Physics Friday 54

Quantum Mechanics and Momentum
Part 11: Angular Momentum Eigenfunctions

In our previous part, we introduced the ladder operators, and showed that they generate a “ladder” of simulataneous eigenstates to L² and L_z. Now, we examine the angular momentum operators in spherical coordinates (r,θ,φ).
Using our conversions and chain rule, we find the linear momentum operator components to be:



From these, we find
,
,
.
Note that all of these are fully independent of the radial coordinate.
Summing the squares of these, we can find that
.

This then gives us our eigenvalue relation as being
.
Compare this to the angular portion of Laplace’s equation in spherical coordinates. We see then that the eigenfunctions of L² are (any radial function times) spherical harmonics:
,
and to make our wavefunctions 2π periodic in φ, we require that j be an integer. This means that the eigenvalues of L_z should be integer multiples of ℏ, ranging from –jℏ to jℏ.

Applying to , we find
,
and so
.
So the spherical harmonic is the simultaneous eigenfunction for L² and L_z, with eigenvalues and , where j and m are integers with j≥0, –j≤m≤j.

Lastly, we note that it can be shown, via more that one method, that if the potential depends purely on the radial coordinate, V(r), so that the problem is spherically symmetric, then the quantum hamiltonian commutes with the angular momentum operators L² and L_z, so that one may simultaneously specify the energy, the square magnitude of the angular momentum, and the projection onto one axis of the angular momentum. This is very useful in determining the electron wavefunctions for the hydrogen atom.