How to think about Quantum Mechanics—Part 8: The quantum-classical limit as music

[Other parts in this series: 1,2,3,4,5,6,7,8.]

On microscopic scales, sound is air pressure f(t) fluctuating in time t. Taking the Fourier transform of f(t) gives the frequency distribution \hat{f}(\omega), but in an eternal way, applying to the entire time interval for t\in [-\infty,\infty].

Yet on macroscopic scales, sound is described as having a frequency distribution as a function of time, i.e., a note has both a pitch and a duration. There are many formalisms for describing this (e.g., wavelets), but a well-known limitation is that the frequency \omega of a note is only well-defined up to an uncertainty that is inversely proportional to its duration \Delta t.

At the mathematical level, a given wavefunction \psi(x) is almost exactly analogous: macroscopically a particle seems to have a well-defined position and momentum, but microscopically there is only the wavefunction \psi. The mapping of the analogyI am of course not the first to emphasize this analogy. For instance, while writing this post I found “Uncertainty principles in Fourier analysis” by de Bruijn (via Folland’s book), who calls the Wigner function of an audio signal f(t) the “musical score” of f.a   is \{t,\omega,f\} \to \{x,p,\psi\}. Wavefunctions can of course be complex, but we can restrict ourself to a real-valued wavefunction without any trouble; we are not worrying about the dynamics of wavefunctions, so you can pretend the Hamiltonian vanishes if you like.

In order to get the acoustic analog of Planck’s constant \hbar, it helps to imagine going back to a time when the pitch of a note was measured with a unit that did not have a known connection to absolute frequency, i.e., to inverse-time units. To my (very limited) understanding, by the 6th century BC it was already understood that an octave was the difference between two notes when one is vibrating twice as fast as another, but the absolute frequency (oscillations per second) of any particular note — say C♯ — was not known.If I’m wrong, it’s not hard to imagine a historical time when this was true, or to imagine a person who was in this particular state of ignorance today.b   Let’s arbitrarily declare \Delta v to be the interval between pitches C♯ and C in the one-lined octave. Today we know that C♯ and C correspond to 277.18 and 261.63 Hz, respectively, so that 1/\Delta v corresponds to 64.31 ms \equiv 1/\Delta C, but in the distant past for humanity (or the very recent past for me), this was unknown.

If I listened very carefully, or at least if I built special equipment, I would find that the purity of a note’s pitch begins to degrade as the duration of the note approaches its inverse frequency; this would be a hint about the location of the acoustic microscopic scale a \equiv \Delta v /\Delta C. That is, I would find it harder and harder to distinguish between the notes C♯ and C as the duration of the notes approached of order 60 ms (although it might happen with even longer durations due to imperfections of my ears/equipment). To confidently establish the relationship between perceived pitch and inverse time, I would probably want to listen to sound made by objects whose frequency of vibration I could measure directly. That would be easy today, but very difficult three thousand years ago.

The analog of \hbar, then, would be the ratio a = \Delta v /\Delta C. Whenever someone says “middle C is 261.63 Hz”, they are effectively setting a=1 (i.e., measuring pitch in units of inverse-time), just as physicists commonly set \hbar=1 (measuring momentum in units of inverse-distance). But crucially, for understanding the history of science, this is not possible until you have equipment that is sensitive to the microscopic scale. Before this, you needed two separate systems of units that could not (then) be connected in a principled manner.

The quantum-acoustic analogy is not just conceptual, it is a mathematically precise correspondence, to the point that there are book-length treatments that apply almost equally well to both.See for instance “Symplectic Methods in Harmonic Analysis and in Mathematical Physics” by de Gosson and especially “Harmonic analysis in phase space” by Folland.c   In particular, the Wigner function (previous posts: 1,2) for simultaneously representing the position and momentum of a particle can be used fruitfully in acoustics for simultaneously representing the duration and pitch of sounds (which is closely related to the short-time Fourier transform). And, importantly, it is true for both quantum mechanics and acoustics that the macroscopic limit (\hbar,a\to 0) is “singular”: Just as only a few \hbar-indexed families of quantum states have a sensible classical limit, only a few a-indexed families of acoustic waveforms have a sensible decomposition into notes (“music”). When this limit fails, it’s not a case of bad music, it’s a case of your speakers blowing out.

What makes quantum mechanics “mysterious” then is clearly not things like the form of the uncertainty principle per se. (There is an acoustic uncertainty principle of identical mathematical form.) Rather, it is the interpretation of the wavefunction in terms of probability amplitudes and our inability to probe it except indirectly through disturbing measurements, as opposed to acoustic waves which can be measured to arbitrary precision.Until you reach the atomic scale where the hydrodynamic approximation breaks down.d   Relatedly, there is, to my understanding, no acoustic analog of a mixed quantum state.EDIT: See one good idea for this in the comments below.e  

Basic harmonic analysis of acoustics is an interesting topic in elementary physics in its own right. Maybe teaching more of it (especially in a phase-space formulation using a Wigner function) before presenting quantum mechanics would help students more easily see what’s truly unusual about quantum mechanics and what’s just an unfamiliar mathematical framework.

