A thin elastic spherical shell (a spherical drum) is set vibrating so that each point…

PHYSICS · Subjective · Fermi University Physics · Quantum Mechanics · Symmetry, angular momentum and spin · Symmetry, angular momentum and spin

The Structure of Quantum Mechanics, Ch Angular momentum from the rotation group

Problem

A thin elastic spherical shell (a spherical drum) is set vibrating so that each point of its undeformed spherical surface moves radially by a small displacement

u(\theta,\phi)

, a single-valued function of the direction

(\theta,\phi)

on the sphere. A particular normal mode has azimuthal dependence

u(\theta,\phi)=f(\theta)e^{im\phi}

for some real number

m

and some function

f(\theta)

. (a) Using the requirement that rotating the shell by

2\pi

about the polar axis must reproduce the same physical displacement pattern, decide which values of

m

are allowed, and justify the answer from the single-valuedness argument used for the orbital case (do not appeal to any result specific to quantum wavefunctions). (b) State explicitly which feature of the argument makes it apply equally to this classical elastic system and to an orbital angular-momentum eigenstate

\psi\propto e^{im\phi}

of a particle, even though the drum involves no

\hbar

and no operators.

Related theory

Quantum Mechanics

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