# A Virtual Challenge

1. May 15, 2006

### reilly

There's a lot of talk about "virtual particles" and of energy conservation within a normal perturbative aproach to QM. I'm struck by the fact that there are precious little concrete examples or illustrations or gedanken experiments to support all the arguments, which go around and around and around ... (I am, indeed, one of the guilty ones.

So here's a challenge:

Work out the complete solution to particle scattering from a finite square well potential by means of perturbation theory. Use any shape that you want.

Show that any finite subset of the series yields a scattering amplitude that does not obey unitary time development under the complete Hamiltonian; does not obey basic probability conservation. Yet, for a weak potential, the non-unitary Born Approx. does a good job of describing the scattering. How can this be?

Examine in detail the dynamical interplay of the pertubative series at a finite time T, when the system is turned on at T=0, with a particle of momentum P. Show how energy flows back-and-forth among the "free particle states", and how the flow diminishes over time, until all the energy is in the set (single) of the usual final states as T->infinity.

(Extra Credit) Extrapolate, and explain the difficulties of asserting energy conservation in, say, Compton Scattering within a finite approximation scheme.

This concludes the final for Advanced Quantum Theory., 303.

Regards,
Reilly

Last edited: May 15, 2006
2. May 15, 2006

### Tom Mattson

Staff Emeritus
Now that the semester's over I have time to play. But I'm a little confused about something. You talk about virtual particles on the one hand, and then QM on the other. Virtual particles suggests QFT to me. Do you mean for the calculation to be done with ordinary QM? Is a nonrelativistic approach satisfactory?

3. May 16, 2006

### reilly

At least in my opinion, virtual particles are simply a covariant virtual state -- just like the intermediate states in ordinary nopn-rel perturbation theory. So, what I'm suggesting is a look at the solution to a 1-D scattering prob with a square well potential, and in particular at the solution expressed in plane waves -- free or virtual particles in rel QM-> free or virtual states in non-rel QM. NRQM is the way to go.

Regards,
Reilly