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Combine the Solutions for Small r and Large r in the Schrödinger Equation

When you apply the quantum mechanical Schrödinger equation for a hydrogen atom, you need to put together the solutions for small r and large r. The Schrödinger equation gives you a solution to the radial Schrödinger equation for a hydrogen atom as follows:

image0.png

where f(r) is some as-yet-undetermined function of r. Your next task is to determine f(r), which you do by substituting this equation into the radial Schrödinger equation, giving you the following:

image1.png

Performing the substitution gives you the following differential equation:

image2.png

Quite a differential equation, eh? But just sit back and relax — you solve it with a power series, which is a common way of solving differential equations. Here’s the power-series form of f(r) to use:

image3.png

Substituting the preceding equation into the one before it gives you

image4.png

Changing the index of the second term from k to k – 1 gives you

image5.png

Because each term in this series has to be zero, you have

image6.png

Dividing by rk–2 gives you

image7.png

This equation gives the recurrence relation of the infinite series,

image8.png

That is, if you have one coefficient, you can get the next one using this equation. What does that buy you? Well, take a look at the ratio of ak/ak–1:

image9.png

Here’s what this ratio approaches as k goes to infinity:

image10.png

This resembles the expansion for ex, which is

image11.png

As for e2x, the ratio of successive terms is

image12.png

And in the limit

image13.png

the ratio of successive expansion coefficients of e2x approaches 2/k:

image14.png

That’s the case for e2x. For f(r), you have

image15.png

Comparing these two equations, it’s apparent that

image16.png

The radial wave function, Rnl(r), looks like this:

image17.png

where

image18.png

Plugging the form you have for f(r),

image19.png

Okay, should you be overjoyed? Well, no. Here’s what the wave function

image20.png

looks like:

image21.png

And substituting in your form of Rnl(r) from this equation gives you

image22.png
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