Scattering partial wave expansion question

[VortexLattice]

Hi, I'm reading about the partial wave expansion in Shankar. In his method, we expand the incident plane wave (he chooses it such that it's coming in along the z axis, and using spherical coordinates) using the Legendre polynomials:

$$e^{ikr cos(\theta)} = \sum _{l = 0} ^\infty i^l (2l + 1) j_l(kr)P_l(cos(\theta))$$

Then he says that "since the potential conserves angular momentum, each angular momentum component scatters independently". I get what he's saying, but my question is: Why couldn't the values of j_l(kr) switch around such that the various angular momentum components switch around, but the total amount is still conserved?

 

[The_Duck]

Suppose we just had a single incoming partial wave instead of a bunch of them superposed. This partial wave has a definite L and m. Since the Hamiltonian conserves angular momentum, the outgoing, scattered wave must have the same L and m, i.e. it must be proportional to the incoming partial wave, since there is only one partial wave for each possible (L,m) pair. When we superpose a bunch of partial waves to get a plane wave, each partial wave still scatters in this simple way, because of linearity/the superposition principle.

 

[VortexLattice]

Suppose we just had a single incoming partial wave instead of a bunch of them superposed. This partial wave has a definite L and m. Since the Hamiltonian conserves angular momentum, the outgoing, scattered wave must have the same L and m, i.e. it must be proportional to the incoming partial wave, since there is only one partial wave for each possible (L,m) pair. When we superpose a bunch of partial waves to get a plane wave, each partial wave still scatters in this simple way, because of linearity/the superposition principle.

I see what you're saying, I think: that when there is a partial wave of only one L and m, the scattered wave has to have that. But when there are a bunch of partial waves with different L's, why can't their amplitudes switch around such that the total L is still conserved?

Thanks!

 

[mfb]

The system is linear - the incoming wave with a superposition of different L can be viewed as the sum of individual incoming waves with fixed L.

 

[VortexLattice]

The system is linear - the incoming wave with a superposition of different L can be viewed as the sum of individual incoming waves with fixed L.

Hmmm, right... Ok, I have another question about the partial wave expansion. He does an example of scattering off a hard sphere, and then shows some qualitative stuff about it at the end. He says about an equation for k -> 0 (low energy): "This agrees with the intuitive expectation that at low energies there should be negligible scattering in the high angular momentum states."

I don't see that intuitively. Why would high angular momentum states scatter less, intuitively?

Thanks!

 

[mfb]

How do you get high angular momentum with low energy => low momentum? Angular momentum is momentum times distance of closest approach. The velocity is fixed, so you need a large distance of closest approach. With a hard sphere, this value is limited by the radius of the sphere.