What are the Eigenvalues and Eigenkets of a Spin-1/2 System in a Magnetic Field?

[phys-student]

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 = (σ_x, σ_y, σ_z) is the Pauli spin operator of the particle. A magnetic field of strength B_z is applied along the z direction and a second field B_x is applied along the x direction. The Hamiltonian of the particles is:
H = H₀ + V
H₀ = -μ_zB_z
V = -μ_xB_x
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₊ = σ_x + iσ_y
σ_- = σ_x - iσ_y
σ₊ = |+><-|
σ_- = |-><+|
σ_z|+> = 1|+>
σ_z|-> = -1|->

The Attempt at a Solution

For part a) I'm pretty sure I did it right
H₀ = (-qB_zħ/2mc)σ_z or H₀ = ε₀σ_z if ε₀ = -qB_zħ/2mc.
The operators H₀ and σ_z commute so they have the same eigenkets |+> and |->
Using the expression for H₀ and the eigenvalue equations for σ_z 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 = (-qB_xħ/2mc)σ_x
Then using equations given above I worked out that σ_x = (σ₊ + σ_-)/2, so I subbed that into the the expression to get:
V = (-qB_xħ/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?

 

[blue_leaf77]

V = (-qBxħ/4mc)(σ+ + σ-)

That looks right.

I start running into problems

What are your problems?

 

[phys-student]

When I try to find the eigenvalues of H I get the expression:

(-ε₀σ_z - (qB_xħ/4mc)(σ₊ + σ_-))|+> = ε|+>

and I don't know how to evaluate it properly. I tried plugging these in: σ+ = |+><-|, σ- = |-><+| but I have no idea what to do after that. Are you supposed to expand it so that you're taking the eigenvalue of the first term and the eigenvalues of the second term?

 

[blue_leaf77]

Write the matrix form of $H$ using the knowledge of the matrix form of Pauli matrices. Then solve the eigenvalue problem in resulting matrix equation.

 

[phys-student]

I'm not very familiar with the matrix form of Pauli matrices, it wasn't covered in this course... Do you know of a source I can read that would help?

 

[DrClaude]

There is an alternative route to the solution. You know that the |+> and |-> kets for a complete basis, therefore the eigenstates of H can be written as a|+> + b|->. Try solving H (a|+> + b|->) = E (a|+> + b|->) for a and b (along with proper normalization).

 

[blue_leaf77]

Pauli matrices are related to the spin matrices of spin 1/2 particles. But if you are not yet familiar with those matrices, DrClaude's suggestion above will also do the job.