much larger than, say, an atom, it behaves like a geometric point-mass/charge/whatever.

Again, light and massive particles are what they are, not what it is convenient for us to imagine them to be.

[Darth Yoshi]

You could equally well view a "particle" as a tight knot of "wave" which happens to be concentrated in a small space- so that if you're concerned with areas much larger than, say, an atom, it behaves like a geometric point-mass/charge/whatever.

That was the understanding that I had after spending some time on quantum mechanics in one of my Astronomy courses.

[Kuroneko]

We're all geeks. We all know that light is a particle and a wave, and subatomic particles are also both particles and waves. So how do you resolve the contradiction?

After wave-particle complementarity, now try to wrap your head around black hole complementarity.

But the the entire issue is a like a linguistic word-game. Not even necessarily in the sense that we're trying to force reality into two predetermined buckets we happen to like while it doesn't completely fit into either one. Because with sufficient layer of interpretation, it can fit into either bucket, straddle them both, or neither. For example: Spoiler

Light comes in discrete chunks. Get a photomultiplier and counts the photons. Increasing the intensity of monochromatic light doesn't change their energy; just their number. Looks like particles.

What about interference? Well, what interferes is the particle state. But confusing a particle with its state is like confusing a balloon with its description (large or small, red or blue, inflated or deflated, etc.). So it's fully consistent to say: light is intrinsically made of particles, not waves; the interference comes from the particle states.

I'm not saying that's the 'best' or 'most intuitive' way to think about it. But it does resolve the apparent contradiction and is fully consistent with both experiment and theoretical formalism of QM, because it goes to an interpretative layer not fully dictated by the mathematical machinery that makes the predictions. [Although we have to be very careful here in regards to photons rather than electrons, because the proper context actually isn't QM, but QED.]

But I think by far the most instructive, though mathematically intensive, way to resolve the issue is this. Spoiler

Consider classical wave mechanics. Through the Hamilton-Jacobi equation, one can show that the limit of short wavelength/high frequency, one obtains classical mechanics with particle trajectories orthogonal to the wavefronts. For example, in standard optics the limits produces geometrical optics, in which light rays follow Fermat's principle of least time. This corresponds to principle of least action in classical mechanics if the wavefronts have velocity E/p.

That's exactly how it works out in QM: matter has a de Broglie wave of phase velocity ω/k = E/p, and in the short-wavelength (high energy) limit a quantum system becomes more classical. But although QM forces one to confront wave-particle duality, the roots of this phenomenon are not even quantum-mechanical in nature!

Asking for the "real," "intrinsic" nature of things is meaningless ... So is light a wave or a particle? The answer is, neither, because the best description of light is not as a wave and is not as a particle. The best description of light is as a ... well, I don't know QED well enough to say for certain. But it's not a wave, and it's not a particle.

Well... kind of. In QED, light is made of particles, more or less by definition--though the way 'particle' is defined is independent of classical intuition, and have some other peculiarities (e.g., particle number is observer-dependent).

But you're right in that there is also a sense which the best description of light is not a wave or a particle. We can motivate it in a much simpler manner than jumping into the nitty-gritty of QED by asking the following:
Q: Why are all photons intrinsically identical?
The particle in question can be replaced by other kinds of elementary particles (why are all electrons identical?). Spoiler

Because of QM, the problem is not as simple as them having the same intrinsic properties (mass, charge, ...). If they just happened to have the same properties, then in general a system of two particles would have the wavefunction in the form
[1] ψ(x1,x2) = φ1(x1)φ2(x2)
by which all we mean is that "particle 1 in state φ1 and particle 2 in state φ2". The fact that they have same mass, etc., would be reflected in the dynamics of the system--how the state changes in time as the two particles interact and so forth. But if there's nothing to distinguish them even in principle, then in general the system would have a wavefunction in the form
[2] ψ(x1,x2) = φ1(x1)φ2(x2) + φ2(x1)φ1(x2),
which is symmetric in the arguments and thus explicitly disallows being able to tell them apart even in principle.

Those two forms lead to very different experimental predictions, and the way electrons and photons behave confirms the latter (with minor corrections to the form due to spin) and falsifies the former.

QED directly provides the reason why: because electrons and photons do not have individually independent existence, but are rather aspects of something else--the Dirac and electromagnetic fields, respectively.

