Spherical Harmonics

This note introduces some of the key facts about spherical harmonics. We try to avoid the approach based on differential equations. Instead, we seek a more intuitive and hence easy-to-remember/apply approach.

Spherical harmonics and rotation

In Dirac notation the spherical harmonics can be denoted as

where and for each , . Here denotes the spherical angle basis, with the direction unit vector characterized by spherical angles and .

Spherical harmonics are simultaneous eigenstates of and ,

where is the angular momentum operator. form an orthonormal basis:

Because any rotation is generated by angular momentum operators , which commute with , any rotation also commutes with , and it will not change the eigenvalue . Hence, using basis, a rotation operator is diagonal with respect to

where is known as the Wigner functions. It is the non-trivial part of the rotation matrix elements in the -subspace.

For a given , a rotation is represented by a matrix and it mixes eigenstates with different ’s. However, any inner product in the -dimensional -subspace is rotationally invariant. For example, consider the inner product

where we used the fact that is normalized. Now, if the inner product is expressed in the spherical angle basis, a.k.a., inserting the identity , it should have no angular dependence:

where is the solid angle element. Therefore we have

which is known as the Unsöld’s theorem (can be viewed as a special case of the addition theorem, see later). Note that can be in any direction, or the angles can take any values.

To relate the Wigner functions to spherical harmonics, consider obtained from rotating by the rotation . We can use the complex conjugate of Eq. (1) to write

To express the spherical harmonics in terms of the Wigner matrix, we insert a completeness relation into Eq. (4):

Now, in the basis, must have eigenvalue , as

So

Combining with the result in Eq. (3), we have

and hence

where we have chosen the phase of to be zero (the Condon-Shortley convention). It should be mentioned at this point that the Legendre polynomial is just defined as

The pre-factor is added to make

Plugging Eq. (6) into Eq. (5), we finally get the relation between Wigner functions and spherical harmonics:

Addition theorem via rotation

Let us prove the famous addition theorem of spherical harmonics. We start by inserting a rotation identity, , and then the completeness relation of spherical harmonics into the definition of the Legendre polynomial (without the constant factor), Eq. (7):

Now let the directions of the rotated vectors be defined as

Recall that the polar angle is the angle between and . Since rotations preserve angles, the angle between and is still . Using the above definitions and restoring the constant factor, Eq. (9) becomes

QED.