Let us generalize the discussion of the simple harmonic oscillator to three dimensions. In this case, the Hamiltonian is:
H=2m1(p12+p22+p32)+21m(Ωx)2,(2.48)
where Ω is a 3×3 symmetric real matrix.
(a) Determine the energy spectrum and eigenvectors of this system.
(b) Suppose the potential is spherically symmetric. Using the equivalent one-dimensional potential approach, find the eigenvalues and eigenvectors of H corresponding to the possible values of orbital angular momentum.