[madd0ct0r]

was I the only one who decided to think of them as a very long wave shaped particle?

Kuroneko, could you talk me through that post a little more? even wiki links would be good.

[HMS Sophia]

My physics teacher could never decide whether they were a pave or a warticle

[Winston Blake]

What I ended up with after reading a few books is the idea that it's particles whose probabalistic distribution in space is described by a wave function. At least, if particles is what your experiment is set up to detect.

This is how I imagine it intuitively. I think it's a very clear, 'no maths', 'pure visualisation' way of explaining it to people. Everything is 'really' 'made of' a ghost wave which propagates, and when anything is 'measured', a particle 'appears' according to the distribution of the ghost wave at that time. It's a sort of a 'ghost-wave musical-chairs game'. Of course, the mechanism by which a particle 'appears' isn't explained, but it's a good initial explanation.

was I the only one who decided to think of them as a very long wave shaped particle?

That's how I learned about refraction when I was young, and it's still the quickest way for me to think about such problems. I.e. as a train of parallel rods flying into a substance, with the 'tip that hits first' speeding up or slowing down.

[Kuroneko]

Kuroneko, could you talk me through that post a little more? even wiki links would be good.

Alright. I'm guessing that you would like to have more information on the connection between wave mechanics and classical mechanics, since that's the most technically involved part. If not, please be more please be more specific. You can probably look up anything I've italicized in wiki or any relevant textbook.

Classically, light is an electromagnetic wave described by Maxwell's equations. But on the other hand, many situations in optics can be modeled very well by treating light as rays, propagating in lines that bend in accordance to the local speed of light in the medium they're traveling in. The specific way rays bend is described by Fermat's principle of least time, which just says the following: the path of a light ray between two points is the path for which the amount of time for the light to traverse is it minimized (or in general, extremized--some situations, e.g., spherical mirrors, can maximize it instead).

To illstrate the principle, here are some toy problems:
0) In a medium in which speed of light is uniform, minimizing time is the same as minimizing length, and length-minimizing curves are straight lines. Thus, light rays go in straight lines.
1) Mirror. Consider a flat mirror, with two points A,B some distances a,b>0 from its surface. Suppose a light ray goes from A, hits the mirror with some angle of incidence α, gets reflected with angle β, and then goes to B. Now, we can reflect the segment from the mirror to B as going to B' as if the mirror wasn't there:

Code: Select all

A_            I intentionally drew the angles
  -_    B     of incidence and reflection as
    -_ /      different.
<-----*---->mirror
       \      Minimizing length A→*→B is the equiv. to
        B'    minimizing length A→*→B'

But that would make A,*,B' collinear, meaning α = β. This is the law of reflection: angle of incidence equals angle of reflection.
2) Transition between two homogeneous media. Instead of a mirror, imagine the horizontal line in the above diagram as the transition between a medium with speed of light v above it and some other speed w below it. It's possible (though rather tedious) to use Fermat's principle to derive Snell's law: sin α/sin β = v/w, though it is usually stated in terms of the refractive index n = c/v.

The geometrical limit in which Fermat's principle is valid is in which the refractive index n varies slowly on the scale of the wavelength. This is derived through the eikonal equation. Spoiler

One can take the wave equation ∂²φ/∂t² = (c²/n²)∇²φ, where c/n is the speed of light in the medium, and plug look for a solution that looks somewhat like a plane wave: φ = exp(A+ik0(S-ct)). If n was constant, it would be a plane wave, but here A,S are arbitrary functions of position. Actually plugging in into the wave equation gets:
[1] ∇²A + (∇A)² + k²0(n²-(∇S)²) = 0
[2] ∇²S + 2∇A·∇S = 0
Under the above assumption of wavelength small compared to the scale on which n varies, the k-term is the most significant, so in the limit of geometric optics, the eikonal equation holds: (∇S)² = n². The surfaces of constant S have the same phase and are therefore wavefronts.

