[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 analogya 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]