Diagonalizing q1ˆ3q2ˆ3 with Degenerate Perturbation Theory

[ThiagoSantos]

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.

 

[dRic2]

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).