sprabimagneticresonance.ppt (Slides 1-8)

• Explain that the first case we will look at in the slides is just the same problem we've been looking at throughout the course but with our heads tilted. The magnetic field now just has an additional x-direction component.
• Now, we will have two Lamor Frequencies since a new x-component of the magnetic field is present, and both of these frequencies show up in our Hamiltonian.
• Perform a brief review on how to solve for the time dependence of states. Ask the class and see if they can give you each step. In particular, the following process should come up:

1. Solve the eigenvalue equation for the Hamiltonian.
2. Find what the state at $t=0$ is in the basis of the hamiltonian.
3. Time-evolve the state with complex phases corresonding to the energy of each eigenvector.

• This process tells us to first solve the eigenvalue equation for the Hamiltonian. Unfortunately, our Hamiltonian is not diagonalized; but, for this problem, we actually don't need to solve for the Hamiltonian because the only difference between the energy eigenvalues for this case and the case $\vec{B}=B_{0}\hat{z}$ is our heads are tipped.
• Point out the energy eigenvalues on slide 2 and how similar they are to the general $\vec{B}=B_{0}\hat{z}$.
• At this point, the eigenvectors would also be found from the Hamiltonian as well. Luckily, our heads are still only tipped, so the eigenvectors will take the form of the general state vectors $\vert+\rangle_{n}$ and $\vert - \rangle _{n}$ that are dependent on the angle between $\hat{z}$ and $\vec{B}$ (see the top of slide 3).
• We now want to take our quantum state and see what the probability is that the states flips at some later time t. This can be written as

$$\vert \langle out\vert in \rangle \vert ^{2} \; = \; ? $$

• Work through the spin flip and probability of a spin flip sections on slides 3 & 4. Important features:

1. The completeness relation is used to change the basis of the initial state vector.
2. Time evolution is done once the initial state is in the basis of the Hamiltonian.
3. The probability is dependent on $\theta$, time, and the difference in energy between the eigenstates of the Hamiltonian.

• This oscillating of the particle's probability of spin flipping (see slide 5) is called Rabi Oscillation.
• Go over the Rabi formula and how the relative magnitudes of the magnetic vector vields in the $\hat{x}$ and $\hat{z}$ direction affect the probability of a flip (slides 5-6).

Providing the slides in a handout form is often beneficial for students so they can discuss and review the calculations at a later time and at their own pace.