Normalization of Radial wavefunction of hydrogen atom

[Arafat Sagar]

All I need to evaluate the normalization coefficient. I need a step by step guide. It will be a great help if someone please tell me where can i get the solution (with intermediate steps). I think the solution can be done using the orthogonal properties of associated Laguerre polynomial. I need the steps. Thanks.

 

[TSny]

You can find the normalization factor using the generating function for the Laguerre polynomials. For example, see Pauling and Wilson's text:

page 131 http://archive.org/stream/introductiontoqu031712mbp#page/n139/mode/2up

Appendix VII: http://archive.org/stream/introductiontoqu031712mbp#page/n459/mode/2up

 

[Arafat Sagar]

thanks a lot. the book contains the solution. but i still have one more confusion. the radial wavefunction (normalized) in the book has a minus sign. but i found many places where the minus sign has not been included. though the probability density requires the mod square of ψ, i want to know the shape of ψ. hence, what is the real value of it? should i include minus sign?

 

[phyzguy]

Wavefunctions are always uncertain up to a phase. So you can multiply the wavefunction by -1, or in fact by any number of the form exp(i * phi) where phi is any real value, without changing any observables.

 

[Arafat Sagar]

i have an equation,
f(ρ) = \rho^{l+1}e^{-\rho}\upsilon(2\rho)

i want to transform it to the following multiplying only the right hand side with (-1)^{2l+1}(2)^{l+1},
f(ρ) = (-1)^{2l+1}(2\rho)^{l+1}e^{-\rho}\upsilon(2\rho)

is it possible?

i want to use (2\rho)^{l+1} instead of \rho^{l+1}, because, the normalization coefficient normalizes with respect to ρ whatever the the function f(ρ) is. and (2)^{l+1} is not a function of ρ.

i want to multiply by (-1)^{2l+1}, because i found that if the associated Laguerre polynomial is AL^{2l+1}_{n+l}(x)=\frac{d^{2l+1}}{dx^{2l+1}}L^{}_{n+l}(x). now, in some places, i found A=1 and other places A=(-1)^{2l+1}. besides, is it something related to Condon-Shortley Phase factor?
as after multiplyng by anything which is not a function of \rho will still satisfy the associated laguerre differential equation, can i do this multiplication of (-1)^{2l+1}(2)^{l+1}? thanks.