So geometrical optics is a problem of path extremization, which is exactly classical mechanics lives. Fermat's principle of least time corresponds to Hamilton's principle of least action. For the simple case of a particle in some potential V, the Lagrangian is L = T - V, where T = (1/2)mv² is the kinetic energy, and the action is the integral of this quantity:
[3] S = Int[ L dt ]
The path the particle takes between two points is the one that minimizes (in general, extremizes) the action. Spoiler

Say you have a particle in static equilibrium: all the forces balance. Recall that the work done by a force is the force times displacement; since the x-component of the force vanishes, then for any infinitesimal displacement δx we imagine the system undergoing, Fxδx = 0. Or in general: F(x0)·δx = 0, for any infinitesimal displacement δx from position x0. For static equilibrium, this sort of "virtual displacement" does no work.

But we don't want static equilibrium; we want dynamics. From F = ma, suppose we "balance" the forces on an arbitrary particle by an "inertial force" -ma:
[4] (F-ma)·δx = 0
So if the force is conservative (∇×F = 0), then F = -∇V for some potential function V. The particle takes some actual path x(t), but we can imagine paths "near" it by having at each point in time some small displacement δx = δx(t), except at the endpoints of the path, which are fixed. Then integration by parts gives:
[5] 0 = Int[ (∇V + mx")·δx dt ] = Int[ (δV - (1/2)mδ(x'·x')) dt ] = Int[ (δV - δ{(1/2)m(v²)}) dt ]
(the boundary terms of int. by parts vanish because displacement goes to zero at the endpoints). Therefore, if T = (1/2)m(v²), then
[6] 0 = δInt[ (T - V) dt ],
i.e., the path the particle takes is a critical point (local extremum) of the action.

The above is somewhat of a handwave, but it turns out that the conclusion applies in general: multiple particles, classical fields, general relativity... and quantum mechanics and quantum field theory just adds some wrinkles to this, but is again very analogous.

Anyway, that's about as deep as I'm willing to go unless you have some specific questions. I hope I've at least motivated why we might expect classical mechanics to be a geometrical-optics limit of wave mechanics: they both turn into a path-extremization problem. Some of thing I'm leaving out are a bit more involved; I'll outline them for the sake of completeness:
1. How the eikonal function corresponds to Hamilton's "W" function in the Hamilton-Jacobi equation: they play the same mathematical role up to a constant of proportionality. The path from Hamilton's principle to the Hamilton-Jacobi equation is mathematically involved.
2. Why that means that if a particle corresponds to a wave of frequency ν, we must have E = hν for some undetermined constant of proportionality h. Hence, the de Broglie relations: λ = h/p, ν = E/h. De Broglie got a Nobel prize for this.

Other than those steps, things are actually pretty simple: if you take the wave equation and separate out a time dependence e[sup]-iωt[/sup], you get:
[7] ∇²φ + k²φ = 0, k = 2π/λ,
so substituting λ = h/p and T = p²/2m = E - V produces:
[8] -(ℏ²/2m)∇²φ + Vφ = Εφ, ℏ = h/(2π)
i.e., the time-independent Schrödinger equation. Of course, classical mechanics also expects h→0 instead, but not taking that limit gives us quantum mechanics instead. See also: Goldstein, Classical Mechanics, esp. Ch 9.


tl/dr: Classical mechanics is the limit of classical wave mechanics in the geometrical regime. So there is a kind of wave-particle duality even in classical mechanics, and we can have a corresponding 'classical mind bender':
Q: Both Hyugens' wave theory of light and Newton's corpuscular (particle) theory of light worked in geometrical optics, despite having apparently contradicting mechanisms. But how could this be?

[Winston Blake]

2. Why that means that if a particle corresponds to a wave of frequency ν, we must have E = hν for some undetermined constant of proportionality h.

This is something I was particularly interested in. Is there a simple way to explain it? On roughly the same level of 'involvedness' as this post?

(I have an engineering degree but I can't understand anything when you and Surlethe get going).

[Surlethe]

In QED, light is made of particles, more or less by definition--though the way 'particle' is defined is independent of classical intuition ...

By "particle" I mean exactly the classical intuition, a time-dependent delta 'function' of mass m.

But it does resolve the apparent contradiction and is fully consistent with both experiment and theoretical formalism of QM, because it goes to an interpretative layer not fully dictated by the mathematical machinery that makes the predictions.

