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Michael Weiss: Anschaulichkeit, Abscheulichkeit

My title comes from a famous pair of quotes, about the love lost between Schroedinger and Heisenberg. The first quote is in a footnote to Schroedinger's "equivalence" paper, where he proved the equivalence of his wave mechanics to Heisenberg's matrix mechanics.
My theory was inspired by L. de Broglie and by brief but infinitely far-seeing remarks of A. Einstein (Berl. Ber. 1925, p. 9ff.) I was absolutely unaware of any genetic relationship with Heisenberg. I naturally knew about his theory, but because of the to me very difficult-appearing methods of transcendental algebra and the lack of Anschaulichkeit [visualizability], I felt deterred, by it, if not to say repelled. [1]
Heisenberg responded in a letter to Pauli:
The more I think about the physical portion of the Schroedinger theory, the more repulsive [abscheulich] I find it....What Schroedinger writes about the visualizability of his theory 'is probably not quite right', in other words it's crap. [2]
Most of the philosophical debates swirling around quantum mechanics have to do with causality. We all know what a wet blanket Einstein was on Las Vegas night. (Probably still is. "Put those dice DOWN! I'm talking to you, God!")

But in the childhood of quantum theory, the matter of visualizability loomed just as large. In his second paper on wave mechanics, Schroedinger wrote:

...it has even been doubted whether what goes on in an atom can be described within a scheme of space and time. From a philosophical standpoint, I should consider a conclusive decision in this sense as equivalent to a complete surrender. For we cannot really avoid our thinking in terms of space and time, and what we cannot comprehend within it, we cannot comprehend at all. There are such things but I do not believe that atomic structure is one of them. [3]
Schroedinger wrote to Willy Wien:
Bohr's standpoint, that a space-time description is impossible, I reject a limine... If [atomic research] cannot be fitted into space and time, then it fails in its whole aim and one does not know what purpose it really serves. [4]
Bohr and Heisenberg of course held a different opinion. The founding papers on matrix mechanics expressed the operational philosophy: "You got your equations, you got your observations, and they match. What more do you want? Shut up and calculate!" Of course, they had to say it more politely, at least in print. For example, here's the abstract, in full, of Heisenberg's famous paper:
The present paper seeks to establish a basis for theoretical quantum mechanics founded exclusively upon relationships between quantities which in principle are observable. [5]
And in the introduction to the "Dreimaennerarbeit", the paper that laid out the whole structure for the first time:
Admittedly, such a system of quantum-theoretical relations between observable quantities...would labor under the disadvantage of not being directly amenable to a geometrically visualizable interpretation, since the motion of electrons cannot be described in terms of the familiar concepts of space and time. [6]
but then they immediately point out what really counts: the equations of motion have the same form as in classical physics. Yet in the same paragraph, they concede:
In the further development of the theory, an important task will lie in the closer investigation of the nature of this correspondence and in the description of the manner in which symbolic quantum geometry goes over into visualizable classical geometry.
Echoes of this argument reverberate faintly today. Textbooks complain about the desire for pictures (see Feynman vol.III pp. 11-4, 11-5 for an example). Pop-science articles glory in "quantum weirdness".

Philosophically, I can't say Schroedinger's position makes sense to me. Our visual systems sport some pretty complicated circuity, a lot of it quite non-intuitive. (Read Hubel's Eye, Brain, and Vision for the nifty details.) It's first rate for watching movies and playing video games, and similar Cro-Magnon activities. It didn't evolve to help us understand quantum field theory!

Historically, common sense intuitions have a pretty poor track record. Who invented statistical mechanics anyway, a masochist during a New England winter? "Brrr, it's cold outside! Hit me harder, molecules, oh, that feels gooood..."

On the other hand, I'm a visualizer myself. I started the "photons, schmotons" thread with one question in mind: just how far can you visualize the QFT description of light? I had in mind something like the balloon analogy in GR: it's wrong because of (a), (b), and (c), but it's still useful because it does capture (e) and (f). Little did I know...

Perhaps by the time the infamous, never-ending, hydra-headed "Photons, Schmotons" thread runs its course, my question will be answered. For the rest of this post though I want to talk about the discarded pictures (discarded by QFT, at any rate). Presumably deadly experimental results could be marshalled to drive stakes through the hearts of all these alternatives, but I'll leave that for someone else to discuss. Pretend we've turned up an old family album in the attic. Each quaint sepia-toned photograph draws our interest and affection. Someone else can recount how prosperous-looking Uncle Max went bankrupt in 1926.

OK, let's say we're determined to visualize wave-particle duality, experiments be hanged! What are our options? I can think of four.

Photons with rhythm

Our photons are little ball-bearings, but with waves etched on the surface like a designer logo. I am reminded of Newton's light corpuscles, with their "Fits of easy Reflection" and "Fits of easy Transmission".

The Ten-Minute History of Science says, "Newton, light particles---BAD! Huyghens, light waves---GOOD!" It comes as a bit of surprise to learn that Newton's Opticks is filled with observations of interference and diffraction phenomena. Newton concluded that his corpuscles had to undergo a periodic change of state, swinging back and forth between Fits of easy Reflection and Fits of easy Transmission.

Newton's theory of light had three characteristics:

(I love the way Newton described polarization: the corpuscles have Sides.)

How does Huyghens stack up?

Yes, no periodicity in Huyghens! [7] Surprised me too. Score two out of three for Newton! [8]

Pilot waves

Of course we think of de Broglie, and this line of thought lead eventually to Bohm's interpretation of QM. But Newton again takes priority. The famous non-framer of hypotheses couldn't refrain from speculating:
...when a Ray of Light falls upon the Surface any pellucid Body, and is there refracted or reflected, may not Waves of Vibrations, or Tremors, be thereby excited... and are not these Vibrations propagated from the point of Incidence to great distances? And do they not overtake the Rays of Light, and by overtaking them successively, do they not put them into the Fits of easy Reflexion and easy Transmission described above? [9]

Continuous waves, discontinuous emission and/or absorption

I call this the microwave popcorn theory: microwaves continuously bath the popcorn, which discontinuously (and unpredictably) pops.

According to the pop-history of science, Planck's theory fell into this category. For example:

Imagine a sponge in a bathtub... According to Maxwell, when a sponge is squeezed it sends out its water in the the usual way and causes waves in the bathtub. Planck's sponge is of a rarer sort. Indeed it is more like a bunch of grapes than a sponge, consisting of myriads of tiny balloons of various sizes, each full of water. When this sponge is squeezed, the balloons burst one after the other, each shooting out its contents in a single quick explosion--- a bundle of water--- and setting up waves... Einstein, however, took the sponge right out of the bathtub... When he squeezed his sponge gently, water fell from it like shimmering drops of rain. [10]
A charming story, but historically all wrong! Kuhn [11] argues persuasively that Planck believed in a completely continuous theory--- continuous waves, continuous emission and absorption--- until 1908, after Einstein put forward his light quantum hypothesis.

However, in 1912 Planck did come up with his so-called "second theory", in which emission is discontinuous, while propagation and absorption remain continuous.

Kuhn's book has full details. Though Planck's second theory never made it to the big time, it did come up with two hits: zero-point energy made its first appearance here, and Bohr got some inspiration for his model of the atom.

Wave packets

Early on, Schroedinger suggested that his wavefunction (for an electron) meant that the charge really was spread out--- space was pervaded with a kind of electron goop. In his equivalence paper (the one with the "Anschaulichkeit" footnote), he notes:
There are today not a few physicists who, exactly in the sense of Mach and Kirchhoff, see the task of physical theory to be merely the most economial description of empirical connections between observable quantities... In this view, mathematical equivalence means almost the same as physical equivalence. [12]
So is matrix mechanics just as good as wave mechanics, or maybe even better, because it doesn't clutter up the story with fairy tales? No, say Schroedinger--- physicists need space-time (i.e., pictorial) descriptions to make progress. He then proposes an interpretation of the wavefunction psi: the real part of (psi d psi/dt) gives the spatial density of electric charge.

