Chapter 7, Problem 7.37P

(a)

To determine

Explain the equation $H\text{'}=\frac{1}{4\pi {\epsilon }_0}\left(\frac{e^2}{R}-\frac{e^2}{R-x_1}-\frac{e^2}{R+x_2}+\frac{e^2}{R-x_1+x_2}\right)$.

Show that $H\text{'}\cong -\frac{e^2{x}_1{x}_2}{2\pi {\epsilon }_0{R}^3}$.

(b)

To determine

Show that the total Hamiltonian separates into two harmonic oscillator.

(c)

To determine

Show that the equation $\Delta V\equiv E-E_0\cong -\frac{h}{8m^2{\omega }_0^3}{\left(\frac{e^2}{2\pi {\epsilon }_0}\right)}^2\frac{1}{R^6}$.

(d)

To determine

The ground state energy of the Hamiltonian using second-order perturbation theory .