Find Eigenvectors for σ⋅n: Solving the Equations

[1missing]

1. n = sinθcosφ i + sinθsinφ j + cos k
σ = σ_x i + σ_y j + σ_z k , where σ_i is a Pauli spin matrix

Find the eigen vectors for the operator σ⋅n
2. Determinant of (σ⋅n - λI), where I is the identity matrix, needs to equal zero
(σ⋅n - λI)v = 0, where v is an eigen vector, and 0 is the zero vector

3.
Not entirely sure how to type out these matrices, but the math is fairly straight forward. If it's needed I can supply pictures of hand written work.

To find the eigenvalues, I took the determinant to get an equation for λ, and I found it was equal to ±1. I then tried to find the eigen vector for the eigenvalue 1. The two equations I ended up with were:

a(cosθ - 1) + b(sinθcosφ - isinθsinφ) = 0

a(sinθcosφ +isinθsinφ) - b(cosθ+1) = 0

No matter how I try to solve for a or b, I end up canceling the unknowns entirely and end up with something like 1 = 1. Is there something I'm not seeing? Not getting a meaningful answer has me questioning my approach, is there something else I could try?

 

[kuruman]

Is there something I'm not seeing?

You are not seeing that the two equations you get are not linearly independent, cannot be linearly independent, because you set the determinant equal to zero in the first place. One normally finds the eigenvectors by getting a as a constant times b from one of the equations and then use the normalization condition. If you give the matter a little thought you will see that you can multiply the eigenvector v by an arbitrary scalar and still satisfy the eigenvalue equation. However there is only one scalar that normalizes the eigenvector.

 

[1missing]

Derp. Once I read normalization the lightbulb went off in my head.

b = a(sinθcosφ +isinθsinφ) / (cos + 1)

1 - |b²| = |a²|

|a²| = (cosθ + 1)² / ((cosθ + 1)² + (sin²θcos²φ +sin²θsin²φ)

|a²| = (cosθ + 1)² / (2cosθ + 2)

a = √(cosθ + 1)/2 = cos(θ/2)Thank you for the help!