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I have a lot of experience with classical scattering theory (acoustic and electromagnetic waves), but I have no experience with quantum scattering theory, so I am trying to understand the similarities between the two fields as I start looking into quantum scattering.

Firstly, I don't understand the notion of scattering from 'potentials'. In, say, acoustics, we have scattering (reflection and transmission of waves) when an incident wave one medium strikes a second medium with different material properties. For example, an acoustic wave in water could strike an air bubble, that has a different density and bulk modulus than water, and hence we will have scattering of waves. Similarly in electromagnetism, except the material parameters are permittivity and permeability this time. In either case the geometry of the second medium is critical to how the waves scatter. We also have boundary conditions such as Dirichlet, Neumann and combinations of both if we want to model transmission of waves.

From the quantum scattering material I've looked at, I don't see much mentioned about geometry as such, I just read that waves scatter off potentials...so what are these potentials (I assume they are different from potentials in classical scattering), are they analagous to different material properties in acoustics/electromagnetism? Surely notions such as angle of incidence and surface geometry from classical scattering are still relevant in quantum scattering? In classical scattering, waves move from one medium to another, such as fluid to air, or air to solid..do waves move between different mediums in quantum scattering..do we need to know properties such as bulk modulus and density/permittivity and permeability? Or do we instead have analogous properties?

Is there an analagous quantum scattering example to the most simple real-world classical scattering situation, that is, the scattering of acoustic waves in water from a gas bubble?

Qmechanic
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ManUtdBloke
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  • Well, for starters, have you seen the Schrodinger equation and compared it with classical wave equations you've studied? – OON Aug 30 '17 at 12:27
  • I think you'd better to put everything you know about classical scattering in a corner of your memory at first, then thoroughly work through the chapter on scattering of a good QM book. Then and only then, you may try to find common patterns. If you try to do it the other way around, you will just confuse yourself. –  Aug 30 '17 at 14:42
  • When a spacecraft is deflected by the gravitational field of a planet, that is scattering from a potential: that's purely classical. See also the classical treatment of Rutherford scattering: it's the same problem. – NickD Aug 30 '17 at 15:37

1 Answers1

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TL;DR: The two are exactly the same, however the evolution equation may differ.

Scattering theory is a very broad term, ranging from classical acoustic scattering to full quantum scattering in quantum field theory. What the OP calls quantum scattering (which is also the nomenclature often adopted in the literature) is the scattering problem on the Schrödinger equation1

$$ \left[-\nabla^2 + V(r)\right]\psi(r,t) = i\frac{\partial}{\partial t}\psi(r,t). $$

This is simply a wave equation, you can then look at scattering at a certain energy, which gives you the time-independent Schrödinger equation. Then you can apply methods such as Green functions and the Lippmann-Schwinger equation to solve the scattering problem on this equation. Even though this is called quantum scattering there is not much quantum about it: you just have to solve a wave equation. Note that $V(r)$ is the potential here. It can originate from a variety of things and its microscopic derivation can be quite complicated.

In what the OP calls classical scattering the problem is can be very similar. E.g. in acoustic scattering one has a Helmholtz equation (correct me if I got this one wrong, I know more about the electromagnetic version)

$$ -\nabla^2 u(r,t) = \varepsilon(r)\frac{\partial^2}{\partial t^2}u(r,t) $$

where $\varepsilon(r)$ encodes the material properties. Then you can solve the scattering problem on this equation, which may give different results since the evolution equations are different, but conceptually it is the same thing.

Is there a 'more quantum' scattering?

Yes there is. E.g. you can canonically quantize the above equations and then you have a Hamiltonian with operators in them. Those systems can have quantum effects, such as non-linear dependence on the number of quanta you are scattering. That is very much not like the wave equations above.

1 I will swipe some constants under the carpet.

Wolpertinger
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  • In the acoustic (or electromagnetic) case the wave function itself is the potential. That is instead of dealing with the velocity (or electric) field we deal with the corresponding scalar potential field instead. In the acoustic case we have $$\Delta \phi + k^2 \phi = 0$$, where $\phi$ is the acoustic potential and $k$ is the wavenumber. It seems strange that in the quantum mechanical version 'potential' instead seems to fulfill the role of the wavenumber instead of being the function we solve for?! – ManUtdBloke Sep 21 '17 at 06:04
  • @eurocoder you're completely right with your observation. This difference can actually be quite significant when it comes to solution methods for the equations, because the acoustic (electromagnetic) differential operator is not as easily written as an eigenvalue problem of a Hermitian operator. there are some tricks to do so for special cases, but in general this makes the acoustic problem harder than the quantum problem in my experience. – Wolpertinger Sep 21 '17 at 06:23