# Is this the correct way to solve this QM integral problem?

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Problem is:

If the behavior of ψ( r,t ) as r->inf is dominated by r-n, what values can n assume if the integral
A(ψ*∇ψ-ψ∇ψ*)⋅nda
taken over the surface at infinity is to vanish.

I considered ψ as ar-n calculate like below
ψ*∇ψ≈ar-n⋅a*(-nr-n-1)=-naa*r-2n-1
ψ∇ψ*≈a*r-n⋅a(-nr-n-1)=-naa*r-2n-1
So... ψ*∇ψ-ψ∇ψ*=0 at anywhere. Thus, n does not affect to integration.

Well, this result is so ridiculous :/

TSny
Homework Helper
Gold Member
Welcome to PF!

Suppose ##\psi(x) = \Large{\frac{e^{ikr}}{r^n}}##. Would this be considered a function that is dominated by r-n? (I think so, but I don't know the precise definition of "dominated by".)

• BvU
Welcome to PF!

Suppose ##\psi(x) = \Large{\frac{e^{ikr}}{r^n}}##. Would this be considered a function that is dominated by r-n? (I think so, but I don't know the precise definition of "dominated by".)
I just tried and got the following result
##\psi^{*}\nabla\psi - \psi\nabla\psi^{*} = \Large{\frac{2ki}{r^{2n}}}##
So, now ##n## must be larger than 0. Quite acceptable result :)

TSny
Homework Helper
Gold Member
Are there any factors of r in the area element da?

Are there any factors of r in the area element da?
Oh. r^2 dependency... So n>1

OMG I've submitted my homework lol

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