Schroedinger also constructed a wave-packet: a well-localized psi function that stays together in time. He did this for a harmonic oscillator potential (just our coherent states, I'll bet!), but he hoped originally to do the same in general.

All waves, no particles anywhere! Can it really be that simple? Schroedinger hoped so.

Schroedinger sent his papers to "grey eminence of theoretical physics", Hendrik Lorentz. (Lorentz incidentally was the first fellow to convince Planck that the black-body formula could not be derived without some sort of discontinuity assumption.)

Lorentz raised several objections [13]. First, he noted that psi is function of (x,y,z) only in the single-particle case. With two particles, psi becomes a function of six variables, the coordinates of both particles:

If I had to choose between wave mechanics and matrix mechanics, I would give preference to the former because of its greater Anschaulichkeit, so long as one is concerned only with the coordinates x,y,z. With a greater number of degrees of freedom, however, I cannot interpret physically the waves and vibrations in q-space and I must decide for matrix mechanics.
Lorentz also pointed out that the harmonic oscillator potential was quite special, and that in the field of a hydrogen atom, the wave packet would spread out rapidly.

I won't go through the rest of Lorentz's criticisms. Schroedinger's biographer notes:

Lorentz belonged to an older generation of physicists, and Schroedinger might have drawn from their discussions the conclusion that his new discoveries cannot be fitted into a classical framework at all.
So all these are wrong! But what's right? Stay tuned...

[1] Ann. Phys., v.79, 734-56; quoted and translated in Schroedinger: Life and Thought, by Walter Moore, CUP, 1989, p.211.

[2] Heisenberg to Pauli, 8 June 1926; quoted and translated in Uncertainty: The Life and Science of Werner Heisenberg, by David Cassidy, W.H. Freeman and Co., 1992, p.215.

[3] Moore, p.208.

[4] Moore, p.226.

[5] Heisenberg, "Quantum-Theoretical Re-interpretation of Kinematic and Mechanical Relations", in Sources of Quantum Mechanics, ed. B.L. van der Waerden, Dover, 1968.

[6] Born, Heisenberg, and Jordan, "On Quantum Mechanics II", in Sources of Quantum Mechanics.

[7] Or so says I. Bernard Cohen in the preface to the Dover edition of the Opticks (see page xlvii).

[8] Why not three out of three, if we believe in quantum mechanics? I side with I.Bernard Cohen: "...we must choose between (1) the historical or (2) the antiquarian approach to the development of science... the antiquarian's sifting of the disjecta membra of the Opticks (often out of context) for an occasional 'precursorship' of one or another 20th-century physical concept." (op. cit.)

[9] Newton, Opticks, III.1 Query 17. And yes, I'm being a bit antiquarian here.

[10] Banesh Hoffmann, The Strange Story of the Quantum, 2nd ed., Dover, 1959, p. 26.

[11] Kuhn, Black Body Theory and the Quantum Discontinuity, 1894-1912, OUP 1978.

[12] Moore, p.212.

[13] See the discussion in Moore, pp. 214-217. This is the source for the two quotes below.

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side! Hit me harder, molecules, oh, that feels gooood..."

On the other hand, I'm a visualizer myself. I started the "photons, schmotons" thread with one question in mind: just how far can you visualize the QFT description of light? I had in mind something like the balloon analogy in GR: it's wrong because of (a), (b), and (c), but it's still useful because it does capture (e) and (f). Little did I know...

Perhaps by the time the infamous, never-endingbaker.htm010064400020410000013000000241360650332206400135230ustar00baezmathprof00004410252514 Michael Weiss: Okay, thanks, Baker, Campbell, and Hausdorff! next up previous
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Michael Weiss: Okay, thanks, Baker, Campbell, and Hausdorff!

Just stopping by for a sec, to drop off some homework. I'll be by again later for a longer chat.

But first--oh professor, I think you took too much off! You say:

This corrects a little mistake of Michael's where he claimed that

e-itH Coh(z) = Coh(e-it z).

It's not quite so simple and nice. Hint: vacuum energy.

But I explicitly said I was redefining the zero-point of energy to get rid of those pesky factors of e-it/2. Hmm, maybe you're saying the formulas are trying to tell me something-- that I shouldn't monkey around with the zero-point? (Ominous music wells up on the soundtrack). (Flash to the final scene in a 50s sci-fi movie, as the grey-haired senior scientist portentously intones, ``There are aspects of Nature that we change at our peril. Let this be a Lesson To Us All...'')

(Actually, the grey-haired senior scientist has already said something about this zero-point stuff.)

Okay.

A) Use the formula

Coh(z) = e(za* - z*a)/ sq2 |0>

to get a curiously similar formula involving an exponential of only creation operators, applied to the vacuum.

Here z=c+ib.

I did almost this computation once before, but let's do a quick recap.

First: we want to show that Coh(z) is an eigenvector of the annihilation operator a. For this we need to compute some commutators, and the slick way is to notice that [a,-] acts like a derivative on many operators. At least this works for power series in a and a*. Say f(a,a*) is a power series with complex coefficients. Since [a,a]=0 and [a,a*]=1, we'll get the right result for [a,f(a,a*)] with this recipe: compute (d/dx)f(a,x) formally, treating a like a constant; then replace x with a* in the final result.

Using this rule on eza* - z*a, we get

[a, e(za* - z*a)/ sq2] = [a, (za*-z*a)/sq2] = (z/sq2) e(za* - z*a)/ sq2

Now put |0> on the right, we get

a Coh(z) = a e(za*-z*a)/ sq2 |0> = (z/ sq2) e(za*-z*a)/ sq2 |0>

since a annihilates |0>. So Coh(z) = z/sq2 Coh(z).

 Next we expand Coh(z) in the basis |0>, |1> ....From the eigenvalue equation, we get immediately:

Coh(z) = C0 (|0> + (z/sq2 |1> + (z/sq2)n sqrt(n!) |n> + ...)

where C0 is the coefficient of |0>.

 We can evaluate |C0| pretty easily. The norm squared of Coh(z)is |C0|2 exp(|z|2/2), from the formula we just got. But Coh(z) has norm 1. How do I know that? Well, e(za*-z*a)/ sq2 is unitary. How do I know that? Well, (za*-z*a)/ sq2 is i times a self-adjoint operator (just take the adjoint and see what you get), so by some theorem or other its exponential is unitary.

So |C0| =  exp(-|z|2/4).  So we've determined e(za*-z*a)/sq2 |0> up to a phase (let's call the phase iota):

Coh(z) = iota  exp(-|z|2/4) SUMn (z/sq2)n /sqrt(n!) |n>

Hmmm, now for a new twist. The professor asked for the answer in terms of a*. Well, a*n |0> = sqrt(n!) |n> --hey, this works out nicely:

e(za*-z*a)/ sq2 |0> = iota exp(-|z|2/4) SUMn  (z/sq2)n /sqrt(n!) a*n |0>

= iota exp(-| z|2/4) eza*/ sq2 |0>

What are we going to do about that phase iota?

Hmmm, let's take another approach. If life was really simple, we could just say that eza*-z*a = eza*e-z*a (it isn't), and since a annihilates |0>, e-z*a|0> = |0> (just expand out e-z*a in a Taylor series). So we'd have:

eza*-z*a|0> = eza*|0>      (NOT!!)

But skimming back over the thread, we get strong hints that

eza*-z*a = enumber eza* e-z*a

Let's ask. Oh, professor!

The Baker-Campbell-Hausdorff formula says that when [A,B] commutes with everything

eA+B = e-[A,B]/2 eAeB
Hey, keen! How do I prove that?
You don't. You thank Baker, Campbell, and Hausdorff for proving it.
Okay, thanks! (They all read the newgroups? I've seen a post from Galileo, so maybe.)

