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So I've been racking my brain around the hydrogen mean values.

[tex]\left\langle \frac{1}{r}\right\rangle=\frac{1}{a_0n^2}[/tex], that I can solve with the recurrence relation in Schaum:

[tex]\left\langle r^k\right\rangle=\int_0^\infty r^{k+2}|R_{nl}(r)|^2dr[/tex]

by simply putting in the radial part of hydrogen wave function. But when I do the exact same thing for

[tex]\left\langle r\right\rangle[/tex]

I get integral:

[tex]\int_0^\infty x^{2l+3}e^{-x}[L_{n-l-1}^{2l+1}(x)]^2dx[/tex]

Which I cannot solve by using the known relations for Laguerre polynomials because the exponent on the x is neither the same or by one greater than Laguerre polynomial (2l+1).

[tex]\int_0^\infty x^{k+1}e^{-x}[L_{n}^{k}(x)]^2dx=\frac{(n+k)!}{n!}(2n+k+1)[/tex]

Am I doing sth wrong? I've checked over and over and cannot find the flaw :\

Mathematica doesn't do these kind of calculations, and explanation in book by Zettili with deriving by l or electric charge and using Kramers recurrence relations aren't helping much :(

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# Mean values for hydrogen atom

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