Assumptions made in stationary-state scattering

[WWCY]

Hi all, I recently started learning about quantum scattering in school and came across a few things I find confusing. Thanks in advance for any assistance!

1. Plane wave approximation to incident waves.

In past QM courses, I kept reading that plane waves were not "physical" since they do not normalise to unity. As such, would using them as a mathematical description for incident waves affect results? I have read that such approximations are valid if the incident waveform is much larger than the scatterer, which i can picture. However, the positional spread of the wavefunction is equal at all points in space, under what conditions would an actual scatterer encounter such a wavefunction?

2. Stationary-state solutions

A stationary-state solution to a scattering Hamiltonian would mean that we are solving a problem in which "everything" (wavefunction of incident and scattered particle) looks the same everywhere, at all points in time. Again, what sort of scattering set-up would allow for such an approximation?

 

[WWCY]

Could anyone assist? Many thanks

 

[DrClaude]

1. Plane wave approximation to incident waves.

In past QM courses, I kept reading that plane waves were not "physical" since they do not normalise to unity. As such, would using them as a mathematical description for incident waves affect results? I have read that such approximations are valid if the incident waveform is much larger than the scatterer, which i can picture. However, the positional spread of the wavefunction is equal at all points in space, under what conditions would an actual scatterer encounter such a wavefunction?

You can always consider that the incoming state is a wave packet. If you decompose it in terms of plane waves, then the result of the scattering will be a superposition of the solutions found for each plane wave, weighted by the proper coefficient. For most well-behaved potentials, the scattering result varies slowly as a function of the incoming wave vector, such that for a narrow enough wave packet in momentum space, there is no significant difference between the full solution and that obtained from a single plane wave corresponding to the peak of the wave packet.

2. Stationary-state solutions

A stationary-state solution to a scattering Hamiltonian would mean that we are solving a problem in which "everything" (wavefunction of incident and scattered particle) looks the same everywhere, at all points in time. Again, what sort of scattering set-up would allow for such an approximation?

The scattering potential has to be independent of time. Otherwise, I don't think that there are any other considerations, as you can write the wave packet as a superposition of these stationary states.

You can also model the scattering process using time-dependent perturbation theory (see for instance the textbook by Sakurai).

 

[WWCY]

Thank you for replying

You can always consider that the incoming state is a wave packet. If you decompose it in terms of plane waves, then the result of the scattering will be a superposition of the solutions found for each plane wave, weighted by the proper coefficient. For most well-behaved potentials, the scattering result varies slowly as a function of the incoming wave vector, such that for a narrow enough wave packet in momentum space, there is no significant difference between the full solution and that obtained from a single plane wave corresponding to the peak of the wave packet.

Apologies, but I'm not sure I follow here. When you mention "scattering result", what do you mean by this? Also, what does it mean to "vary slowly" as a function of an incoming wave-vector?

 

[DrClaude]

Apologies, but I'm not sure I follow here. When you mention "scattering result", what do you mean by this? Also, what does it mean to "vary slowly" as a function of an incoming wave-vector?

There are different ways to represent the outcome of a scattering event, but I was thinking specifically about the scattering amplitude $f(\theta, \phi)$, which gives the angular distribution of the scattered particle and can be used to calculate the scattering cross section. In many cases, $f(\theta, \phi)$ is a function of the wave vector $k$, but if that function is slowly varying with respect to $k$, then the difference between the scattering of a plane wave of wave vector $k$ and a wave packet of width $\delta k$ centered on $k$ will be minimal.