Quantum Analogs

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