# 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