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.

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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/introducti.../n139/mode/2up

Appendix VII: http://archive.org/stream/introducti.../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?

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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) = [itex]\rho^{l+1}[/itex][itex]e^{-\rho}[/itex][itex]\upsilon(2\rho)[/itex]

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

is it possible?

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

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