Spin–Orbit Interaction

Consider in three dimensions an electron in a harmonic oscillator potential which is perturbed by the spin–orbit interaction

(i)

(ii)

(iii)

(iiii)

a) What are the eigenvalues of the ground state and the lowest excited states of the three-dimensional harmonic oscillator?

b) Use perturbation theory to estimate how these eigenvalues are altered by the spin–orbit interaction.

SOLUTION

a) In three dimensions the lowest eigenvalue of the harmonic oscillator is (3/2)hω, which can be viewed as hω/2 from each of the three dimensions. The ground state has s-wave symmetry. The lowest excited states have eigenvalue (5/2)hω. There are three of them. They have p-wave symmetry and are the states L = 1 and M_L = (1,0, -1).

b) In the spin–orbit interaction we take the derivative ∂V/∂r and find

(1)

(2)

The matrix element ∆ is a constant, which simplifies the calculation. We evaluate the factor L . S by defining the total angular momentum J as

(3)

(4)

(5)

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For the ground state of the harmonic oscillator, L = 0 and J = S = 1/2. The above expectation value of L . S is zero. The ground state is unaffected by the spin–orbit interaction, although it is affected by relativistic corrections as well as by other states.

The first excited states have L = 1, S = 1/2 so that J = 3/2, 1/2. For J = 3/2 we find that

(7)

For J = 1/2 we find that

(8)