Finding the Schrödinger Equation for the Hydrogen Atom
Finding the Total Energy Equation for Three-Dimensional Free Particle Problems
Solving the Wave Function of Small r and Large r Using the Schrödinger Equation

How an Increase in r Affects the Appearance of Hydrogen Wave Functions

In a hydrogen atom, the wave functions change as you change the orbital radius, r. So what do the hydrogen wave functions look like? Given that


looks like this:


Here are some other hydrogen wave functions:


Note that


behaves like rl for small r and therefore goes to zero. And for large r,


decays exponentially to zero. So you’ve solved the problem of the wave function diverging as r becomes large — and all because of the quantization condition, which cut the expression for f(r) from an exponent to a polynomial of limited order. Not bad.

The radial wave function R<sub>10</sub>(<i>r</i>).
The radial wave function R10(r).

You can see the radial wave function R10(r) in the first figure. R20(r) appears in the second figure. And you can see R21(r) in the last figure.

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