# How to Find the Eigenfunctions of L^{2} in Spherical Coordinates

Your quantum physics instructor may ask you to find the eigenfunctions of L^{2} in spherical coordinates. To do this, you start with the eigenfunction of

given that in spherical coordinates, the L^{2} operator looks like this:

That’s quite an operator. And, given that

you can apply the L^{2} operator to

which gives you the following:

And because

this equation becomes

Wow, what have you gotten into? Cancelling terms and subtracting the right-hand side from the left finally gives you this differential equation:

Combining terms and dividing by

gives you the following:

Holy cow! Isn’t there someone who’s tried to solve this kind of differential equation before? Yes, there is. This equation is a Legendre differential equation, and the solutions are well known. (Whew!) In general, the solutions take this form:

where

is the *Legendre function*.

So what are the Legendre functions? You can start by separating out the *m* dependence, which works this way with the Legendre functions:

where P* _{l}*(

*x*) is called a

*Legendre polynomial*and is given by the Rodrigues formula:

You can use this equation to derive the first few Legendre polynomials like this:

and so on. That’s what the first few P_{l}* _{ }*(

*x*) polynomials look like. So what do the associated Legendre functions, P

_{lm}*(*

_{ }*x*) look like? You can also calculate them. You can start off with P

_{l0}

_{ }(

*x*), where

*m*= 0. Those are easy because P

_{l0}

_{ }(

*x*) = P

_{l}*(*

_{ }*x*), so

Also, you can find that

These equations give you an overview of what the P* _{lm}* functions look like, which means you’re almost done. As you may recall,

is related to the P_{lm}* *functions like this:

And now you know what the P* _{lm}* functions look like, but what do C

*, the constants, look like? As soon as you have those, you’ll have the complete angular momentum eigenfunctions,*

_{lm}You can go about calculating the constants C* _{lm}* the way you always calculate such constants of integration in quantum physics — you normalize the eigenfunctions to 1.

that looks like this:

(Remember that the asterisk symbol [*] means the complex conjugate. A complex conjugate flips the sign connecting the real and imaginary parts of a complex number.)

Substitute the following three quantities in this equation:

You get the following:

so this becomes

You can evaluate the integral to this:

So in other words:

Which means that

which is the angular momentum eigenfunction in spherical coordinates, is

The functions given by this equation are called the *normalized spherical harmonics*. Here are what the first few normalized spherical harmonics look like:

In fact, you can use these relations to convert the spherical harmonics to rectangular coordinates:

Substituting these equations into

gives you the spherical harmonics in rectangular coordinates: