Homework Help: Energy Eigenvalues for Ion with Spin

1. The problem statement, all variables and given/known data
An ion has effective spin ħ. The spin interacts with a surrounding lattice so that: H_spin = A S² _z.

I first had to write H as a matrix. Then i had to find the energy eigenvalues.

2. Relevant equations


3. The attempt at a solution
I figured j=1 and mj = 1,0,-1

S² _z = ħ²(1 0 0; 0 0 0; 0 0 1)

H=Aħ²(1 0 0; 0 0 0; 0 0 1)


I think this is right so far but do not know how to progress further. Any help would be appreciated.

 

Hi,
yes it is.

I have an attempt at the eigenvalues only.
I tried following the instructions here https://www.khanacademy.org/math/li.../v/linear-algebra-eigenvalues-of-a-3x3-matrix
Using their notation i got
Hv = λv for the eigenfunction equation
(λI₃ - H)v = 0

λI₃-H = (λ-ħ²A 0 0; 0 λ 0; 0 0 λ-ħ²A)

from this i got the characteristic polynomial λ³ - λ²ħ²2A + λħ⁴A² = 0

i put that into wolfram alpha and got the results λ=0 and λ = Aħ²

 

In my original post i wasnt aware how i could find the eigenvalues. In the time since posting i came across that idea. It also didnt help that when i posted i was under the impression A was a matrix not a constant. Sorry if i was not clear in my original post. Does this seem correct to your eye then, i have very little confidence in my quantum work.