Shankar, R. (1994), *Principles of Quantum Mechanics*, Plenum Press. Chapter 13, Exercise 13.1.3.

[If some equations are too small to read easily, use your browser’s magnifying option (Ctrl + on Chrome, probably something similar on other browsers).]

The wave function for the hydrogen atom can be obtained by a series solution of the differential equation, leading to the result (which I’ve rewritten in Shankar’s notation, although my original post used Griffiths’s notation):

Here, we have

The energy levels of the hydrogen atom are

where . The coefficients in 3 are given by a recursion relation

Combining and , the formula becomes, for a given

The coefficient which starts everything off is determined by normalization.

As an example, we can find the wave function . In this case and so the first term in the recursion, with gives and . The full wave function is then

To evaluate we use the energy for :

This gives

where is the Bohr radius

Plugging everything into 8, using , we have

Normalizing gives the condition

Working out the integral (using software or tables) gives

So the final wave function is

which agrees with Shankar’s equation 13.1.27.

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