Spin Interaction

Consider a spin-1/2 particle which is bound in a three-dimensional harmonic oscillator with frequency ω. The ground state Hamiltonian H₀ and spin interaction are

(i)

(ii)

(iii)

where is a constant and σ = (σ_x, σ_y, σ_z) are the Pauli matrices. Neglect the spin–orbit interaction. Use perturbation theory to calculate the change in the ground state energy to order O(λ²).

SOLUTION

In first-order perturbation theory the change in energy is

(1)

where

(2)

since H' = λr . σ and the matrix element of is zero for the ground state |0⟩. The first excited state is three-fold degenerate: denote these states as

(3)

(4)

(5)

In this notation the matrix elements are

(6)

In second-order perturbation theory

(7)

where σ² = 3Ĩ where the unit matrix is Ĩ. Each spin state has the same energy, to second order.