Calculating Energy Levels in a Ni2+ Ion Crystal

[Nico045]

Hello, I am stuck at the beginning of an exercise because I have some trouble to understand how are the energy level in this problem :

In a crystal we have Ni²⁺ ions that we consider independent and they are submitted to an axial symmetry potential. Each ion acts as a free spin S=1. We have the hamiltonian :

H₀=C(S_z²-S²/3)
C is a constant >0

- I need to know the energy level of each ion and the eigenstates.

After that, we add a magnetic field B oriented on the z axis (in interaction with the ion) which is given by the hamiltonian :
H₁=2u_BBS_z

- Here again I have to find the levels of energy for H = H₀+H₁

 

[DrClaude]

H₀=C(S_z²-S²/3)
C is a constant >0

- I need to know the energy level of each ion and the eigenstates.

So what are the eigenvalues and eigenstates of that Hamiltonian? (Hint: think in terms of the quantum numbers for operators that commute with H.)

 

[Nico045]

I haven't studied much the eigenvalues of the spin but I can try. (I am not sure at all) :

S_z = +1/2 , -1/2
S=1

Then H₀=C(1/4-1/3) = -C/12

eigenvalues K=-C/12

and I should look for something who satisfy :

H₀Ψ=KΨ

 

[DrClaude]

I haven't studied much the eigenvalues of the spin but I can try. (I am not sure at all) :

S_z = +1/2 , -1/2
S=1

The eigenvalue of the $\hat{S}^2$ is $S (S+1)$.

and I should look for something who satisfy :

H₀Ψ=KΨ

Looking at it in terms of a wave function is not necessary. Do you know Dirac notation?

 

[Nico045]

So the eigenvalue of S² should be only 2 ? (since it is given that S=1)

Yes i know Dirac notation

 

[DrClaude]

So the eigenvalue of S² should be only 2 ? (since it is given that S=1)

Correct.

Edit: assuming ħ = 1.

Yes i know Dirac notation

Then you should be able to write the eigenstate as a ket,

 

[Nico045]

I have been a little busy, I'm sorry for not answering sooner.

It should be on this form then :

| eigenvalue of S² , eigenvalue of S_z >
$\frac{C}{3}$ for |2,1>

$\frac{C}{3}$ for |2,-1>

$\frac{-2C}{3}$ for|2,0>

And if we add a magnetic field B oriented on the z axis (in interaction with the ion), the hamiltonian is $H=C(S_z^2 -\frac{1}{3}S^2) + 2 \mu_B B S_z$ and the eigenvalues are :

$\frac{C}{3}+ 2 \mu_B B$ for|2,1>
$\frac{C}{3} - 2 \mu_B B$ for |2,-1>
$\frac{-2C}{3}$ for |2,0>

Is that right ?

 

[DrClaude]

Looks fine, except

| eigenvalue of S² , eigenvalue of S_z >

is not conventional. Normally, one would use $| S, M \rangle$ (in other words, use the value of the spin $S$, not the eigenvalue of $\hat{S}^2$).