A particle of mass moves in an isotropic three-dimensional harmonic oscillator potential

PHYSICS · Multi-part · Fermi University Physics · Quantum Mechanics · The formalism of quantum mechanics · The formalism of quantum mechanics

The Structure of Quantum Mechanics, Ch Compatible observables and uncertainty as a theorem

Problem

A particle of mass

m

moves in an isotropic three-dimensional harmonic oscillator potential

V(r)=\tfrac12 m\omega^2 r^2

, with

\hat H=\hat H_x+\hat H_y+\hat H_z

in Cartesian form. ((a)) Name a complete set of commuting observables built from

\hat H_x,\hat H_y,\hat H_z

, and say how many labels a state needs. ((b)) For the shell with total quantum number

N=n_x+n_y+n_z=4

, count the number of independent states sharing that energy.