- #1

Xyius

- 508

- 4

## Homework Statement

*The paramagnetic resonance of a paramagnetic ion in a crystal lattice is described by*

the spin Hamiltonian

the spin Hamiltonian

[tex]\hat{H}=aH\hat{S}_z+bH\hat{I}_z+2D(3\cos^2\theta-1)\left[ \hat{S}_z^2-\frac{1}{3}S(S+1) \right]+\frac{1}{2}D \sin 2\theta \left[ \hat{S}_+\left( \hat{S}_Z + \frac{1}{2} \right) +\hat{S}_-\left( \hat{S}_z-\frac{1}{2} \right) \right]+ \\ \frac{1}{4}D\sin^2 \theta (\hat{S}_+^2+\hat{S}_-^2)+A\hat{S}_z\hat{I}_z+\frac{1}{2}A(\hat{S}_+\hat{I}_-+\hat{S}_-\hat{I}_+)[/tex]

*##\hat{\vec{S}}## and ##\hat{\vec{I}}## are the electronic and nuclear spin operators, respectively. a, b, A, and D areconstants (a ≪ b), ##\theta## is the angle between the crystal symmetry axis and the direction*

of the magnetic field ##\vec{H}##. For simplicity we take ##\hbar=1##

(i) Assuming that A,D ≪ a, compute the corrections to the energy levels of the unper-

turbed Hamiltonian ##\hat{H}_0=aH\hat{S}_z+bH\hat{I}_z## up to second order in perturbation theory.

Solve this problem for arbitrary values of S and I.

(ii) Assuming that A,D ∼ a, how do you calculate the first order corrections to the

energy levels of the unperturbed Hamiltonian ##\hat{H}_0=aH\hat{S}_z##? While you should not

attempt to solve this problem for arbitrary values of S and I, make sure you report all

matrix elements involved in the calculation.

of the magnetic field ##\vec{H}##. For simplicity we take ##\hbar=1##

(i) Assuming that A,D ≪ a, compute the corrections to the energy levels of the unper-

turbed Hamiltonian ##\hat{H}_0=aH\hat{S}_z+bH\hat{I}_z## up to second order in perturbation theory.

Solve this problem for arbitrary values of S and I.

(ii) Assuming that A,D ∼ a, how do you calculate the first order corrections to the

energy levels of the unperturbed Hamiltonian ##\hat{H}_0=aH\hat{S}_z##? While you should not

attempt to solve this problem for arbitrary values of S and I, make sure you report all

matrix elements involved in the calculation.

## Homework Equations

1. The first order energy correction is:

[tex]\Delta^1=<n,l_n|\hat{V}|n,l_n>[/tex]

Where ##|n,l_n>## are the degenerate kets of the unperturbed Hamiltonian and ##\hat{V}## is the perturbation and must be diagonal.

2. The second order energy correction is:

[tex]\Delta^2=\sum_{k \notin S_n}\frac{|<k|\hat{V}|n,l_n>|^2}{E_n^0-E_k^0}[/tex]

Where ##|k>## are the non-degenerate kets, ##E_n^0## is the nth energy level of the unperturbed Hamiltonian (same deal for ##E_k^0##). And ##S_n## is the degenerate subspace.

## The Attempt at a Solution

As much as I hate to say this, I seem to be a bit too confused to confidently know where to start. I realize I have a few problems / confusions about some things and I am wondering if anyone can clarify. Maybe after I get some clarification on what is confusing me, I can make some real progress on this problem. (I will only deal with part 1 for now.)

**1.)**So calculating the first order correction to the energy amounts to using equation 1 from above on the perturbation. I am assuming the perturbation is

[tex]\hat{V} = 2D(3\cos^2\theta-1)\left[ \hat{S}_z^2-\frac{1}{3}S(S+1) \right]+\frac{1}{2}D \sin 2\theta \left[ \hat{S}_+\left( \hat{S}_Z + \frac{1}{2} \right) +\hat{S}_-\left( \hat{S}_z-\frac{1}{2} \right) \right]+ \\ \frac{1}{4}D\sin^2 \theta (\hat{S}_+^2+\hat{S}_-^2)+A\hat{S}_z\hat{I}_z+\frac{1}{2}A(\hat{S}_+\hat{I}_-+\hat{S}_-\hat{I}_+)[/tex]

Now this is obviously a mess. My first thought is to find a basis in which this perturbation is diagonal, but I cannot do that because I have ladder operators in there! I feel like my only choice of basis are either the ##|j,m>## basis, or the ##|m_1,m_2>## basis.

**2.)**The problem says to solve for any value of S and I. But this is confusing because don't the value of S and I dictate how large the matrix is? How can I find an arbitrarily sized matrix, diagonalize it, and then find the eigenvalues if the size can be anything?

If anyone can help, I would really appreciate it!