Well, that makes short work of this half of the problem. Let's set:

A = za*/ sq2

B = -z*a/ sq2

[A,B] = -(1/2) zz* [a*,a] = zz*/2

which is a number and so commutes with everything, so

eA+B = e-zz*/4 eAeB

so

Coh(z) =  exp(-|z|2/4) eza*/ sq2  =  exp(-|z|2/4) eza*/ sq2 |0>

so the factor iota is 1.

Whew! Heavy firepower, just to determine that measly little phase factor iota! But then, rumor has it that Gauss spent two years of Sundays just trying to determine the sign of a certain square root.


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Michael Weiss

3/10/1998 beethovn.htm010064400020410000013000000252210650352012200142400ustar00baezmathprof00004410252514 Michael Weiss: Roll over Beethoven next up previous
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Michael Weiss: Roll over Beethoven

John Baez writes:
Before I dig into the business of working out what coherent states look like in the basis of eigenstates of the harmonic oscillator Hamiltonian, let me comment on one thing:
Uh-oh, better get the rest of my homework in before the professor goes over the assignment in class.

OK, last time I figured out that the coherent state you get by sliding the Gaussian bump c units to the right:

e-icp |0>

is proportional to this, in the basis of eigenstates of the Hamiltonian, aka ``particle representation'':

exp(-D2/2) SUMn (Dn/ sqrt(n!)) |n>

where D=c/ sq2  [sq2 being our symbol for sqrt(2)]. This time I've included the factor exp(-D2/2)  so as to get a normalized state-vector. Since e-icp is a unitary operator, by some theorem or other, e-icp |0> is also normalized. So we've expressed the coherent state in the particle representation, up to a phase.

The probability distribution of this state-vector is a Poisson distribution with mean value D2:

Prob(we're in state |n> ) = exp(-D2)  ((D2)n/n!)

Mean value = SUMnProb(we're in state |n>)

(To do the mean-value sum, notice that the n=0 term politely disappears, and if you pull out a factor of D2, what's left is just the sum of all the probabilities-- which had better add up to 1.)

Now this is nice. The original question was:

``If an electromagnetic wave has amplitude A and angular frequency w, how many photons does it contain per unit volume, on the average?''
and dimensional analysis said: A2/w. Well, presumably c and D will end up being proportional to A, once we actually start talking about electromagnetic waves, and not this mickey-mouse-dot racing around a circle! And the mean value should be proportional to the photon density. So we have our explanation for the A2 factor.

But to make this convincing, we have to know that D really does correspond to the radius of the dot's racetrack. So we need the time-evolution of the coherent state.

Now here I got stuck for a while, for I ignored the old adage: ``Never compute anything in physics unless you already know the answer!'' (Who said that, Wheeler?) I was trying to show that e-icp |0> would evolve to e-icp cos t|0>, since that's the formula for simple harmonic motion.

No good! The coherent state wavefunction has both position and momentum encoded in it. You can't expect the position to change without the momentum also changing.

OK, let's start over. First, we have to consider a more general coherent state, say one with momentum b and position c, thus:

eibq e-icp |0>

which I'll call Coh1(c+ib). [ Coh1 because it's ``coherent state, take 1'': John Baez will shortly introduce a better choice. --ed.]

Now we want to express this in the particle representation, because we know how the states |n> evolve: |n> evolves to e-int |n> (if we factor out the common e-it/2, i.e., redefine the energy zero-point).

I did this for the special coherent state Coh1(c) a while back, but I was working entirely too hard. I used the fact:

[q,pn] = inpn-1

Now this looks a lot like a derivative formula: [q,pn] = i (dpn/dp). (Ignore the fact that I haven't defined d/dp.) Actually, this shouldn't be surprising: we know that

[p,A] = -i (dA/dx)

i.e., bracketing with p is just about the same as taking derivatives with respect to x. Now q and x are pretty closely related, and what holds for q ought to hold for p, using Fourier transforms and all.

So we should have:

[p, f(q)] = -i df/dq

[q, g(p)] = i dg/dp

at least if f(q) is a power series in q, and g(p) is a power series in p.

As a check, let's verify the product rule. If that works, then we should have our result for all powers of p and q just by induction, and then for all power series by continuity arguments. The continuity arguments might take up a chapter or two in a functional analysis textbook, but hey, I'm sure the kindly moderator will cut us some slack. (Pause for thunderbolts to dissipate.)

[s, AB] = sAB - ABs

[s, A] B + A [s, B] = (sA - As) B + A (sB - Bs)

                          = sAB - AsB + AsB - ABs

It works!

So:

[q, e-icp] = i(-ic) e-icp = c e-icp

[p, eibq] = -i(ib) eibq = b eibq

[(q+ip), eibq e-icp] = (c+ib) eibq e-ipc

You have to be a little bit careful here. eA eB is not generally equal to eA+B if A and B don't commute.

If that last equation doesn't leap out at you from the previous two, well it's just a bit of straightforward grinding.

And so:

So Coh1(c+ib) is an eigenvector of q+ip with eigenvalue c+ib. It then follows that, up to a phase (call it iota):

Coh1(c+ib) = iota e-(c2+b2)/2 SUMn ((c+ib)n/ sqrt(n!)) |n>

Remember now that we want to time-evolve this. (So much physics, so little e-iHt...) As I said earlier, |n> evolves to e-int |n> (if choose our energy zero-point so as to get rid of the vacuum energy).

So the n-th term of our formula for Coh1(c+ib) will acquire the factor e-int in t-seconds. But we can absorb this into the factor (c+ib)n, just by replacing c+ib with (c+ib)e-it. So:

e-iHt Coh1(c+ib) = Coh1( e-it (c+ib) )

Roll over Beethoven, how classical can you get! If I told you that Coh1(c+ib) was my symbol for a dot in the complex plane at position c+ib, you'd say the equation I just wrote is obvious.

[Moderator's note: The aphorism, ``Never calculate anything until you know the answer,'' is indeed due to Wheeler. It appears in Taylor and Wheeler's book Spacetime Physics under the name of ``Wheeler's First Moral Principle.'' No other moral principles are mentioned, so maybe it's Wheeler's Only Moral Principle. --Ted Bunn]


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Michael Weiss

3/10/1998 boston.htm010064400020410000013000000137270666531052500137570ustar00baezmathprof00004410252514 John Baez, Michael Weiss: In Boston, you should definitely concentrate on driving. next up  previous
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John Baez, Michael Weiss: In Boston, you should definitely concentrate on driving.

[All you web-TV viewers out there, remember: Michael Weiss has been studying the wave equation in 1+1-dimensional spacetime, preparing to quantize it. The resulting "fotons" - massless spin-0 particles in 1+1 dimensions - will be but a pale imitation of Nature's wondrous "photons" - massless spin-1 particles in 3+1 dimensions. Nonetheless, once Michael understands fotons, photons will be pretty easy.

In the last episode, Michael and John worked out the complex structure for the Hilbert space describing a single foton. We now eavesdrop on an email exchange between them!]

Michael Weiss writes:

I may be a little distracted from QFT for a bit.

At odd moments, though, like sitting in traffic, I find myself wondering "OK, that's the complex structure, now what's the inner-product? JB gave a formula to get the inner-product from the symplectic form. Well, what's the symplectic form?"

Are these the right things to be wondering about? Or should I concentrate on driving?

John replies:

In Boston, you should definitely concentrate on driving.

However, when you are sure nobody is about to plow into you, here are some things to ponder:

First, I've already told you the whole Hilbert space structure - in Fourier transform land, it's just L^2 of the forwards lightcone - so the question is really what we want to do with it.

Michael interjects:

What about the backwards lightcone? Seems a shame, after going to all that work to figure out a complex structure for *both* halves of the lightcone, just to throw away the bottom half.

Of course, for the subspace of real functions, the values on the bottom half are determined by the values on the top half. But I thought the idea was that our complex vector space of real-valued functions sits inside the complex vector space of complex-valued functions in a copacetic way. I assume this remains true when we add in the Hilbert space structure.

