# Hydrogen atom - radial wave function normalization

R(r)=Nr^l e^(-Zr/na) âˆ‘_(j=0)^(n-l-1)â–’ã€–b_j r_j ã€—

Finding the normalization constant:

Rodriguez formula for associated Laguerre formula is:

(e^x x^(-k))/n! d^n/(dx^n ) (ã€–e ã€—^(-x) x^(n+k) )=(e^x x^(-k-n))/n! x^n d^n/(dx^n ) (ã€–e ã€—^(-x) x^(n+k) )

R(r)=âˆ«_0^âˆžâ–’(Nr^l e^(-Zr/na) (e^x x^(-k-n))/n! x^n d^n/(dx^n ) (ã€–e ã€—^(-x) x^(n+k) ))^2 dr=1 (a)

Can you find N for me by using the formula for R(l) in (a)

Answer is in the purple box:

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#### Solution Preview

Hello,

I am recommending you next book where you can find maybe more details about hydrogen atom and other stuff.

https://physicsdemocracy.files.wordpress.com/2011/05/principles-of-quantum-mechanics-as-applied-to-chemistry-and-chemical-physics-1999.pdf

R(r)=Nr^l e^(-Zr/na) âˆ‘_(j=0)^(n-l-1)â–’ã€–b_j r_j ã€—

Finding the normalization constant:

Rodriguez formula for associated Laguerre formula is:

(e^x x^(-k))/n! d^n/(dx^n ) (ã€–e ã€—^(-x) x^(n+k) )=(e^x x^(-k-n))/n! x^n d^n/(dx^n ) (ã€–e ã€—^(-x) x^(n+k) )

R(r)=âˆ«_0^âˆžâ–’(Nr^l e^(-Zr/na) (e^x x^(-k-n))/n! x^n d^n/(dx^n ) (ã€–e ã€—^(-x) x^(n+k) ))^2 dr=1 (a)

Can you find N for me by using the formula for R(l) in (a)

Answer is in the purple box:

Solution.

This is not an easy task. I will start with defining Laguerre and associated Laguerre polynomials. In the literature it can be find that the Laguerre polynomials are defined in two ways and with respect to that the normalization factor can be slightly different.

I way

Laguerre polynomials are defined by relation

(1)

Another definition involves the use of a generating function, for Laguerre polynomials,

(2)

This identity in the ...

#### Solution Summary

The task was to derive the normalization factor for the hydrogen atom radial wave function. In the first part we defined Laguerre and associated Laguerre polynomials. Second part was to solve one particular type of integral which includes associated Laguerre polynomials and which we need to find the normalization factor. Third part was to find the normalization factor for the hydrogen atom radial wave function when we had all tools needed.