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Eigen functions/values for manybody Hamiltonian with creation/annihilation operators 
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#1
Jun311, 04:17 PM

P: 11

Problem:
 I’m trying to understand how to generally find Eigen functions/values (either analytically or numerically) for Hamiltonian with creation/annihilation operators in manybody problems. Procedures:  1. I setup a simple case of finitepotential well with two indistinguishable fermions (ignore spin for the moment). 2. I got the creation/annihilation field operators. 3. I secondquantized all the elements of the Hamiltonian (kinetic energy, well/external potential, and electronelectron electrostatic interaction). My question is how to proceed next? I have the Hamiltonian which is now function of the creation/annihilation operators. How can I solve for the manybody Eigen function/values after that? Thanks so much. 


#2
Jun311, 09:54 PM

Sci Advisor
P: 1,909

ranytawfik,
It would be easier to respond if you post some of your math that you've already obtained in the first 3 steps. [ If you're not familiar with Latex on this forum, try this thread: http://www.physicsforums.com/showthread.php?t=8997 ] 


#3
Jun411, 08:28 AM

Sci Advisor
Thanks
P: 4,160

In general, the procedure is as follows:
1) Try to find the normal modes of your system, that is, transform your creation/annihilation operators to a basis in which the Hamiltonian is diagonal: H = ∑ a_{k}*a_{k}. 2) Find the states for each normal mode as a quantized harmonic oscillator. The ground state is 0_{k}> such that a_{k}0_{k}> = 0. The excited states are n_{k}> = (a_{k}*)^{n}n_{k}>. If you can't do step (1), you'll have to use perturbation theory. 


#4
Jun411, 10:38 PM

P: 11

Eigen functions/values for manybody Hamiltonian with creation/annihilation operators
Thanks strangerep and Bill. I'll try to do what you suggested, Bill, and post a follow up with the detailed Latex equations and procedures.



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