John concurs:

Right, that's true. If we're studying *real* solutions of the wave equation in the Fourier transform picture, we only need to worry about their values on the forwards light cone - since that determines their values on the backwards light cone. If we're studying *complex* solutions we need to keep track of both the forwards and backwards lightcone. In that case, the Hilbert space is L^2 of the forwards lightcone with its usual complex structure, direct summed with L^2 of the backwards lightcone with *minus* the usual complex structure.

Anyway, there are various things we could do.

One thing we could do is see what the Hilbert space of our "foton" looks like in terms of initial data - meaning the field u and its first time derivative udot at t = 0. The symplectic structure, the complex structure, the real inner product - the whole schmear! To figure this out, we need to do little Fourier transforming. Good exercise, that.

Michael responds:

Sounds like a good project. Is this going to help me understand "non-localization of fotons"?

John:

Hmm. I'm actually not an expert on the nonlocalizability of massless particles. To creep up on *that* issue, it would probably be better to first study *massive* particles, where the Newton-Wigner localization *does* make sense, and then see how it blows up in our face when we set the mass to zero.

The above project would be a good warmup exercise for all that stuff - it'd get our Fourier transform skills back up to shape. But the more immediate payoff of understanding the Hilbert space structure in terms of initial data is that we'd understand some of the funky-looking equations people write down when they talk about "canonical commutation relations" in quantum field theory. After all, the commutators come straight out of the symplectic structure!

Also, we would get ourselves into the position of being able to understand coherent states of fotons, which is a good warmup for understanding coherent states of actual photons. We know all there is to know about coherent states of the harmonic oscillator; once we understand exactly how the Hilbert space of a single foton works, we'll be able to put that knowledge to good use!

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cinema.htm010064400020410000013000000050540650332240200136650ustar00baezmathprof00004410252514 Cinematic Interlude next up previous
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Cinematic Interlude

Camera pans in on John Baez, strapped in a chair. A maniacal Michael Weiss hovers over him, holding a dentist's drill, whose tip contains, instead a diamond, a gleaming photon. Michael asks with a grating voice that sends phonons racing up and down one's spine (simultaneously):

``Is it real?''

Baez groggily looks around. ``Huh. Most people just post to sci.physics.research and hope for someone to answer, not kidnap the moderator and strap him to a dentist's chair! I know I'm overdue for a checkup, but this is ridiculous. Are you getting back at me for avoiding questions about ontology, or something? Is what real, anyway?''

``Shaddup, wiseguy.'' Weiss clobbers Baez with a cosh.

After a few more bizarre special effects, changes of scene, and a whole lot of dreamy music, Baez wakes up. All he remembers is a question...



Michael Weiss
3/10/1998
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A@@ 9re_$,obz%ٱh-s;x"\oS("@NHS~&9yA$w+ _7!JQ0VEFӢp @ѐ JW4T05i ӚS6AӞ> j1B John Baez: A Gaussian bump with a corkscrew twist! next up previous
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John Baez: A Gaussian bump with a corkscrew twist!

OK, last time I figured out that the coherent state you get by sliding the Gaussian bump c units to the right ...
Very nice. I would like to understand this better and think more about the best way of deriving it. The way I suggested to you was stupid and grungy, but you seem to have fought your way through and then discovered some much nicer approaches. I haven't thought about this stuff enough, so I'd like to polish your work to a fine sheen before moving on.

So...let me go back to our general picture of coherent state.

What's the best quantum approximation to a classical particle on the line with specified position and momentum? A Gaussian bump with a corkscrew twist! We will only be interested here in bumps that have equal uncertainty in position and momentum:

Delta p = Delta q = 1/2

The simplest case is when

<p> = <q> = 0

Then we use the ground state of the harmonic oscillator:

|0> = exp(-x2)

where I left out the normalization factor to reduce clutter.

To get coherent states with other expectation values of position and momentum, say

<p> = b,    <q> = c

we can take our ground state, translate it in position space by an amount c, and then translate it in momentum space by an amount b:

eibq e-icp |0> = exp(ibx - (x-c)2)

where I have attempted to make an even number of sign errors. [An allusion to Dirac's comment on the first presentation of the Klein-Nishina formula (by Nishina). See Gamow, Thirty Years That Shook Physics. --ed.]

But wait! We could also have translated it first in momentum space and then in position space, getting

e-icp eibq |0> = exp(ib(x-c) - (x-c)2)

How does this answer fit with the other? Well, it differs only by a phase.

``Only a phase''...ah, what an understatement! When physicists and mathematicians mutter darkly about ``cocycles'', ``projective representations'', ``double covers'', ``central extensions'', and even more intimidating things like ``anomalies'', ``the Virasoro algebra'' and ``affine Lie algebras'', they are secretly complaining about the many subtleties that can caused by a mere phase!

So let us think about this a little bit. The two coherent states above differ by the phase e-ibc. That should be no surprise; the Heisenberg commutations relations

pq-qp = -i

lead directly -- with a dose of mathematical optimism -- to the exponentiated version called the ``Weyl commutation relations''

e-icp eibq = e-ibc eibq e-icp

which describe how translations in position space and momentum space commute only up to a phase. Actually, mathematical physicists of the rigorous variety prefer to take the Weyl relations as basic and derive the Heisenberg relations as a consequence. But we are being relaxed here so we can think of them as two ways of saying the same thing.

Now, Michael has taken

Coh1(c+ib) = eibq e-icp |0>

as his definition of a coherent state with expected momentum b and expected position c. This is fine...up to a phase...but it's slightly annoying how one needs to ``break the symmetry'' between momentum and position in this definition. Why not

Coh2(c+ib) = e-icp eibq |0>    ?

Or even better, how about some choice that treats position and momentum even-handedly! ``Mind your p's and q's!'' There's much wisdom in that phrase...

Here's a nice way to mind our p's and q's; we make the following new definition:

Coh(c+ib) = e-icp + ibq |0>

Here we ``simultaneously translate in position and momentum space'' instead of favoring one or the other. This state is not equal to either of the two choices listed above, but again it differs only by a phase.

Why?

Well, one can show that

e-icp+ibq = e-ibc/2 eibq e-icp

               = e+ibc/2 e-icp eibq

at least if I've not made a sign error. So this new definition ``steers a middle course'' between the other two choices, phase-wise.

(Also, fans of symplectic geometry will appreciate the funny skew- symmetric quality of the expression -icp + ibq in our new definition. But let's not get into that.)

Here's a little assignment for Michael, or any other students willing to pitch in! Remember that c represents position and b above represents momentum, so (c,b) represents a point in phase space. Also remember that it's good to think of phase space here as the complex plane. So let's define

z = c+ib.

(Don't confuse this lower-case z with the upper-case Z we had before. The upper-case Z was an exhalted operator; the lower-case z is just a lowly complex number.)

Now: take the expression

e-icp + ibq

and rewrite it in terms of z and its complex conjugate z*, while simultaneously rewriting p and q in terms of annihilation and creation operators. Remember that

a = (q+ip)/ sq2     q = (a+a*)/ sq2

a* = (q-ip)/ sq2     p = (a-a*)/i sq2

Some nice stuff should happen.

If we do this, we will get a cool expression for our coherent state in terms of annihilation and creation operators applied to the vacuum state. This won't immediately solve all our problems, but it should help us understand a lot about how our coherent states of the harmonic oscillator evolve in time.


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Michael Weiss

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Michael Weiss: Seeing Double

Michael Weiss: Seeing Double

OK, John gave us the diagram:
 \    /
  \  /  <------ forwards lightcone (w >= 0)
   \/
   /\
  /  \  <------ backwards lightcone (w <= 0)
 /    \
where pure waves exp(i(kx - wt)) live. Let me draw in the w and k axes:
    w
 \  |  /
  \ | /            
   \|/
---------k
   /|\
  / | \
 /  |  \    
("Rule, Britannia! Britannia rules the waves" - which I guess she did when Maxwell was alive...)

