The three-dimensional isotropic harmonic oscillator is both three independent one-dimensional oscillators and a central-potential problem. These two descriptions must give the same energies and degeneracies.
Because \(r^2=x^2+y^2+z^2\), the potential is central.
Define the shell number \(N=n_x+n_y+n_z\).
Write \(\Psi=R_{Nl}(r)Y_l^m(\theta,\phi)\). The radial equation contains
The oscillator length b controls the size of the low-energy wave functions.
Because the Hamiltonian separates as \(H_x+H_y+H_z\), Cartesian products are immediate eigenstates. But rotational symmetry also guarantees simultaneous eigenstates of \(H,L^2,L_z\). Within each degenerate energy shell, unitary combinations of Cartesian states create the spherical basis. Neither basis is more physical in general; the useful choice is set by the measurement or perturbation.
Each independent Cartesian oscillator contributes \(\hbar\omega/2\), so the ground energy is \(3\hbar\omega/2\). The ground wave function is a spherically symmetric Gaussian, corresponding to N = 0, l = 0, m = 0. This provides a direct check that the radial derivation and the sum of three 1D oscillators agree at the bottom of the spectrum.
Use the following chain when working with Three-dimensional harmonic oscillator. First identify the symmetry, prepared state and observable being discussed; this determines which basis and quantum numbers are meaningful. Next write the operator or differential equation before substituting eigenvalues. Apply boundary conditions, commutators or normalization one step at a time, keeping dimensions and allowed quantum-number ranges visible. Only then simplify to the highlighted result. Finally translate the mathematics into a measurement statement: specify which outcomes are possible, which quantities remain uncertain, how degeneracy is counted, or where probability is concentrated. This sequence prevents a correct formula from becoming disconnected from the physical assumptions that justify it.
Self-check: Cartesian and spherical descriptions, central confinement and the effective radial potential. The final answer should explicitly connect this objective to at least one equation on the page and one experimentally meaningful prediction.
Cartesian states \(|n_xn_yn_z\rangle\) and spherical states \(|Nlm\rangle\) are different bases of the same degenerate energy subspaces.
This page is designed to support short guided exercises on: Cartesian and spherical descriptions, central confinement and the effective radial potential.
Use these anchors to design compact exercises. The exercise should be answerable from this page plus standard algebra, while longer derivations, full worked examples and broader context should point back to the original book.