The three-dimensional isotropic harmonic oscillator has energy eigenvalues

PHYSICS · Subjective · Fermi University Physics · Quantum Mechanics · The formalism of quantum mechanics · Central potentials and canonical three-dimensional systems

The Structure of Quantum Mechanics, Ch The postulates, measurement and collapse

Problem

The three-dimensional isotropic harmonic oscillator has energy eigenvalues En=(n+32)ℏωE_n=(n+\tfrac32)\hbar\omega, with degeneracy gn=(n+1)(n+2)/2g_n=(n+1)(n+2)/2 at level nn (so g0=1g_0=1, g1=3g_1=3). Using the Cartesian basis ∣nx,ny,nz⟩|n_x,n_y,n_z\rangle of energy eigenstates, the oscillator is prepared in the normalised state ∣ψ⟩=12∣0,0,0⟩+12∣1,0,0⟩+12∣0,1,0⟩+12∣0,0,1⟩.|\psi\rangle = \tfrac12|0,0,0\rangle + \tfrac12|1,0,0\rangle + \tfrac12|0,1,0\rangle + \tfrac12|0,0,1\rangle. (a) Identify which of the four basis kets belong to the n=0n=0 eigenspace and which belong to the n=1n=1 eigenspace. (b) Write the two projectors P^E0\hat P_{E_0} and P^E1\hat P_{E_1} in terms of these kets. (c) Compute P(E0)P(E_0) and P(E1)P(E_1), and check that they sum to 11.

Related theory

Browse the problem library