Although the Schrdinger equation for helium itself cannot be solved exactly, there exist helium-like systems that do admit exact solutions. A simple example is rubber-band

Although the Schrödinger equation for helium itself cannot be solved exactly, there exist “helium-like” systems that do admit exact solutions. A simple example is “rubber-band helium,” in which the Coulomb forces are replaced by Hooke’s law forces:

(a) Show that the change of variables from r_1,r₂, to

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

(b) What is the exact ground state energy for this system?
(c) If we didn’t know the exact solution, we might be inclined to apply the method of Section 8.2 to the Hamiltonian in its original form (Equation 8.78). Do so (but don’t bother with shielding). How does your result compare with the exact answer?

H = - 2 - - (V + V) + {ma (r +r) ^\_m6|r - 1/ 2m (8.78)