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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:

image0.png

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:

image1.png

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

image2.png

And dividing this equation by

image3.png

gives you

image4.png

This equation has terms that depend on either

image5.png

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

image6.png

Multiplying

image7.png

gives you

image8.png

And multiplying

image9.png

gives you

image10.png

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

Solving for small r

The Schrödinger equation for

image11.png

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

image12.png

is pretty close to

image13.png

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

image14.png

You can break the solution,

image15.png

into a radial part and an angular part:

image16.png

The angular part of

image17.png

is made up of spherical harmonics,

image18.png

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

image19.png

where

image20.png

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,

image21.png

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

image22.png

Note that

image23.png

diverges as r goes to infinity because of the

image24.png

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

image25.png
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