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

[Azruine]

< Mentor Note -- thread moved to HH from the technical physics forums, so no HH Template is shown >

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]

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".)

 

[Azruine]

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".)

Thanks for reply!
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]

Are there any factors of r in the area element da?

 

[Azruine]

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