Spin Interaction
المؤلف:
Sidney B. Cahn, Gerald D. Mahan And Boris E. Nadgorny
المصدر:
A GUIDE TO PHYSICS PROBLEMS
الجزء والصفحة:
part 2 , p 68
22-8-2016
1541
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 = (σx, σy, σz) 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.
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