Partial wave analysis - incoming/outgoing?

[JoePhysicsNut]

In the chapter on partial wave analysis in Griffiths's Introduction to Quantum Mechanics, he considers a spherically symmetric potential and says that for large r, the radial part of Schrödinger's equation becomes,

\frac{d^{2}u}{dr^{2}}≈-k^{2}u

with a general solution of

u(r)=C\exp{ikr}+D\exp{-ikr}.

He then says the first term represents the outgoing wave and the second term is the incoming wave. Why is that the case? These two differ by \pi in the complex plane, but I don't see how that enables one to make the "incoming"/"outgoing" distinction.

 

[fzero]

Act on the solution with the radial momentum operator $\hat{p}_r = -i\hbar \partial_r$ and compare the signs of the two terms.

 

[JoePhysicsNut]

Act on the solution with the radial momentum operator $\hat{p}_r = -i\hbar \partial_r$ and compare the signs of the two terms.

Thanks! The operator yields the momentum for the first term and (-1)*momentum for the second. A negative magnitude for momentum does not make sense, so therefore it is to be evaluated for times t<0 making it the incoming wave. Is that the argument?

 

[fzero]

Thanks! The operator yields the momentum for the first term and (-1)*momentum for the second. A negative magnitude for momentum does not make sense, so therefore it is to be evaluated for times t<0 making it the incoming wave. Is that the argument?

We're computing the momentum, not just the magnitude. If $k$ is real, then the magnitude of momentum is always positive. We're also not discussing time dependence here. The argument is simply that, if $k>0$, then one solution carries momentum in the $+r$ direction, while the other carries it in the $-r$ direction. If we did add in the time dependence, we could see this more explicitly, but it isn't necessary.

 

[Meir Achuz]

A factor e^{-i\omega t} is understood. This makes one form outgoing and the other ingoing.