Interesting point: we have to "interpret" the mathematical machinery that makes the predictions because it's not intuitive at all. Compare to classical mechanics, where the mathematical machinery is quite intuitive: it makes perfect sense to integrate a vector field. But there's no correspondingly easy intuition behind the evolution of an expectation value. We can visualize a vector-valued function of time, its derivatives, and its acceleration. We can't as easily visualize operators on Hilbert spaces.

[Surlethe]

To add another layer to Kuroneko's more recent post (for a general audience), there's a similar field patching together classical mechanics as the limit of quantum mechanics as h goes to zero or frequency (energy) goes to infinity. It's very similar to what Kuroneko was talking about: we're still dealing with eigenfunctions of operators, but the relevant operator is no longer the wave operator but the Laplacian. The relevant limit can be expressed two ways: wavelength going to 0 or h going to 0. Both give (some version of) classical mechanics in the limit.

Example result: Egorov's theorem, which can be expressed morally as "the time evolution of a QM observable* corresponds to the time evolution of its symbol**."

* (i.e. Hermitian operator on the appropriate Hilbert space)

** (the corresponding a classical observable, i.e., fn. on cotangent bundle)

[madd0ct0r]

oh christ I wish i hadn't asked.

give me a few days to work through that lot. It's been a long time since I did paths outside of structural mechanics.


EDIT: meant to say 'maths' but paths is probably as good a word as any given the context.

[Kuroneko]

This is something I was particularly interested in. Is there a simple way to explain it? On roughly the same level of 'involvedness' as this post?

Eh... well, if you're willing to simply accept the Hamilton-Jacobi equation, it's not too bad. Getting it from the action principle is kind of tedious, but is present in every advanced classical mechanics book, so I'll simply state the result: solving the physical system is equivalent to solving
[HamJac] H(x,∂S/∂x,t) + ∂S/∂t = 0,
where H is the Hamiltonian (total energy as a function of position,momentum,time). If the Hamiltonian is actually time-independent, then the equation must be separable, so S = W - Et, where W is time-independent.

This holds in general, but for the simple case of a particle in a time-independent potential V, H = T + V, where T = p²/2m is the kinetic energy. Note we're plugging in ∂S/∂x for momentum, so:
[HamJac] (1/2m)[ (∂S/∂x)² + ... ] + V = -(∂S/∂t) = E,
where the ellipsis represents any other coordinates that may be present, i.e.,:
[HamJac] (∇S)² = (∇W)² = 2m(E - V)
Aha! That's exactly the eikonal equation (∇W)² = n² for a ray of light traveling through a medium with refractive index n = sqrt[2m(E-V)].

As discussed in a previous post, we should suspect that Hamilton's principle of least action corresponds to Fermat's principle of least time in geometric optics. Let's check to see if that's sensible. The speed of the particle is v = ds/dt, where ds is the displacement. Therefore, rearranging T = (1/2)m(ds/dt)² gives dt² = (m/2T)ds², and so the action is:
[1] Int[ (T-V) dt ] = Int[ sqrt[2mT] ds ] = Int[ sqrt[2m(E-V)] ds ]
But if integrand is the refractive index, then the speed of the corresponding light ray is v = c/n, so:
[2] Int[ n ds ] = Int[ c dt ],
so minimizing action of a particle really is the same thing as minimizing time of a light ray. The connection between Hamilton's principle and Fermat's principle is explicit: the dynamics of a particle of total energy E in a potential V are exactly those of a light ray in refractive index sqrt[2m(E-V)].

In actual geometric optics, interference and other such effects are suppressed by taking the limit of zero wavelength and the story ends in the simple moral "geometrical optics is just like classical mechanics." But suppose we don't do that. Then the eikonal S [(∇S)² = n²] is proportional to the particle W [(∇W)² = 2m(E-V)]; the phase part of the wave, k0(S-ct) ~ k0W-k0ct, but k0c ~ ν, so we need E ~ ν to make the connection exact.

