**Required math: calculus**

**Required physics: 3-d Schrödinger equation**

References: Griffiths, David J. (2005), Introduction to Quantum Mechanics, 2nd Edition; Pearson Education – Problem 4.11.

A few more examples of working out the hydrogen atom wave functions. Using the formulas in the last example, we can get . The recursion formula for , is

The series has 2 terms, and we get , so

To find we normalize the radial function:

So and is

The complete wave function is then

For , we have

This time, there is only a single term in the series, so we have

Doing the normalization integral for gives which gives the final result

There are 3 wave functions corresponding to , for which we need the spherical harmonics

The three wave functions are thus

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