# Breit equation/relativistic QM and simplifications

1. Feb 24, 2012

### Amok

Hello there,

http://en.wikipedia.org/wiki/Breit_equation

And I'm having a hard time understanding a few thing about this equation. The first thing is that, from what I can gather, the Breit Hamiltonian is basically a Dirac Hamiltonian, plus a the coulomb interaction operator (in the coulomb gauge), plus a correction that accounts for the fact that electron-electron interaction is felt in a retarded fashion. Why is it that this equation is/was useful? Is it not possible to write the exact relativistic Hamiltonian for a multi-electron system?

I'm reading a book where it is stated that to generalize the Dirac equation to a two electron system you basically "do a tensor product" of the two Hamiltonians with the 4x4 Id matrix then add them up. One issue that arises is that you have different time for each Hamiltonian. A way to simplify this is to assume that both times are the same. So far so good. The second issue that arises is the choosing of (A,$$\phi$$) for each one of the electrons. As I understand it the Breit operator is a way of doing this (although exactly how this happens still eludes my understanding). Isn't there a way to write out these potentials exactly?

My second doubt concerns the wikipedia article. The section entitled "Breit Hamiltonians" seems misleading to me. The Hamiltonians described in that section are the so-called two component Hamiltonians, which come from the reduction of the dirac-coulomb-breit hamiltonian into "quasi-relativistic" forms. But this can be done (and I have done it by hand) with the simple Dirac Hamiltonian, and also leads to Darwin, Mass-velocity, SO terms. Am I wrong about this?

I have third doubt. This is related to the statement that the Breit hamiltonian accounts for the fact that interactions between electrons are retarded. This statement is made time and time again in many sources, however I've come across one book that states this the last term in hamiltonian (called B in the wiki article) is basically the quantization of the classical expression for the interaction energy between two charged particles in which retardation is explicitly neglected!

Last edited: Feb 24, 2012
2. Feb 25, 2012

### juanrga

That inconsistent equation is useful to study certain aspects of stationary states of few-electron systems, where retardation and other effects are small.

It is not possible to write the exact relativistic Hamiltonian for a multi-electron system, because the field theory does not exist. Already in his textbook on classical electrodynamics Jackson devotes a section to explain why the full relativistic Lagrangian of a two-electron system does not exist. In short, field theory fails.

You can write down the exact potentials for a single electron. The problem start when you consider a second electron. See Jackson textbook.

The simple Dirac Hamiltonian only contains a kinetic term. The Dirac-Coulomb Hamiltonian contains a Coulomb interaction term and the Dirac-EM Hamiltonian contains Coulomb (plus retardation) and magnetic interactions.

Yes there is lots of confusion about retardation. I recommend you to take a look to the Darwin Lagrangian in Jackson book. The Breit Hamiltonian can be obtained from there.

3. Feb 25, 2012

### Amok

Thank you for replying. Can you give me the full name of the reference?

4. Feb 28, 2012

### juanrga

Classical Electrodynamics by John David Jackson (Wiley; 1962).

In my copy the relevant section is the 12.6 (this can vary in other editions).