**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 *L*^{2} 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 *L*^{2} 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 –*j*ℏ to *j*ℏ.

Applying to , we find

,

and so

.

So the spherical harmonic is the simultaneous eigenfunction for *L*^{2} 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 *L*^{2} 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.

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Tags: Angular Momentum, Eigenfunction, Friday Physics, physics, Quantum Mechanics, Spherical Coordinates, Spherical Harmonics

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February 20, 2009 at 2:24 am |

[…] in spherical coordinates, with the angular components being the spherical harmonics (see here and here). As in here, when we perform the spherical coordinate separation , we obtain radial equation […]