The diagram stretches out to infinity in all directions, of course, but let's say this piece just goes from w=-1 to w=+1, and k=-1 to k=+1. At the four corners we *could* put the four functions:

exp(i(-x -t))        exp(i(x - t))
exp(i(-x +t))        exp(i(x + t))

which span a 4-complex-dimensional subspace of the space of *all* complex-valued functions on R. (Nice thing about this post - I can just talk about complex vector spaces, and fuggedabout inner-products, measures, and so forth for the nonce.)

Here I'm using the *wrong* complex structure - the one where to compute i times f(x), you just multiply using ordinary complex arithmetic. With that definition, I'm sure our four functions are linearly independent.

With John's *clever* complex structure, I get a little confused. Do we really have four independent vectors, or am I seeing double?

Let me think about the *real-valued* functions first. I've been working with these four till now, two that travel rightward, two leftward:

leftward             rightward
---------            --------
sin(-x -t)           sin(x - t)
cos(-x -t)           cos(x - t)

Since these are all real-valued, we can form a real vector space with them, no sweat. 4-dimensional as a real vector, natch.

Where do these go on my "Rule Britannia" diagram? Well, = that's a bit ambiguous. Does cos(x - t) has positive frequency, or should I think of it as cos(-x + t) with negative frequency? Both and neither!

Blame Euler, who noticed that cos(x - t) is the average of *two* of our exp waves, the positive frequency exp(i(x - t)) and the negative frequency exp(i(-x +t)). We have the problem deciding whether sin(x - t) has positive or negative frequency.

Anyway, following John's instructions, we now define the operation of J on sin(x - t) by the simple rule: "roll back 1/4 period". This gives cos(x - t), either way you look at it. Likewise J applied to cos(x - t) gives -sin(x - t) (as it must, if we're gonna have J^2 = -1).

And that turns our 4-dimensional *real* vector space into a 2-dimensional *complex* vector space! After all, if you know what r times v means, for any real number r and vector v, *and* you know what i times v means, then you *ought* to know what (r+is) times v means--- or, to be crystal clear, (r+Js) times v. Our two rightward waves now become linearly dependent; one's just J times the other.

How about our exp waves? Hmm, Euler always said, "shut up and calculate". (Or something like that.) Let's try this:

   exp( i(x - t)) = cos(x - t) + J sin(x - t) = 2 cos(x - t)

   exp(-i(x - t)) = cos(x - t) - J sin (x - t) = 0

Seems a little odd. But OK, I guess. I suppose I'm trying to find a complex-linear map from:

the space of complex-valued functions on R, with the ordinary unclever boring "i" complex structure
to:
the space of real-valued functions on R, with the clever interesting "J" complex structure
and I've found one! And it maps the 4-complex-dimensional space spanned by the four corners of the British Empire, down to the 2-complex-dimensional space spanned by our trig waves.

This map has a couple of nice properties. First, say f belongs to the first space, but just happens to be real-valued. Then the map leaves f alone - it just maps f to itself. Next, the *negative* frequency waves just get annihilated! How's *that* for the power of positive thinking!

(Hmm, you think I can throw away my glasses?)

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John Baez: The Most Enlightening Set of Names

Before I dig into the business of working out what coherent states look like in the basis of eigenstates of the harmonic oscillator Hamiltonian, let me comment on one thing:
Hmmm. John Baez promised that the plain old harmonic oscillator would have something to do with E and B. Hey wait, the energy for the harmonic for the harmonic oscillator is (p2 + q2)/2; for the electromagnetic field, (E2 + B2)/2. Coincidence? You be the judge.
Of course it's no coincidence; the electromagnetic field is just a big bunch of harmonic oscillators, one for each ``mode'', and the formula for the energy is just like that for the harmonic oscillator. B is sort of like the ``position'' and E is sort of like the ``momentum''. But let's leave the details for later on.

What we're doing now is like finishing school. Everyone bumps into the harmonic oscillator in quantum mechanics, but rather few get to see its full beauty. There are three main representations to learn:

1.
the Schrödinger representation
2.
the Heisenberg (or Fock) representation
3.
the Bargmann-Segal representation
The first should really be called the ``wave'' representation. In this representation we diagonalize the position operator, thinking of the state as a function on position space.

Similarly, the second should be called the ``particle'' representation. In this we diagonalize the energy, thinking of the state as a linear combination of states | n> having n ``quanta'' of energy in them.

The equivalence of these first two representations is the basis of ``wave-particle duality'' in quantum field theory.

The third representation could be called the ``complex wave'' representation. At least that's what it's called in the book Introduction to Algebraic and Constructive Quantum Field Theory, where the first representation is called the ``real wave representation''. In some rough sense, this representation diagonalizes the creation operators. Of course, not being self-adjoint, the creation operators can't be diagonalized in the usual sense. There is a nice substitute, however. In the complex wave representation, we think of phase space as a complex vector space using the trick

Z = q + ip

and then think of states as analytic functions on phase space. Then the creation operator becomes multiplication by Z.

Perhaps this would be the most enlightening set of names for these three representations:

1.
the configuration space representation
2.
the particle representation
3.
the phase space representation
Everyone who studies quantum mechanics learns about the first two representations. The third, while in many ways the most beautiful, is somewhat less widely known. We won't get into it much here. The only reason I mention it is that it's lurking in the background whenever you relate quantum mechanics to the classical phase space and drag in the Z = q + ip trick.

Anyway, if you master these three basic viewpoints on the harmonic oscillator, it's a snap to generalize to quantum field theory, at least for free quantum fields, which are just big bunches of harmonic oscillators.


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Michael Weiss

3/10/1998 ="next" HREF="beethovn.htm"> John Baez: Michael has roamed ergodically next up previous
Next: Michael Weiss: Okay, thanks, Up: Schmotons Previous: John Baez: My lysdexia

John Baez: Michael has roamed ergodically

Okay, after having caught various slips by the absentminded professor, Michael has shown the following:

Let z = c+ib be a point in the phase space of a particle on the line, corresponding to position c and momentum b.

The coherent state with average position c and momentum b is

Coh(z) = e-icp + ibq |0>

                    = e(za* - z*a)/ sq2 |0>

Very good!

Here's some more homework. Actually, looking back over this thread, I see that Michael has roamed ergodically over the space of ways of thinking of this stuff, and has come very close to almost all possible ways, so this homework is not terribly novel.

A) Use the formula

Coh(z) = e(za* - z*a)/ sq2 |0>

to get a curiously similar formula involving an exponential of only creation operators, applied to the vacuum.

The formula is something like

Coh(z) = exp(-|z|2) eza* |0>

but you'll need to stick in a couple of constant factors here and there.

(Actually Michael has already done something like this, starting from a different angle.)

B) Use the commutation relations between H and a* to work out

eitH a* e-itH

and then

eitH eza* e-itH

Together with A), use this to work out the time evolution of the coherent state Coh(z).

 C) Show that if we evolve a coherent state over one period of our oscillator -- i.e., take t = 2 pi -- it does not return to the same wavefunction, unlike for the classical oscillator.

This corrects a little mistake of Michael's [HA! see below. --ed.] where he claimed that

e-itH Coh(z) = Coh(e-it z).

It's not quite so simple and nice. Hint: vacuum energy.


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Michael Weiss

3/10/1998 flooding.htm010064400020410000013000000233170706231034100142340ustar00baezmathprof00004410252514 Michael Weiss: Flooding the spacetime plain next  up  previous
Next:  Michael Weiss: Anschaulichkeit Up:  Schmotons Previous:  John Baez: Fourier transform land

Michael Weiss: Flooding the spacetime plain

Professor Wizard wants to talk about massive particles, where Newton-Wigner localization provides an approximate sort of position operator. I'm tempted--- I've had my sights on Newton-Wigner localization for a while. But not just yet.

He's also referred to the massive literature on massless position operators. Does it harbor the clues to the riddles that befuddle me? Perhaps, but I have my doubts...

