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.