Footnotes

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  1. I am of course not the first to emphasize this analogy. For instance, while writing this post I found “Uncertainty principles in Fourier analysis” by de Bruijn (via Folland’s book), who calls the Wigner function of an audio signal f(t) the “musical score” of f.
  2. If I’m wrong, it’s not hard to imagine a historical time when this was true, or to imagine a person who was in this particular state of ignorance today.
  3. See for instance “Symplectic Methods in Harmonic Analysis and in Mathematical Physics” by de Gosson and especially “Harmonic analysis in phase space” by Folland.
  4. Until you reach the atomic scale where the hydrodynamic approximation breaks down.
  5. EDIT: See one good idea for this in the comments below.
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6 Comments

  1. Jess, you might have a look at my “An algebraic approach to Koopman classical mechanics”, Annals of Physics 2020, arXiv:1901.00526 (DOI there). One thing to look for there is the list of four YouTube videos I link to that tell just this story, but I also note that the wave/audio analogy is problematic insofar as it is not probabilistic. The Wigner function is also mostly used in signal analysis as an essentially deterministic tool, for which I quite warmly commend the paper by Leon Cohen in Proc.IEEE 1989 that I cite there.
    Classical Mechanics is usually said to have a commutative algebra of observables, functions on phase space, but we can use Currying of the Poisson bracket to naturally construct a noncommutative algebra of observables, so that CM has an algebraic structure isomorphic to that of QM, instead of there being only the relatively imprecise relationship of “quantization”. Other consequences abound.

    • > I also note that the wave/audio analogy is problematic insofar as it is not probabilistic.

      I agree, and I mention this in the penultimate paragraph of the post.

  2. I discover that I already mentioned that paper to you in a comment responding to a post of yours in February 2020, when it had only just come out. Sorry.

  3. Could we maybe say that a musical score represents a kind of mixed state for an acoustic signal? Suppose you represent two frequencies played at the same time. The score doesn’t include the cross term that the coherent Wigner function would include. But you could maybe think of this as not specifying the phase of the signals (coming from the combination of say, two piano keys)?
    Isn’t this the same sense in which we talk about a mixed state in quantum mechanics? There is some uncertainty in exactly what state is there, but it’s not coming from quantum mechanics, but from some other reason. In any given situation, we assume that the actual state is a pure state and not a mixed state (unless you have ignored the environment).

    I’ve studied Wigner functions in the context of optics, and one uses statistical optics, where uncertainty comes from lack of knowledge of the precise details of the system that is creating the optical signal — my context is an electron bunch emitting synchrotron radiation.

    • So the idea here would be that any actual performance of the music would be a “pure” state (since the actual acoustic signal would be some air pressure as a function of time), but that the musical score, which does not specify the phase offset between two different notes, is an indifferent “mixture” of the various possible choices of phase. Yes, that does make sense to me.

      The only issue is that, although I think there’s a reasonable sense in which a musical score is a mixed state, it wouldn’t be the case that all mixed states could be interpreted as a musical scores, and in particular I wouldn’t know how to think musically about, say, a partially decohered cat stat.

    • Boaz, that seems to me a wonderfully stated comment. In any case, it resonates with me. I would take the musical score as a multi-channel specification for an orchestral performance: a design for an experiment, so to speak, not the experiment itself. For a given ensemble of performances by the same orchestra and conductor in the same konzerthalle, the variation from performance to performance might be closer to a pure state or closer to a mixed state, partly depending on the nature of the interactions between the musicians, their instruments, and everything else. That is, although the score doesn’t specify phase differences, the musicians may fall into something close to the same phase differences even over multiple performances, just because, inter alia, a particular choice feels “right”. More in a physics parlance, we have Huygens’ discovery that a pendulum will synchronize with others over even quite a small number of cycles if interactions are strong enough.

      There are different ways in which the same mixture might arise, so that one could consider different subensembles of the same ensemble of performances and find that some of those performances are closer to a pure state than others. If we can identify details of the performances that correlate with a particularly close to pure choice of subensembles, such as whether the conductor and the first violin were both drunk the night before, et many ceteræ, then we can further specify, beyond just the score, that we require the conductor and the first violin to be drunk the night before any performance. The difference resonances of a different konzerthalle and of a different audience can change the feel of the music altogether.

      A lot depends on how we measure a performance. We could use just one microphone, or hundreds, and we could for different performances install sound barriers and reflectors that use many different materials and geometries. We could measure and record the signals out of hundreds of microphones up to the MHz range and discover unexpected signal correlations in different parts of the concert hall that would be erased if we focused only on the human auditory range. We could instead go for the low frequency of just a few binary interviews with a few audience members after each movement of each performance, as we do in quantum computation: “was it good or bad?” One could say that a musical score’s purpose is to produce a particular sequence of atmospheric compressions, but one could also say that its purpose is to produce a particular shared emotional response in the audience, which only a very sophisticated analysis indeed of the output of microphones can measure. At the end of a sequence of performances, such analyses of the whole ensemble and of subensembles are essentially global, akin to time-frequency analysis, but with the higher order mathematical complexity of probability/statistics added. It’s hardly surprising we are still unraveling it after almost a century.

      Finally, as far as quantum field theory is concerned, Planck’s constant determines the amplitude of a Poincaré invariant noise. A quantum field theory in the absence of thermal noise (and its random field equivalent if there is one) applies multiplicative modulations to that Poincaré invariant noise, then we can use both mixtures and superpositions of those modulations. Then a lot more can be said, unsurprisingly, but I will spare you. Thank you for your comment!

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