Let me begin with some less refined fare. I'll begin with a famous old experiment, nearly bereft of mathematics, but elegant in its sheer simplicity. Then I'll shift gears abruptly to mathematics--- rather naive mathematics, but (I'm convinced) central to the unravelling of my confusions.

In 1910 or thereabouts, Geoffrey Taylor placed a very weak light source in a box on one side of a two-pinhole screen. He placed a photographic film on the other side, then he went sailing for a few months. He calculated that there was (with overwhelming probability) never more than one photon at a time in the box. Nonetheless, when he returned from his topsails and spinnakers to develop the film, he found the familiar two-point-source diffraction pattern--- the very same diffraction pattern that Thomas Young observed over a hundred years before.

Thomas Young's results we explain using field amplitudes. We square the electric field, and use that as a measure of light intensity. Two months before the mast, and the intensity has transformed itself into a photon density.

This is old hat for Photon, Schmoton fans. The whole schmoton thread got its start with the question: what is the relation between the average photon density and the field amplitude? After puttering around with some easy dimensional analysis, we launched into coherent states and concluded that if a field has amplitude c, then the matching coherent state has a Poisson distribution of quanta with expectation value c^2/2. (This was a zero-dimensional field, i.e., a field defined just at one point.)

All well and good--- but for a diffraction pattern, we'll need at a two-dimensional field, at least! And for Taylor's experiment, we'd like to say that the probability of detecting a photon at a given spot on the film is proportional to the intensity you'd calculate using Young's familiar field mathematics.

And Taylor's experiment is but a prototype. Leaf through the literature of classical wave optics. Pick a page, and take the low-intensity limit. Does the result survive, after replacing all squared amplitudes (intensities) with photon detection probabilities? I certainly hope so!

But this means that the probability of detecting a photon at a given position has to have some meaning in the brave new world of quantum optics --- massless or no.

Do I need a photon position operator? Well, maybe. Or maybe I need a quick course in scattering theory. After all, the schmoton thread has resolutely stuck to non-interacting schmotons. But all those non-interacting photons just slipped right through Taylor's box with nary a trace....

I do see one out, of a sort. Taylor's experiment involved a quasi-static quantum state of the electromagnetic field. After all, no need to go sailing for months if your weak light source burns out in two minutes! Perhaps for this special case, a spatial photon probability density makes sense.

'Nuff physics, let's do some math. A while ago I asked, with an aggrieved sense of injustice, why no one bats an eyelash at a photon momentum operator. Surely we can Fourier transform our way from k to x and back again to our heart's content? I did have a bad feeling about x's without t's--- I've been chided before for leaving out the 'c' in 'Schmoton' [1]. Still, how come k's without w's raise no eyebrows?

I now have a glimmer.

Fourier transforming u(x) into U(k) is a fine and dandy activity for a lazy non-relativistic afternoon. Put that way, the solution is obvious. Let u flood the spacetime plane, and Fourier transform u(t,x) into U(w,k)!

U(w,k) = (const) integral exp i (kx-wt) u(t,x) dt dx
u(t,x) = (const) integral exp -i (kx-wt) U(w,k) dw dk

integrating over all spacetime, or over all of (w,k) space.

Now the great thing about dt dx is that it's relativistically invariant. Ditto dw dk. I despair of rendering this in ascii-art, but if you can't picture a nice square piece of dx dt stretching itself area-preservingly into a rhombus, like one of those novelty finger handcuffs, ---well, you need a good dose of Taylor and Wheeler's Spacetime Physics.

Our u(t,x) does indeed flood the plain--- I can picture it shimmering out in green and silver ripples as far as the eye can see. U(w,k) is another school of fish entirely. We demanded that

u_tt = u_xx

so differentiating under the integral sign in our second Fourier fourmula above,

w^2 U(w,k) = k^2 U(w,k)

or w^2 = k^2. (Not a rigorous proof, but who cares?) (I think the official lingo says that U is "supported on the light-cone".)

In other words, u(t,x) is made up of various sized dollops of waves travelling left and right, but all with speed 1. (For more about the plusses and minusses of +-k, +-w, see Seeing Double.)

Earlier I wondered what the formula for the inner product of u and v could be, for u and v in our vector space of foton wavefunctions. In non-relativistic QM, once you have an inner-product, you're half-way to a probability measure.

I guessed:

integral (something with u(x)'s and v(x)'s) d(something)

but now a better bet looks like:

integral u(t,x)* v(t,x) dx dt (over all t,x)

or maybe

integral U(w,k)* V(w,k) dw dk (over all k^2 = w^2)

I guess I should pick one of these, Fourier transform it, and see what I get...

But not now! (After all, what I really should be doing is my taxes.)

But before signing off, let me pick some low-hanging from the previous equations.

It's easy to see how we could scrounge a probability measure for k from the second formula. After all, w^2=k^2, so w is virtually a function of k! If we assume u is real (standard so far for fotons), then U is completely determined by its values on the upper half (w>=0) of the light cone. (U(-w,-k) = U(w,k)*.) We can restrict attention to non-negative w, and set w=|k|.

So it would be natural to rewrite integral U(w,k)* V(w,k) dw dk as an integral just over k. U(w,k) = U(|k|,k) = U(k), say. The integrand becomes |U(k)|^2. The probability that a foton with momentum wavefunction U(k) has momentum between A and B is proportional to the integral of |U(k)|^2 dw dk, restricting attention to k's between A and B. (You'll notice I'm being a bit coy about w. More on that in a moment.)

For the position wavefunction u(t,x), we have no such reduction-- no obvious way to get rid of t. It sticks to x like gum on its shoe. Though I suppose in the static case (u(t,x) independent of t), we luck out...

To finish the reduction to k alone--- to obliterate all traces of w--- we need to replace dw dk by a differential involving k alone.

Perplexing! After all, the light cone is a one-dimensional locus! Its two-dimensional (dw dk) measure is zero. Talk about no visible means of support! Of course this is one of those Dirac delta thingies, but how to handle it?

After mulling and chewing and cogitating, I came up with this:

integral U(k)* V(k) dk/|k|

There's some geometric intuition behind this, but my post is long enough already, and besides, Schedule B awaits...

Perhaps Professor Wiz would like to deduce something before he's up to his neck in deductions!


[1] Nonetheless, Professor Wiz appears to have overstated his case, Yiddish-wise.

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fourier.htm010064400020410000013000000156040723761502000141140ustar00baezmathprof00004410252514 John Baez: Fourier transform land ... it's much less crowded there next up  previous
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Michael Weiss: Seeing Double

John Baez: Fourier transform land ... it's much less crowded there

Michael Weiss wrote:
the four functions:
exp(i(-x -t))        exp(i(x - t))
exp(i(-x +t))        exp(i(x + t))
[...] span a 4-complex-dimensional subspace of the space of *all* complex-valued functions on R.
I'm not sure why you're talking about complex-valued functions on R here. There are different R's you might mean - the x axis, the t axis - but what's really staring us in the face are some complex-valued functions on R^2 - the plane where the variables (t,x) live.
Here I'm using the *wrong* complex structure - the one where to compute i times f(x), you just multiply using ordinary complex arithmetic. With that definition, I'm sure our four functions are linearly independent.

With John's *clever* complex structure, I get a little confused. Do we really have four independent vectors, or am I seeing double?

Your four functions are all of the form

exp(-i(wt - kx))

where w is the energy and k is the momentum, as usual. Let me Fourier transform and think of them as delta functions in (w,k) space, living at the following four points:

(w,k) = (1,1)
(w,k) = (1,-1)
(w,k) = (-1,1)
(w,k) = (-1,-1)
These span a little 4-complex-dimensional subspace of the space of complex-valued functions on the lightcone in Fourier transform land. [1]

Now, there are two interesting complex structures on this vector space.

The *stupid* complex structure is just the obvious one: multiplication by i.

