Graduate Selection rules using Group Theory: many body

[SteveP]

Hello, I am newish in group theory so sorry if anything in the following is not entirely correct.
In general, one can anticipate if a matrix element <i|O|j> is zero or not by seeing if O|j> shares any irreducible representation with |i>.

I know how to reduce to IRs the former product but I cannot see how it would be done for, lets, say, two particle states. This is, how can one anticipate the result of <i,j| K |m,l> (the integral K_{ijml}) provided that K is a two-particle operator (for example the coulomb potential 1/|r1-r2|)?

Can anyone tell me any reference on group theory where this is treated?

Thanks in advance,

Steven

 

[Charles Link]

$\\$ I had 3 books handy that might have the result you are looking for: $\\$
1) Peskin and Schroeder treat a few things on Lie groups, (Group theory), that you can find in the index,
but not the result you are looking for. $\\$
2) Fetter and Walecka has nothing in the index that is a match. $\\$
3) Michael Tinkham's Group Theory and Quantum Mechanics has a section on Selection Rules for Vibrational Transitions,
(pp. 248-250), that may be a very good fit. Tinkham's entire book, in general, is not very difficult reading, and the topics can often be understood if you make up arrows, etc. to represent the function of interest and see how the arrow transforms under the various group operations which are usually rotations or reflections.

 

[DrDu]

You could try "Morton Hamermesh, Group theory and its application to physical problems, Dover Pubs" or "Eugene Wigner, Group Theory".
While 1/ r_ij will have little symmetries, $\sum_{ij} 1/r_{ij}$ will be totally symmetric, so you will have to look whether your two wavefunctions transform as the same irrep.