# Calculate the Wave Function of a Hydrogen Atom Using the Schrödinger Equation

If your quantum physics instructor asks you to find the wave function of a hydrogen atom, you can start with the radial Schrödinger equation, R* _{nl}*(

*r*), which tells you that

The preceding equation comes from solving the radial Schrödinger equation:

The solution is only good to a multiplicative constant, so you add such a constant, A* _{nl}* (which turns out to depend on the principal quantum number

*n*and the angular momentum quantum number

*l*), like this:

You find A* _{nl}* by normalizing R

*(*

_{nl}*r*).

Now try to solve for R* _{nl}*(

*r*) by just flat-out doing the math. For example, try to find R

_{10}(

*r*). In this case,

*n*= 1 and

*l*= 0. Then, because N +

*l*+ 1 =

*n*, you have N =

*n*–

*l*– 1. So N = 0 here. That makes R

*(*

_{nl}*r*) look like this:

And the summation in this equation is equal to

And because *l* = 0, *r** ^{l}* = 1, so

Therefore, you can also write

where *r*_{0} is the Bohr radius. To find A_{10} and *a*_{0}, you normalize

to 1, which means integrating

over all space and setting the result to 1.

and integrating the spherical harmonics, such as Y_{00}, over a complete sphere,

gives you 1. Therefore, you’re left with the radial part to normalize:

Plugging

into

gives you

You can solve this kind of integral with the following relation:

With this relation, the equation

becomes

Therefore,

This is a fairly simple result. Because A_{10} is just there to normalize the result, you can set A_{10} to 1 (this wouldn’t be the case if

involved multiple terms). Therefore,

That’s fine, and it makes R_{10}(*r*), which is

You know that

And so

becomes

Whew. In general, here’s what the wave function

looks like for hydrogen:

where

is a generalized Laguerre polynomial. Here are the first few generalized Laguerre polynomials: