Diagonalizing q1ˆ3q2ˆ3 with Degenerate Perturbation Theory

  • #1
ThiagoSantos
2
1
Homework Statement
Determine the first order correction of a system of two identical harmonic oscilators
Relevant Equations
Hˆ =(p1ˆ2 + p2ˆ2+q1ˆ2 +q2ˆ2)/2+fq1ˆ3q2ˆ3. where f is the coupling constant
I tried to use the degenerated perturbation theory but I'm having problems when it comes to diagonalizing the perturbation q1ˆ3q2ˆ3 which I think I need to find the first order correction.
 
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  • #2
I am rusty, but I try.

According to Wikipedia the first-order correction is

(assuming you want to calculate the correction to the ground state ). Your ground state is the vacuum for both oscillators so . Where with .(here you have two identical oscillators so ). So you just have to calculate:

which (if I am not mistaken) will result in integrals of the form

you can solve this by putting (), which will yield the Gamma function (https://en.wikipedia.org/wiki/Gamma_function).
 

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