11.6: Spherical Harmonics

The key issue about three-dimensional motion in a spherical potential is angular momentum. This is true classically as well as in quantum theories. The angular momentum in classical mechanics is defined as the vector (outer) product of \(r\) and \(p\),

\[L=r \times p .\]

This has an easy quantum analog that can be written as

\[\hat{L}=\hat{r} \times \hat{p}\]

After exapnsion we find

\[\hat{L}=-i \hbar\left(y \frac{\partial}{\partial z}-z \frac{\partial}{\partial y}, z \frac{\partial}{\partial x}-x \frac{\partial}{\partial z}, x \frac{\partial}{\partial y}-y \frac{\partial}{\partial x}\right)\]

This operator has some very interesting properties:

\[[\hat{L}, \hat{r}]=0 .\]

Thus

\[[\hat{\boldsymbol{L}}, \hat{H}]=0!\]

And even more surprising,

And even more surprising,

\[\left[\hat{L}_x, \hat{L}_y\right]=i \hbar \hat{L}_z .\]

Thus the different components of \(L\) are not compatible (i.e., can't be determined at the same time). Since \(L\) commutes with \(H\) we can diagonalise one of the components of \(L\) at the same time as \(H\). Actually, we diagonalsie \(\hat{L}^2, \hat{L}_z\) and \(H\) at the same time!

The solutions to the equation

\[\hat{L}^2 Y_{L M}(\theta, \phi)=\hbar^2 L(L+1) Y_{L M}(\theta, \phi)\]

are called the spherical harmonics.

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