Approximating H Wavefunction Circular State for Large n

[teroenza]

1. Homework Statement
We are studying circular states of the hydrogen atom (states where the l quantum number is = n-1). We are asked to evaluate [itex] \langle \Psi_{n,n-1,n-1}| r_{n,l=n-1}|\Psi_{n,n-1,n-1}\rangle [/itex]. The wave function is that of the hydrogen atom, and the thing we are taking the expectation value of is the radial portion of the hydrogen wave function.

We are also told that n>>1. 2. Homework Equations 3. The Attempt at a Solution
I'm confused about how to construct the wavefunction with n being left general (not a numeric value). Forming the spherical harmonics requires using the associated Legendre polynomial [itex] P_{n-1,n-1}(\theta) [/itex], but the derivative [itex] \frac{d^{2n}}{d\theta^{2n}}(\theta^2-1)^{n-1} [/itex] necessary to do that is where I'm stuck.

I feel that using the fact that n>>1 will help simplify this, but I'm not sure how.

 

[Dr. Courtney]

You could do a few test cases, n = 20, 30, 40, 50, ..., 100.

Then look for the trend or fit the values you determine for the expectation value for r as a function of n.

 

[teroenza]

I see. In this case I can just use the orthonormality of the spherical harmonics to integrate away the angular dependence.