The *clever* complex structure is multiplication by i in the cases where (w,k) lives in the forwards lightcone, and multiplication by -i on the backwards lightcone:

 \    /
  \  /  <------ forwards lightcone (w >= 0)
   \/
   /\
  /  \  <------ backwards lightcone (w <= 0)
 /    \
In both cases we have a 4-dimensional complex vector space: only the complex structure is different.

Now, however, let us turn to the physics. We are studying the wave equation, but we have to decide what we are studying: *real* solutions of the wave equation, or *complex* solutions. The advantage of talking about *real* solutions is that it's it's obvious that we have to do something clever to get the right complex structure: ordinary multiplication by i isn't even available. So let's see what the above has to do with *real* solutions of the wave equation.

First of all, not every function in our little four-dimensional complex vector space corresponds to a real solution of the wave equation! There's a subspace of functions that do, which has *real* dimension four. It has a basis given by these functions:

leftward             rightward
---------            --------
sin(-t - x)           sin(-t + x)
cos(-t - x)           cos(-t + x)
Now, if you hit one of these functions with the *stupid* complex structure - that is, if you just multiply it by i - it won't be real any more, obviously. But if you hit it with the *clever* complex structure it stays real, I claim.

Why?

Well, being quantum field theorists, we go to Fourier transform land to do our serious thinking... it's much less crowded there. If you take the Fourier transform of a real function f(t,x), its value at (w,k) is the complex conjugate of its value at (-w,-k). If we take a function on the lightcone with this property and hit it with the stupid complex structure, it loses this property! But if we hit it with the clever complex structure, it still has this property. QED.

That was supposed to make it all obvious, but if it didn't, let's do an example. Take one of our real solutions of the wave equation, say

f = sin(-t + x)

and write it out using our basis of complex exponentials:

sin(-t + x) = exp(i(-t + x))/2i - exp(-i(-t + x))/2i

Now let's hit it with the clever complex structure, which we call J. The first term is the Fourier transform of a guy living on the forwards lightcone, while the second is the Fourier transform of a guy living on the backwards lightcone, so we multiply the first by i and the second by -i, and we get:

Jf = exp(i(-t + x))/2 + exp(-i(-t + x))/2
   = cos(-t + x) 
Yes! A real-valued solution! And it's a wave moving in the same direction, rightward.

That's how it always goes, even in higher dimensions. When we we take a real solution of the wave equation and hit it with the clever complex structure, it stays real. If we start with a sine or cosine wave moving in a given direction, hitting it with the clever complex structure gives a similar wave moving in the same direction, with the same energy and momentum.

We already knew that, actually: on eigenstates of the Hamiltonian, the complex structure is just evolution backwards in time a quarter period. For example, if we evolve

sin(-t + x)

backwards in time by a quarter period, we get

cos(-t + x)


[1] If you complain that a delta function is not a function, let me remind you that this is physics. Either learn to live with inaccuracy, or take "function" to mean "distribution".

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Michael Weiss: Seeing Double gaussian.htm010064400020410000013000000216171020750202200142400ustar00baezmathprof00004410252514 John Baez: A Gaussian Bump next up previous
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John Baez: A Gaussian Bump

Let me sketch step 1 here and see if I can get Michael to do the actual work.

We can think of states of the harmonic oscillator as wavefunctions, complex functions on the line, but they have a basis given by eigenstates of the harmonic oscillator Hamiltonian H. We call these states | n> where n = 0,1,2,3, etc., and we have

H |n> = (n + 1/2) |n>

As wavefunctions, |0> is a Gaussian bump centered at the origin. This is the ``ground state'' of the harmonic oscillator, the state with least energy. The state |n> is the same Gaussian bump times a polynomial of degree n, giving a function whose graph crosses the x axis n times:

We can think of |n> as a state with n ``quanta'' in it. Quanta of what? Quanta of energy! This is a little weird, but it'll come in handy to think of this way later. By the time we get to step 6, these ``quanta'' will be honest-to-goodness photons.

So it's nice to have operators that create and destroy quanta. We'll use the usual annihilation operator a and creation operator a*,  given by

a |n> = sqrt(n) |n-1>

and

a* |n> = sqrt(n+1) |n+1>

One can relate these guys to the momentum and position operators p and q, which act on wavefunctions as follows:

p = -i (d/dx)

q = x

In the latter equation I really mean ``q is multiplication by the function x''; these equations make sense if you apply both sides to some wavefunction.

So maybe Michael can remember or figure out the formulas relating the p's and q's to the a's and a*'s.

Once we have those, there's something fun we can do.

To translate a wavefunction psi to the right by some amount c, all we need to do is apply the operator

e-icp

to it. The reason is that

((d/dc) e-icp psi) (x) = (-ip e-icp psi) (x)

=  -((d/dx) e-icp psi) (x)

so the rate at which e-icp psi changes as we change c is really just minus the derivative of that function...meaning that it's getting translated over to the right. (We'll give some more detail for this step at the end of this section.) Folks say that the momentum operator p is the ``generator of translations'':

So we can get a wavefunction that's a Gaussian bump centered at the point x = c by taking our ground state |0> and translating it, getting:

e-icp|0>

This is called a ``coherent state''. In some sense it's the best quantum approximation to a classical state of the harmonic oscillator where the momentum is zero and the position is c. (We can make this more precise later if desired.)

If we express p in terms of a and a*, and write

e-icp = 1 - icp + ((icp)2/2!) + ...

we can expand our coherent state in terms of the eigenstates |n>. What does it look like?

If we figure this out, we can see what is the expected number of ``quanta'' in the coherent state. And this will eventually let us figure out the expected number of photons in a coherent state of the electromagnetic field: for example, a state which is the best quantum approximation to a plane wave solution of the classical Maxwell equations. It looks like there should should be about c2 ``quanta'' in the coherent state e-icp|0>. This should shed some more light (pardon the pun) on why our previous computations gave a photon density proportional to the amplitude squared and thus the energy density.

The thing to understand is why, even when we have a whole bunch of photons presumably in phase and adding up to a monochromatic beam of light, the amplitude is only proportional to the square root of the photon number. You could easily imagine that a bunch of photons completely randomly out of phase would give an average amplitude proportional to the square root of the photon number, just as |heads - tails| grows on average like the number of coins tossed (for a fair coin).

A few more details:

Suppose we have a wavefunction psi. What is e-icp psi? The answer is: it's just psi translated c units to the right.

Why? If we take psi and translate it c units to the right we get

psi(x - c)

so we need to show that

(e-icp psi)(x) = psi(x - c).

To show this, first note that it's obviously true when c = 0. Then take the derivative of both sides as a function of c and note that they are equal. That does the job.

We are assuming that if two differentiable functions are equal somewhere and their derivatives agree everywhere, then they can't ``start being different'', so they must be equal everywhere.

Or if that sounds too vague:

Technically, we are just using the fundamental theorem of calculus. Say we have two differentiable functions f(s) and g(s). Then

f(x) = f(0) + integral0x  f'(s) ds

It follows from this that if f(0) = g(0) and f'(s) = g'(s) for all s, then f(x) = g(x) for all x.


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Michael Weiss

3/10/1998 I>)

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Next: John Baez: My lysdexia Up: Schmotons Previous: John Baez: A Gaussian

Michael Weiss: Now for a bit of straightforward grinding.

OK, so let's set

z = c+ib

Coh(c+ib) = e-icp + ibq |0>

Now for a bit of straightforward grinding. We want to express -icp+ibq in terms of z and a.

-icp = -i  (z+z*)/2  (a-a*)/i sq2

                      = (-1/2 sq2) (za + z*a - za* - z*a*)

ibq = (z-z*)/2  (a+a*)/sq2

                   = 1/2 sq2 (za + za* - z*a - z*a*)

-icp + ibq = (1/sq2) (za* - z*a)
 


Michael Weiss

3/10/1998 hw-week.htm010064400020410000013000000374771020750247600140250ustar00baezmathprof00004410252514 Michael Weiss: Hmm, homework on the weekend! next up previous
Next: John Baez: Ever wonder Up: Schmotons Previous: John Baez: A Gaussian

Michael Weiss: Hmm, homework on the weekend!

