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