Abstract
Many physical phenomena have mathematically equivalent derivations even
when they describe completely different principles. Spherical harmonics are
typically first encountered in an introductory upper-division quantum mechanics
course while covering the Schrodinger equation in three dimensions, but they
also exist in the Helmholtz equation and other branches of physics which incorporate
the Laplacian operator in spherical coordinates. In this experiment, sound
waves are emitted into a spherical cavity and the resonant frequencies are
recorded. The wave amplitude at each resonant frequency is compared with its
polar angle in the cavity and the shape is found to take on the form a
spherical harmonic function. Eight total resonant frequencies are measured from 2.00000 kHz to 9.00000 kHz with an average error ranging
from 1 Hz, up to 3Hz at higher frequencies. Of the
eight, six are found to be spherical harmonic states. An analytic solution to
the resonant states is given and analogous n, l, and m resonating numbers similar to the
hydrogen atom’s quantum numbers are found. The theoretically predicted resonant
frequencies are compared to the best fit measured states across a frequency
spectrum of 110 Hz to 10000 Hz using the SpectrumSLC quantum
analogs software and other supporting software. Six n=0 states are found near
experimentally measured frequencies and come to agree with their theoretical
value with an average error of 1.14% under the calculated value. Finally,
degenerate resonance states are split apart by breaking the spherical symmetry
inside the wave emitting chamber. Selected state functions corresponding to m=±1, m=±2, and m=±3 orbitals are measured. Their
accuracy to their theoretical expressions exhibits some strong divergence, but the
range still allows qualitative analysis for future experiments.
Link:
Paper