The constant of proportionality is completely undertermined. To make it consistent with the usual notation, let's call it h: E = hν. The de Broglie relations follow.
As a summary, here's a diagram of some of the conceptual connections:

Code: Select all

  Wave      charge-free  Maxwell's
Equation <-------------- Equations
    |         medium
    | 
    v         de Broglie
.Eikonal <------------------>  Quantum
.Approx.           ^          Mechanics <---+
    |       eikon.=|=Ham-Jac     |          |
    |λ→0     eqn  =|=  eqn       |h→0       |Deformation
    |              v             |          |[x,p]=ih/2π
    v       Fermat = Hamilton    v          |
Geometric  <-----------------> Classical ---+
 Optics                        Mechanics

In the nineteenth century, both the fact that wave optics had a limit of geometric optics and that geometric optics is analogous to classical mechanics was well known. What was unappreciated, however, is that one can think of the transition between waves and geometric optics as a two-step process:
(1) Properties of the medium don't change much on the scale of the wavelength.
(2) Wavelength goes to zero.
Of course, the latter implies the former, but taking (1) but not (2) and taking the correpondence seriously is precisely Quantum Mechanics.

Interesting point: we have to "interpret" the mathematical machinery that makes the predictions because it's not intuitive at all.

In the sense that we have a natural urge for visualization it in order to make things as intuitive as possible, yes. But the lesson is the reverse: you don't have to to do this, and doing it too much is what frequently gets you into trouble.

... there's a similar field patching together classical mechanics as the limit of quantum mechanics as h goes to zero or frequency (energy) goes to infinity. ... The relevant limit can be expressed two ways: wavelength going to 0 or h going to 0. Both give (some version of) classical mechanics in the limit.

Yes, depending on which version of quantum mechanics you start with. From the path-integral formulation, the action-extremizing paths reinforce each other in the limit, making them dominate, recovering Hamilton's principle of least action. In the Hamiltonian formulation, writing the wavefunction explicitly in terms of amplitude and phase, Ψ(x,t) = AeiS, through some tedium it is possible deduce the following from Schrödinger equation with ∇Ψ and ∇²Ψ:
∂S/∂t + (∇S)²/2m + V = (ℏ²/2m)(∇²A)/A
In the limit of ℏ→0, we drop the right side and rearrange:
(∇S)² = 2m(-(∂S/∂t) - V)
That's the Hamilton-Jacobi equation of classical mechanics.

As an aside, there's an additional payoff to the above exercise of being able to write the squared amplitude in terms of the phase, giving as a sensible physical interpretation for it: as per the Born rule, A² is a probability density, (A²/m)∇S is its probability current. Though there's a much easier way to define the probability current in QM, but in this form the role of the phase of the wavefunction is explicit.

Now the above involves some algebraic gymnastics, but it's conceptually simple, though perhaps it's not as abstractly satisfying as...

Example result: Egorov's theorem, which can be expressed morally as "the time evolution of a QM observable* corresponds to the time evolution of its symbol**."

Oho! That's some heavy stuff. Though now that you mention it, it's possible to motivate at least the background for it in a simple manner that also makes the following intuitive:

But there's no correspondingly easy intuition behind the evolution of an expectation value. We can visualize a vector-valued function of time, its derivatives, and its acceleration. We can't as easily visualize operators on Hilbert spaces.

I agree with this as literally stated, but not necessarily its spirit. Visualization does suffer a some problems in QM, but that's not quite the same thing as an irreparable crippling of intuition altogether, and in general lacking having a picture is not the same thing as not lacking an intuitive reason for why something is the case. To pick on your example, I think the following about the time evolution of expectations are fair to say:
(1) From Shcrödinger, it's trivial, though not necessarily intuitive.
(2) From classical mechanics, it's a warped but straightforward reflection.
(3) From Heisenberg, it's intuitive.
I'll come back to this in a bit.

[Kuroneko]

But there's no correspondingly easy intuition behind the evolution of an expectation value. We can visualize a vector-valued function of time, its derivatives, and its acceleration. We can't as easily visualize operators on Hilbert spaces.

I agree with this as literally stated, but not necessarily its spirit. Visualization does suffer a some problems in QM, but that's not quite the same thing as an irreparable crippling of intuition altogether, and in general lacking having a picture is not the same thing as not lacking an intuitive reason for why something is the case. To pick on your example, I think the following about the time evolution of expectations are fair to say:
(1) From Scrödinger, it's trivial, though not necessarily intuitive.
(2) From classical mechanics, it's a warped but straightforward reflection.
(3) From Heisenberg, it's intuitive.
I'll come back to this in a bit.