Hmm, homework on the weekend! (grumble, grumble) Oh, well, beats cleaning out the attic (``Whaddya mean we have wasps!? Sure they're not photons?'')

John Baez assigns:

So maybe Michael can remember or figure out the formulas relating the p's and q's to the a's and a*'s.
Hey, I know that! Basic QM...just toss off the answer, q+ip, q-ip,...No, wait, what does it say here: ``Show all work!!''

All righty, then. Actually this is kind of fun. I still remember how slick and utterly unintuitive this seemed when I first encountered it in Dirac's book.

Moreover, this slick calculation evokes even earlier memories, of two jewels of high-school algebra: (a+b)(a-b) = a2-b2, and (a+bi)(a-bi) = a2+b2. If these have lost their luster over the years, try them out on your favorite bright 8-year old. I think of (a+b)(a-b) as the primordial instance of that staple of mathematical physics: cancellation of terms, leaving a simple final result.

Well, if having the cross-term cancel is cool, having them not cancel turns out to be even cooler!

Dirac does offer one slender reed of motivation, a single sentence: ``The above equations are all as in the classical theory.'' The ultimate incarnation of Bohr's correspondence principle. In fact, Olivier Darrigol wrote a book about that a few years back:

From C-Numbers to Q-Numbers: The Classical Analogy in the History of Quantum Theory. California Studies in the History of Science, University of California Press, December 1992.
OK, so let's start with a classical dot racing around in a circle. Projecting onto the x-axis gives simple harmonic motion. We'll use q for the x-coordinate and p for the y-coordinate, following Hamiltonian tradition. (Why p's and q's? Because Hamilton hung out in Irish pubs, minding his pints and quarts?)

Now as complex number, q+ip is that racing dot. Everything flows from q+ip. Let me be ruthless with factors of m and the spring constant and the like-- set them all to 1. The total energy is H=(q2+p2)/2, or (1/2)(q+ip)(q-ip). As a function of time, q+ip = e-it (for clockwise motion). The velocity vector is -i(q+ip).

OK, so let's do the same thing in Quantum Land. Now q and p no longer commute; in fact, qp-pq = i. So we set Z=q+ip and try to compute H, which was ZZ*/2 last time. But this time the cross-terms don't cancel, and we get:

ZZ*/2 = (q2 + p2 + 1)/2 = H + 1/2

Z*Z/2 = (q2 + p2 - 1)/2 = H - 1/2

I guess I won't go through the whole development. To get the annihilation operator a and the creation operator a*, we just need to absorb that pesky factor of 2 into a and a*: a = Z/sqrt(2), a* = Z*/sqrt(2).  [Sqrt(2) will be turning up a lot in these notes, so for the HTML version I will use the abbreviation sq2.  --ed.] So we get:

      aa* = H + 1/2     [a,a*] = aa* - a*a = 1

a*a = H - 1/2      [q,p] = qp - pq = i

a  = (q+ip)/sq2    p = (a - a*)/ i sq2

a* = (q-ip)/sq2      q = (a+a*)/ sq2

I think that will do it for HW, part 1.

Hmmm. John Baez promised that the plain old harmonic oscillator would have something to do with E and B. Hey wait, the energy for the harmonic oscillator is (p2 + q2)/2; for the electromagnetic field, (E2 + B2)/2.  Coincidence? You be the judge.

OK, part 2 of the homework. First John Baez defines a coherent state to be e-ipc|0>, then asks:

If we express p in terms of a and a*, and write

e-ipc = 1 - ipc + ((ipc)2/2!) + ...

we can expand our coherent state in terms of the eigenstates |n>. What does it look like?

Since p = (a-a*)/i sq2, -ipc = -c(a-a*)/ sq2. Let's write D for c/sq2, so -ipc = -D(a-a*). So our coherent state is:

|0> - (D(a-a*)/1!) |0> + (D2(a-a*)2/2!) |0> - (D3(a-a*)3/3!) |0> + ...

The problem is to figure out the coefficient of |n> , i.e., express the coherent state in the form:

C0 |0> + C1 |1> + C2 |2> + ...

[At this point, the poor student went off on a long detour. Inspired by John Baez's remarks about coin-tossing, he rambled through the theory of random walks-- at random! Alas, this lead nowhere, but (by a process to tortuous to summarize) it did inspire his next step. --ed.]

So I guessed that

Cn = (D/ sqrt(n)) Cn-1

so of course

Cn = (Dn/ sqrt(n!)) C0

Hmmm. Let's use this guess for the Cn, and apply a to SUMn Cn |n>; to make things a little simpler, let's divide through by C0 first. We get:

 a (|0> + (D/1) |1> + D2/(sqrt(2!) |2> + ... ) =

0 + (D/1) |0> + (D2/sqrt(2!)) sqrt(2) |1> + ... =

D ( |0> + (D/1) |1> + ... )

I.e., the coherent state would be an eigenstate of the annihilation operator a!

So I tried to prove this with combinatorial tricks, but no luck. Then I had the idea of going back to using q and p, instead of a and a*. With hindsight, the suggestion to rewrite 1 - ipc + ((ipc)2/2!) - ... in terms of a and a* looks like a false lead. Say it ain't so, John!

So here's how my solution goes: [slicker solutions will be found later in these notes. --ed.]

First off, we need a generalization of [q,p] = i. Here it is:

[q,pn] = q pn - pn q = nipn-1
or

qpn = pn q + nipn-1

which allows us to move q's past p's.

Next we apply this rule to q e-ipc   [You'll find the two lines of computation in the postscript version, but I'm not up to translating them into HTML.  --ed.]

or q e-ipc = e-ipc q + c e-ipc. So:

(q+ip)e-ipc |0> = e-ipc(q+ip) |0> + ce-ipc |0>

= ce-ipc |0>

since q+ip does annihilate |0>. So e-ipc|0> is an eigenstate of a, with eigenvalue c/ sq2 (=D).

And now for the payoff!

If the coherent state is proportional to:

|0> + (D/sqrt(1!)) |1> + (D2/sqrt(2!)) |2> + ...

then the probability distribution over the energy eigenstates is:

  • |0>    exp(-D2) 1
  • |1>    exp(-D2) D2/1!
  • |2>    exp(-D2) D4/2!
  • .....
  • |n>    exp(-D2) D2n/n!
that is, a Poisson distribution with parameter D2. The exp(-D2) is just the normalization factor, to make the probabilities add up to 1.

The expectation value for this Poisson distribution is D2. In other words, the average number of photons--err, quanta-- in the coherent state is D2.

Hmm, Baez has already pointed out how the coherent state looks like a bump displaced c units from the origin.

Suddenly the ``classical dot'' racing in a circle has a whole new lease on life. There was something pretty unsavory about what Dirac did to it. It reminds me of those ``before and after'' pictures-- are they really the same person? Before: the dot races in a circle of any radius. After: nothing races around at all, instead we have a bunch of wavefunctions that just sit there. Yeah, I know-- only after we've factored out the time-dependent part e-i(n+1/2)t. But this kind of motion is in an imaginary direction, and doesn't affect the wherabouts of the particle, if you know what I mean.

Could these utterly different kinematics really come from the same cinematographer?

Sure! Just use coherent states. As D increases continuously, the average energy increases continuously, as does the range of motion of the bump.

Hmm. Guess I should check that last assertion.

Hokay, we want the time evolution of the bump. Should be a piece of cake. The time evolution of |n> is just e-i(n+1/2)t |n>, and we've expressed the coherent state as a sum of energy eigenstates.

Let's redefine the zero-point of energy to get rid of that annoying 1/2. So e-iHte-ipc|0> is equal to C0 times:

|0> + e-it (D/ sqrt(1!)) |1> + e-2it (D2/ sqrt(2!)) |2> + ...

that is, exp(-ip e-itc)|0>, since basically we've just replaced D with e-itD, and c</