Solving the Wave Function of Small r and Large r Using the Schrödinger Equation

Your quantum physics instructor may ask you to solve for the wave function for a made-up particle of mass m in a hydrogen atom. To do this, you can begin by using a modified Schrödinger equation that solves for large and small r:

Because the Schrödinger equation contains terms involving either R or r but not both, the form of this equation indicates that it’s a separable differential equation. And that means you can look for a solution of the following form:

Substituting the preceding equation into the one before it gives you the following:

And dividing this equation by

gives you

This equation has terms that depend on either

but not both. That means you can separate this equation into two equations, like this (where the total energy, E, equals E_R + E_r):

Multiplying

gives you

And multiplying

gives you

Now you can solve for r, both small and large.

Solving for small r

The Schrödinger equation for

is the wave function for a made-up particle of mass m (in practice,

is pretty close to

so the energy, E_r, is pretty close to the electron’s energy). Here’s the Schrödinger equation for

You can break the solution,

into a radial part and an angular part:

The angular part of

is made up of spherical harmonics,

so that part’s okay. Now you have to solve for the radial part, R_nl(r). Here’s what the Schrödinger equation becomes for the radial part:

where

To solve this equation, you take a look at two cases — where r is very small and where r is very large. Putting them together gives you the rough form of the solution.

Solving for large r

For very large r,

Because the electron is in a bound state in the hydrogen atom, E < 0; thus, the solution to the preceding equation is proportional to

Note that

diverges as r goes to infinity because of the

term, so B must be equal to zero. That means that