Required math: calculus
Required physics: 3-d Schrödinger equation
Reference: Griffiths, David J. (2005), Introduction to Quantum Mechanics, 2nd Edition; Pearson Education – Problem 4.43.
If we combine all the results from the solution of the angular and radial equations for the hydrogen atom, we get a formula for the spatial wave function, which is given in Griffiths’s book as eqn 4.89:
where is an associated Laguerre polynomial and is a spherical harmonic.
The version in Griffiths sets (so it is that version that is used in the above formula for ) while some other sources use .
For the spherical harmonics, the formulas are
with the being the associated Legendre function:
As an example of using this formula, we’ll construct . We get and we worked out in the previous post as
Plugging these into the overall formula gives
To check the normalization, we do the integral:
using Maple for the integral.
The expectation value of is (using Maple)
The gamma function is infinite at all non-positive integral arguments, so this value is finite for and all non-integer values less than . The smallest integer value for which this is finite is . Note in particular that if , the formula reduces to as it should.
The expectation value of itself is
This is quite a bit bigger than the value for the ground state, which is .