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Quantum Mechanics - quick integral question.

ψnlm, ψnlm', ψnlm'' are energy eigenfunctions of hydrogen, and a function of r, θ and Ø.

(they all have the same n and l, but different m's)

I am given ψ(r,t=0)=ψnlm+ψnlm'+ψnlm'' .

I am after the probability density as a function of θ, and so need to integrate over r and Ø.

I have been introduced to ψnlm=Rnl(r)Θlm(θ) ϕm(Ø)

I am checking my understanding of integration tecniques

First of all, energy eigenfunctions are orthogonal when integrated over all space in r/θ/Ø.

So I integrate over Ø and attain |RnlΘlm|^2+|RnlΘlm'|^2+|RnlΘlm''|^2

Which is fine, so now I need to integrate over r.

∫ ( |RnlΘlm|^2+|RnlΘlm'|^2+|RnlΘlm''|^2 ) r^2 dr. *

But I know that Rnl(r),Θlm(θ), ϕm (Ø) are separately normalised.

So ∫ (Rnl)^2 r^2 dr =1 ( Here Rnl is real ! )

So from * I atttain :

|Θlm|^2+|Θlm'|^2+|Θlm''|^2

which gives me the wrong answer.

If anyone could help shed some light on this, that would be greatly appreciated !

The correct solution uses the orthogonal tecnique, and then continues explicitly , as a pose to using the normalization technique, so I suspect this may be were I am going wrong...

(edited 10 years ago)

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