Interacting Electrons

Consider two electrons bound to a proton by Coulomb interaction. Neglect the Coulomb repulsion between the two electrons.

a) What are the ground state energy and wave function for this system?

b) Consider that a weak potential exists between the two electrons of the form

(1)

where V₀ is a constant and s_j is the spin operator for electron j (neglect the spin–orbit interaction). Use first-order perturbation theory to estimate how this potential alters the ground state energy.

SOLUTION

a) The wave function for a single electron bound to a proton is that of the hydrogen atom, which is

(1)

where a₀ is the Bohr radius. When one can neglect the Coulomb repulsion between the two electrons, the ground state energy and eigenfunctions are

(2)

(3)

(4)

The last factor in (3) is the spin-wave function for the singlet S = 0 in terms of up α and down spin states. Since the spin state has odd parity, the orbital state has even parity, and a simple product function ѱ(r₁)ѱ(r₂) is correct. The eigenvalue is twice the Rydberg energy E_R.

b) The change in energy in first-order perturbation theory is δE = ⟨i|V|i⟩. The orbital part of the matrix element is

(5)

where the final integration variable is x = r/a₀.

Next we evaluate the spin part of the matrix element. The easiest way is to use the definition of the total spin S = s₁ + s₂ to derive

(6)

(7)

where for spin-1/2 particles, such as electrons, s₁ . s₁ = s(s + 1) = 3/4. Since the two spins are in an S = 0 state, the expectation value ⟨s₁ . s₁⟩ = -3/4. Combining this with the orbital contribution, we estimate the perturbed ground state energy  to be

(8)