- #1
phys-student
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Homework Statement
Consider a spin system with noninteracting spin 1/2 particles. The magnetic moment of the system is written as:
μ = (ħq/2mc)σ
Where σ = (σx, σy, σz) is the Pauli spin operator of the particle. A magnetic field of strength Bz is applied along the z direction and a second field Bx is applied along the x direction. The Hamiltonian of the particles is:
H = H0 + V
H0 = -μzBz
V = -μxBx
a) Find the eigenvalues and eigenkets of H0
b) Express V in terms of σ+ and σ-
c) Find the eigenvalues and eigenkets of H
Homework Equations
σ+ = σx + iσy
σ- = σx - iσy
σ+ = |+><-|
σ- = |-><+|
σz|+> = 1|+>
σz|-> = -1|->
The Attempt at a Solution
For part a) I'm pretty sure I did it right
H0 = (-qBzħ/2mc)σz or H0 = ε0σz if ε0 = -qBzħ/2mc.
The operators H0 and σz commute so they have the same eigenkets |+> and |->
Using the expression for H0 and the eigenvalue equations for σz given above the eigenkets of H0 are ε0 and -ε0.
Part b I'm not so sure. I wrote the expression for V in the same way that I did for H0:
V = (-qBxħ/2mc)σx
Then using equations given above I worked out that σx = (σ+ + σ-)/2, so I subbed that into the the expression to get:
V = (-qBxħ/4mc)(σ+ + σ-)
When I try to do part c I start running into problems and I think it is because I did something in part a or b wrong. Can anyone tell me if I've made any mistakes in part a or b?