Question

Two identical bosons, each of mass m, move in the one-dimensional harmonic oscillator potential V = 1/2 mw²x². They also interact with each other via the potential

\(V_{int} (x_1 , x_2) = \alpha e^{-\beta(x_1 - x_2)^2}\)

where \(\beta\) is a positive parameter. Compute the ground state energy of the system to first order in the interaction strength parameter α.

Answer 1

Being bosons the two particles can simultaneously stay in the ground state. Taking V_int as perturbation, the unperturbed wave function of the ground state for the system is

The perturbation energy to first order in α is then

where the integration has been facilitated by the transformation

The ground state energy of the system is therefore