Thermodynamics, Statistical Physics, and Quantum Mechanics

5.37 Helium Atom (Tennessee)

In the ground state of the two-electron system, both orbitals are in 1s states.

So the spin state must be a singlet with The spin plays no role

in the minimization procedure, except for causing the orbital state to have

even parity under the interchange of spatial coordinates. The two-electron

wave function can be written as the product of the two orbital parts times

the spin part:

where is the Bohr radius and is the variational parameter. The orbitals

are normalized to unity. Each electron has kinetic (K) and potential

(U) energy terms which can be evaluated:

where is the Rydberg energy. The difficult integral is that

due to the electron–electron interaction, which we call V:

First we must do the angular integral over the denominator. If is the

larger of and then the integralover a solid angle gives

I n the second integral w e have set and which
makes the integrals dimensionless. Then we have split the into
two parts, depending on whether is smaller or greater than The first
has a factor from the angular integrals, and the second has a factor

286 SOLUTIONS