Although the Schrodinger equation for helium itself cannot be solved exactly there exist helim-like systems that do admit exact solutions. A simple example is a rubber-band helium, in which the Coulomb forces are replaced by Hookes law forces:

H = -²/(2m)(₁² +₂²) + mw²(r₁² +r₂²) lmw²|r₁ + r₂|²

(a) show that the change of variables from r₁, r₂ to

u = (1/2)(r₁ + r₂) v = (1/2)(r₁ - r₂)

turns the Hamiltonian into two independent three-dimensional harmonic oscillators.

(b) What is the exact ground state energy for this system? (6 pts)