3.6 Orbital a ngular momentum

1. Exercise: Show this identity. 3.6 Orbital angular momentum • Orbital angular momentum as rotation generator • 1. One can show that the orbital angular momentum obeys 2. We have shown earlier that L does generate expected rotations if p generates translation

2. Exercise: Show this identity.

3. Exercise: Show this identity.

4. z θ y r φ x Angular momentum in spherical polar coordinates

5. Exercise: Show these identity. L2 in spherical polar coordinates

6. Eigenvalues and eigenfunctions Now look for eigenfunctions of L2, in the form Eigenvalue condition becomes

7. The Legendre equation Make the substitution

8. Legendre polynomials and associated Legendre functions In order for solutions to exist that remain finite at μ=±1 (i.e. at θ=0 and θ=π) we require that the eigenvalue satisfies (like SHO, where we found restrictions on energy eigenvalue in order to produce normalizable solutions) The finite solutions are then the associated Legendre functions, which can be written in terms of the Legendre polynomials: Merzbacher, 247-8 where m is an integer constrained to lie between –l and +l.

9. Spherical harmonics The full eigenfunctions can also be written as spherical harmonics: Because they are eigenfunctions of Hermitian operators with different eigenvalues, they are automatically orthogonal when integrated over all angles (i.e. over the surface of the unit sphere). The constants C are conventionally defined so the spherical harmonics obey the following important normalization condition: First few examples (see also 2B21):

10. The orbital angular momentum To summarize: l is known as the principal angular momentum quantum number: determines the magnitude of the angular momentum m is known as the magnetic quantum number: determines the component of angular momentum along a chosen axis (the z-axis) These states do not correspond to well-defined values of Lx and Ly, since these operators do not commute with Lz. Semiclassical picture: each solution corresponds to a cone of angular momentum vectors, all with the same magnitude and the same z-component.

11. Spherical Harmonics

13. Spherical harmonics as rotation matrices

14. z z’ z”z’” y”’ y’y” x y x’ x”’ x” x – x’ – x”—x”’

16. z z’ z”z’” y”’ y’y” x y x’ x”’ x” x – x’ – x”—x”’