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Average of r^s for hydrogenic wavefunctions

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The wavefunction psi_{n,l,m}(r, theta, phi) is given by:

psi_{n,l,m} = R_{n,l}(r)Y_{l,m}(theta, phi)


R_{n,l} = -(2/n a)^(3/2) sqrt{(n-l-1)!/[2n (n+l)!^3 ]} exp[-rho/2] rho^(l) L^{2 l +1}_{n+l} (rho)

Here rho = 2r/(n a) and

L^{(2 l +1}_{n+l} (rho) is the associated Laguerre poynomial which is given by

L^(2 l +1)_n+l (rho) = sum over k from zero to n-l-1 of (-1)^(k+1) (n+l)!^2/[(n-l-1-k)! (2 l+1+k)!] rho^k/k!

Y_{l,m}(theta, phi) is the spherical harmonic which can be expressed for positive m as:

Y_{l,m}(theta, phi) = (-1)^m sqrt{(2l+1) (l-m)!/[4 pi (l+m)!]} P^{m}_{l}(cos(theta)) exp(i m phi)

Here the function P^{m}_{l}(cos(theta)) is the associated Legendre polynomial, which is given by:

P^{m}_{l}(w) = (1-w^2)^(m/2) d^m/dw^m P_{l}(w)

and P_{l}(w) is the ordinary Legendre polynomial which can be exprssed as:

P_{l}(w) = 2^(-l) 1/l! d^l/dw^l (w^2 - 1)

Let's work out psi_{3,2,1}(r, theta, phi) using these ...

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A detailed solution is given.