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

On microscopic scales, sound is air pressure fluctuating in time . Taking the Fourier transform of gives the frequency distribution , but in an eternal way, applying to the entire time interval for .

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 of a note is only well-defined up to an uncertainty that is inversely proportional to its duration .

At the mathematical level, a given wavefunction is almost exactly analogous: macroscopically a particle seems to have a well-defined position and momentum, but microscopically there is only the wavefunction . The mapping of the analogy^{a } is . 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 , 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.,… [continue reading]