How to Keep a Function of r Finite as r Goes to Infinity
Working with Three-Dimensional Harmonic Oscillators
Finding the Schrödinger Equation for the Hydrogen Atom

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:

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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:

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

image2.png

And dividing this equation by

image3.png

gives you

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This equation has terms that depend on either

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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,

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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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How to Work with a Cubic Potential
Solving the Wave Function of R Using the Schrödinger Equation
Determining the Angular Part of a Wave Function
Applying the Radial Equation Outside the Square Well
When to Use Spherical Coordinates Instead of Rectangular Coordinates
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