Diagonalizing q1ˆ3q2ˆ3 with Degenerate Perturbation Theory

ThiagoSantos
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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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I am rusty, but I try.

According to Wikipedia the first-order correction is
$$\bra{GS} H_{int} \ket{GS}$$
(assuming you want to calculate the correction to the ground state ##\ket{GS}##). Your ground state is the vacuum for both oscillators so ##\ket{GS} = \ket{0}\ket{0}##. Where ##\ket{0} \propto \exp(- \omega_i q_i^2)## with ##i = 1,2##.(here you have two identical oscillators so ##\omega_1 = \omega_2 = 1##). So you just have to calculate:
$$\bra{0}\bra{0} q_1^3 q_2^3 \ket{0} \ket{0}$$
which (if I am not mistaken) will result in integrals of the form
$$\int dq_i q_i^3 e^{-2q_i^2} $$
you can solve this by putting ##t=x^2## (##dt = 2xdx##), which will yield the Gamma function (https://en.wikipedia.org/wiki/Gamma_function).
 
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The value of H equals ## 10^{3}## in natural units, According to : https://en.wikipedia.org/wiki/Natural_units, ## t \sim 10^{-21} sec = 10^{21} Hz ##, and since ## \text{GeV} \sim 10^{24} \text{Hz } ##, ## GeV \sim 10^{24} \times 10^{-21} = 10^3 ## in natural units. So is this conversion correct? Also in the above formula, can I convert H to that natural units , since it’s a constant, while keeping k in Hz ?

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