Thermodynamics, Statistical Physics, and Quantum Mechanics

The last factor in (S.5.47.3) is the spin-wave function for the singlet

in terms of up and down spin states. Since the spin state has odd parity,

the orbital state has even parity, and a simple product function

is correct. The eigenvalue is twice the Rydberg energy

b) The change in energy in first-order perturbation theory is

The orbital part of the matrix element is

where the final integration variable is

Next we evaluate the spin part of the matrix element. The easiest way

is to use the definition of the total spin to derive

where for spin-1/2 particles, such as electrons, Since

the two spins are in an state, the expectation value

Combining this with the orbital contribution, we estimate the perturbed

groundstateenergy to be

5.48 Stark Effect in Hydrogen (Tennessee)

We use the notation to describe the four orbital states: thes-state
is and the three are Spin is not affected
by this perturbation and plays no role in the calculation. For degener-
ate perturbation theory we must evaluate the 10 different matrix elements
which occur in the symmetric 4 × 4 matrix. The interac-
tion potential is One can use parity and other group theory
arguments to show that only one matrix element is nonzero, and we call it

QUANTUM MECHANICS 299