Finding the magnetization in nuclear magnetic resonance

[David DCruz]

Homework Statement

J-coupling term between two spins is

H_J = ħJ/4 σ_z⁽¹⁾ σ_z⁽²⁾

In the measured magnetization spectrum of the spins, this leads to the splitting of the individual
spin lines by frequency J, which we’ll now derive. We can write the magnetization of spin 1 as:

<M₁(t)> = tr(ρ(t)σ₊⁽¹⁾) = tr[ρ(t)σ₊⁽¹⁾⊗(e₊⁽²⁾+e_-⁽²⁾)]

where e₊⁽²⁾ = matrix(1 0; 0 0 )
e_-⁽²⁾) = matrix(0 0;0 1)
σ₊ = σ_x + i σ_y
(1) refers to 1st qubit; (2) refers to 2nd qubit

Assume ρ(t) evolves according to U(t)=exp(-iH_Jt/ħ)

Show that
<M₁(t)> = exp(iJt/2) tr[ρ(0)σ₊⁽¹⁾e₊⁽²⁾] + exp(-iJt/2) tr[ρ(0)σ₊⁽¹⁾e_-⁽²⁾]

Homework Equations

Mentioned above

The Attempt at a Solution

I expressed ρ(t) = U(t) ρ(0) U⁺(t)

Then I wrote <M₁(t)> = tr[ρ(0)exp(iH_Jt/ħ)σ₊⁽¹⁾exp(-iH_Jt/ħ)⊗(e₊⁽²⁾+e_-⁽²⁾)]
I expanded out the exponential hamiltonian to get

<M₁(t)> = tr[ρ(0)exp(-iJtσ_z⁽²⁾/2) σ₊⁽¹⁾⊗(e₊⁽²⁾+e_-⁽²⁾)]

I'm not sure how to proceed from here

 

[Charles Link]

I think you only need to use the fact that $\sigma_z^{(1)}$ and $\sigma_z^{(2)}$ have eigenvalues of $1$ and $-1$. I do't think it requires a lengthy derivation to show what you are trying to show. Once you assign the eigenvalues, it shows what the possible energies are, and thereby the energy differences between the two states. I think I get $J/2$ as the frequency difference.

 

[David DCruz]

Do you mean to use the eigen values of σ_z⁽¹⁾ and σ_z⁽²⁾ right from the beginning of the solution instead of what I did or from where I am currently stuck. Also, I'm confused as to what happens with the tensor product. How does it vanish in the final answer. As far as I understand it, if an operator acts on a tensor product (where the operator can be broken into operators that act in only one of the local Hilbert spaces), the result is the tensor product of the states got by acting the local operators on the corresponding states in the local Hilbert spaces. I'm taking a shot in the dark here but can you please tell me if the following is correct just from a mathematical point of view.

<M₁(t)> = tr[ρ(0)exp(-iJtσ_z⁽²⁾/2) σ₊⁽¹⁾⊗ (e₊⁽²⁾+e_-⁽²⁾)]
=tr[ ρ(0) σ₊⁽¹⁾ ⊗(exp(-iJtσ_z⁽²⁾/2)e₊⁽²⁾ + exp(-iJtσ_z⁽²⁾/2)e_-⁽²⁾) ]

 

[Charles Link]

In the way I'm suggesting, you would use the eigenvalue result immediately. Your calculations involving the spin operator acting on two separate possible spin states with a tensor product is considerably different from the elementary quantum mechanics that I am familiar with. Perhaps there are others who might be able to work the problem as well with what my be a more advanced approach. @bhobba Might you be able to assist here? I would simply use the known eigenvalues from the beginning. $\\$ Editing: Also, in studying your original post in more detail, perhaps the problem you are trying to solve is much more detailed than the solution that I presented in post 2. As I understood the original post, the goal was to find the splitting of the spectral lines. $\\$ Additional comment: Without the spin-spin coupling, the energy of a given spin state in the magnetic field $B$ is given by $E=-\mu \cdot B$ , where $\mu=\frac{g \mu_N \sigma}{2}$, when working with nuclear spins. (For nuclear spins, the Bohr magneton (which applies to electrons) is replaced by the nuclear magneton $\mu_N$). $\\$ The magnetization $M_z$, (a macroscopic property), is related to the average value of the spin operator, ($M_z=A(n_+-n_- )$ where $A$ is a proportionality constant, and $n_+$ is the density of spins in the spin up state etc.), but I don't think that is needed to solve the problem.