1.22 Solution of the 3D HO Problem in Spherical Coordinates
As and example of another problem with spherical symmetry, we solve the 3D symmetric harmonic
oscillator (See section 17) problem. We have already solved this problem in Cartesian coordinates.
Now we use spherical coordinates and angular momentum eigenfunctions.
The eigen-energies are
E=(
2 nr+ℓ+3
2
)
̄hωwherenris the number of nodes in the radial wave function andℓis the total angular momentum
quantum number. This gives exactly the same set of eigen-energiesas we got in the Cartesian
solution but the eigenstates are now states of definite total angular momentum and z component of
angular momentum.
1.23 Matrix Representation of Operators and States
We may define thecomponents of a state vectorψas the projections of the state on a complete,
orthonormal set of states, like the eigenfunctions of a Hermitian operator.
ψi ≡ 〈ui|ψ〉
|ψ〉 =∑
iψi|ui〉