(I know much of this will be a repeat for you, but in the interests of any other mathematically-inclined members...) First, though simple is necessarily not the same thing as intuitive, the time evolution of expectation can be found just by the product rule and plugging in Schrödinger's equation iℏ|ψ'〉 = H|ψ〉:
d〈A〉/dt = (d/dt)〈ψ|A|ψ〉 = 〈ψ'|A|ψ〉 + 〈ψ|A'|ψ〉 + 〈ψ|A|ψ'〉 = 〈(H/iℏ)ψ|A|ψ〉 + 〈ψ|∂A/∂t|ψ〉 + 〈ψ|A|(H/iℏ)ψ〉
Since H is Hermitian, 〈ψ|A|Hψ〉 = 〈ψ|AH|ψ〉, and similarly for the first term, so in terms of the commutator [A,H] = AH-HA,
Quantum: d〈A〉/dt = 〈∂A/∂t〉 + 〈[A,H]〉/iℏ, for any observable A
Time evolution is a partial plus some Lie bracket acting with Hamiltonian... where has this been seen before?

Let's back up to classical mechanics, in which the state of the system is a point in the phase space. As the system evolves in time, the state traces out a curve, but at any time, you can think of the direction the system is taking as vector field over the phase space. That's the Hamiltonian vector field {·,H} (given by a Poisson bracket with the usual Hamiltonian, though that part's not essential here; it's just a repacking of the equations of motion in Hamiltonian mechanics). A vector field acting on a function is a directional derivative, and that's the time evolution of any classical observable:
Classical: df/dt = ∂f/∂t + {f,H}, for any observable f
Time evolution is a partial plus some Lie bracket acting with the Hamiltonian... now if you want to be really cute, represent the state as a vector from an origin, an call the vector |ψ〉, and rename the above directional derivative operator Ĥ = {·,H}. Then the time derivative of the classical state is given by |ψ'〉 = Ĥ|ψ〉. Unwarped Schrödinger.

Back in QM, the state is a unit vector, and as time goes on it rotates about this way or that. And that's naturally the way it looks in a fixed coordinate system. But how does it look if our point of view "ride along" with the vector? In other words, in a rotating coordinate system such that the state vector is fixed? Well, the Hamiltonian is the generator of time evolution, |ψ〉 = e-i(t/ℏ)H|ψ0〉, so: 〈A〉 = 〈ψ|A|ψ〉 = 〈ψ0|ei(t/ℏ)HAe-i(t/ℏ)H|ψ0〉.
There are therefore two equivalent pictures:
(Schrödinger) The coordinate system is fixed, the operators are fixed, and the state evolves as |ψ〉 = e-i(t/ℏ)H|ψ0〉.
(Heisenberg) The coordinate system rotates with the state, keeping it fixed, and the operators evolve according to A = ei(t/ℏ)HA0e-i(t/ℏ)H.
Apply product rule:
Quantum: dA/dt = ∂A/∂t + [A,H]/iℏ
Suddenly, observables are behave nearly identically as they do in classical mechanics, and taking expectation value gets us what we got from the Schrödinger equation. The reason the above is true is intuitive: we're just traveling with the state, and its particular form is easy to find: product rule. In the Heisenberg picture, the time evolution of expectations is obvious.

So the 'moral meaning' of Egorov's theorem is already seen: quantum observables act like classical observables with Poisson brackets replaced by commutators. What remains, of course, is formalizing and quantifying that relationship, because without further analysis, "act like" is still a weasel-word description.

[starslayer]

A couple of pedantic corrections, Kuroneko:

-When you've said potential, you've really meant potential energy.

-The Hamiltonian is not simply the total energy in all cases. That is only true under two conditions: 1. The potential energy is velocity independent. 2. The transformation between a given coordinate system and the generalized coordinates of the system does not explicitly depend on time. The general formulation of the Hamiltonian is H = Σpjdqj/dt - L, where the pj's are the generalized momenta and the qj's are the corresponding generalized coordinates.

Neither of these really has any effect on the results you've presented, but they are things for the audience to keep in mind.

[Winston Blake]

Kuroneko, thank you for your time in writing that post (and for all your participation in this thread). I am still processing parts of it, but I am amazed by much of it, for example the fact that classical mechanics can be derived from Maxwell's equations.