Re: [Phys-L] what's quantized and what's not

[Moses Fayngold <moshfarlan@yahoo.com>]

On Sunday, April 24, 2016 2:00 PM, John Denker <jsd@av8n.com> wrote (responding to my comments):



> I disagree with most of that.  I disagree with the overall drift
> and with many of the details.

Disagreement "...with the overall drift" warrants going through it point by point.
> Quantum mechanics is right, so far as we know.  However, a great
> deal of what is /said/ about quantum mechanics is wrong.  
Let us see it. 

> > Actually, quantization is known already in CM, e.g., frequency
> > quantization of a finite string or elastic rod.
(Citing from my previous statement)
> That seems kinda inconsistent with the previous warning about
> arguments based on classical mechanics.
There are no inconsistencies here. The frequency f of a vibrating string in the sufficiently high frequency range may be practically continuous observable in the sense that the ratio (delta f) / f = (f(n+1)-f(n)) / f(n) <<1 at  n >>1. A measuring device designed for this region of spectrum may be totally insensitive to see delta f , so f here is continuous for all practical purposes. And yet it does not conflict with the fact that all actual set of harmonics is quantized. 
> In fact if you look at the oscillations of a string, rod, organ pipe,
> et cetera, you find that the resonances are not particularly sharp.
> There is a nontrivial linewidth.  
This fact implicitly assumes dissipation, which was not assumed in the initial discussion. Dissipation is (almost!) always present, but it can be negligible in many situations. The line width is there in all Spectroscopy, and yet we distinguish between discrete and continuous spectra. The same is true about atomic energy spectra, which can be explained only by QM. But what is really important here, is the fact that even with the broad lines, the corresponding radiated field is quantized, with the only distinction that each respective photon is a non-monochromatic wave packet. So while we can, strictly speaking, dispute energy quantization in the narrow sense (as a discrete set) by saying that observed values are merely spectral maxima, it does not eliminate the reality of quanta. As JD himself admits below, the corresponding Hilbert space is is not restricted to the basis of pure energy eigenstates. And as of angular momentum, even mathematical purists cannot deny today its strict quantization as a discrete set. 
> In accordance with Floquet's theorem, 
> if you drive the thing at frequency f, it will respond at frequency f,
> whether or not that's one of the resonant frequencies.

Floquet's theorem has nothing to do with driving oscillator. The latter involves an external source and is described by an inhomogeneous equation; it is quite natural that its solution (for linear systems) has the frequency of the source. As to Floquet's theorem, it is about the Bloch waves in periodic structures - totally beyond the current discussion.
> The same is true of atoms (whether or not you consider this analogous
> to classical resonances).  I've never seen an atom emit a photon
> "instantly".  I've never seen an atomic transition with zero linewidth.
> There is not "overwhelming" evidence of this;  indeed there is no
> evidence at all.

This is an example of the widely spread confusion between instant emission, on the one hand, and finite lifetime of an excited system, on the other. The former is a "quantum jump"  (the outcome of projective measurement); the latter is statistical characteristic of the Gamow states, associated with impossibility to predict the exact moment of the jump. Yes, you can prepare an atom in the excited state and wait million years until the nearby detector clicks informing you about the long-waited photon emission. Will you call this an "overwhelming evidence" that atomic transition takes one million years? I doubt, since the next identically prepared atom may do the same immediately after preparation. And the latter outcome is overwhelmingly more probable than the former one. The broader the line width, the more probable! This is a compelling evidence that considered quantum transition is an instant process, regardless of its linewidth. The only "trouble" is that QM does not predict when exactly it will occur. Hence the concept of the average lifetime which is inversely proportional to the respective line width. I've never seen an emission /absorption process as something extended in time. I've never seen an electron in PEE gradually climbing up the energy scale, sucking in the energy from the incident photon, making it smaller and smaller, then triumphantly emerging from "macroscopic atom" (metallic slab) and finally swallowing all of the photon to gain kinetic energy necessary to reach the detector. But I've always seen (more honestly, read) about instant disappearance of the photon and simultaneous emergence of the photoelectron from the slab. And the same can be said about lots of other quantum transitions. Instant photon emission/absorption is everyday QM. 
> An experiment optimized to look for "instant" or near-"instant" transitions
> would be incompatible with assigning a definite energy to the photons.
> Here "incompatible" has a precise technical meaning, as explained by
> Heisenberg.
False.  Time-energy relationship has not the same meaning as momentum-coordinate relationship, because time (even in SR!) is not the same as space. In (Delta E)(delta t) in the Heisenberg relationship delta t ) has the meaning of the average lifetime of the system for the given transition, not the transition time.

> One of the most fundamental principles of quantum mechanics says that
> it doesn't suffice to /talk/ about the energy.  If you want to know
> the energy, you have to measure it. 
True. But, as Landau used to say, "I can measure the energy and look at the watch." So one can know the energy at any moment. And for a stationary system, the energy can be known exactly at any moment.   
> The same goes for other observables
> such as the spin components Sx, Sy, and Sz.
> *IF* you measure Sx, the result is quantized.
> *IF* you measure Sy, the result is quantized.
> *IF* you measure Sz, the result is quantized.
True. So, the angular momentum is quantized!

> However, you can't measure all three of them at the same time.  You can't
> measure any one of them without greatly disturbing the others.  If you
> measure Sx, there are some things you can say about Sy and Sz, but quite
> a few things that you cannot say.  
True.
> You ought no even imagine that they
> are quantized.  
False. If I have an electron with its spin up, I know that it is an equally-weighted superposition of eigenstates |Sx+> and |Sx->, or |Sy+> and |Sy->. The exact value of the respective net spin component in such superposition is indeterminate, but In no way does it undermine quantization of the corresponding eigenvalues. 
> We see this All The Time in atomic physics.  An atomic electron has
> a perfectly well defined position.  
False. An atomic electron with "a perfectly well defined position" is no longer atomic, since a perfectly defined position involves infinite energy due to QM indeterminacy which JD himself so vividly describes.  
> You can measure the position if
> you want.  However such a measurement is incompatible with measuring
> the energy, incompatible with the spectroscopic N,l,m quantum numbers.
> If you measure position you forfeit the option of measuring the energy
> and vice versa.
True, but, using JD's own expression,"That seems kinda inconsistent with the previous warning..."

Moses Fayngold,NJIT
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