Energy Eigenvalues for Ion with Spin

[Mastern00b]

Homework Statement

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.

Homework Equations

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.

 

[Goddar]

Hi. Is your Hamiltonian entirely given by H_spin?
If so, what are the eigenvalues of the matrix you found?

 

[Mastern00b]

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

 

[Goddar]

Well, you can easily find this without Wolfram: your characteristic equation is λ(λ–Aħ²)² = 0,
so the eigenvalues are easy to read-off.
So what are you missing? The Hamiltonian operator returns energy eigenvalues and you found the Hamiltonian's eigenvalues...

 

[Mastern00b]

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.

 

[Goddar]

A should be a constant unless otherwise specified in your problem's statement; if it were a matrix, you couldn't get very far into the problem without knowing more about it anyway.
The one important thing to note though, is that from the characteristic equation you can see that one of your eigenvalues is degenerate (it appears more than once): Aħ². It corresponds to two different states of the system with the same energy.