**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 spherical harmonics can be calculated from a standard formula, as can the associated Laguerre polynomials. The forms of these functions vary according to the normalization. My version is

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 .

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