How Do Heisenberg Equations Describe Spin Dynamics in a Magnetic Field?

[BOAS]

Homework Statement

Consider a spin $\frac{1}{2}$ particle at rest in a B-field $\vec B = B_0\vec e_z$.

(a) Find the Heisenberg equations for $\hat S_x$ and $\hat S_y$.

(b) Obtain from the Heisenberg equations two decoupled second-order differential equations for $\langle \hat S_x \rangle_{\psi}$ and $\langle \hat S_y \rangle_{\psi}$ for a general state $| \psi \rangle$ of the particle.

(c) Solve the equations for a particle that initially (at t = 0) is in a state of $S_x = \frac{\hbar}{2}$

Homework Equations

The Attempt at a Solution

For part (a) I have done the following;

$\langle \psi (t) | \hat B_S | \psi (t) \rangle = \langle e^{-\frac{i \hat H t}{\hbar}} \psi (0) | \hat B_S | e^{-\frac{i \hat H t}{\hbar}} \psi (0) \rangle = \langle \psi (0) | e^{\frac{i \hat H t}{\hbar}} \hat B_S e^{-\frac{i \hat H t}{\hbar}} | \psi (0) \rangle$

$B_H = e^{\frac{i \hat H t}{\hbar}} \hat B_S e^{-\frac{i \hat H t}{\hbar}}$

where the H and S subscripts correspond to the Heisenberg and Schrödinger pictures respectively.

The hamiltonian that I have is $\hat H = - \gamma B_0 \hat S_z$

which leads me to the conclusion that my spin operators are unchanged in the Heisenberg picture.

for part (b) I use the Heisenberg equation of motion on both the $\hat S_x$ and $\hat S_y$ operators

$\frac{d}{dt} \hat S_x = \frac{i}{\hbar} [\hat H, \hat S_x] = \gamma B_0 \hat S_y$

$\frac{d}{dt} \hat S_y = \frac{i}{\hbar} [\hat H, \hat S_y] = - \gamma B_0 \hat S_x$

I then differentiate both again, and substitute the original equations into the result to find

$\frac{d^2}{dt^2} \hat S_y = -(\gamma B_0)^2 \hat S_y$

$\frac{d^2}{dt^2} \hat S_x = -(\gamma B_0)^2 \hat S_x$

These two equations are SHM equations with solutions of the form $y = A \cos \omega t + B \sin \omega t$

solving these I find that $\hat S_y (t) = \hat S_y (0) \cos (\gamma B_0 t) - \hat S_x (0) \sin (\gamma B_0 t)$ and $\hat S_x (t) = \hat S_x (0) \cos (\gamma B_0 t) + \hat S_y (0) \sin (\gamma B_0 t)$

I don't really understand what the question is trying to get me to do here. Any guidance would be much appreciated.

 

[DrDu]

Looks all ok, I think the only thing you are assumed still to do is to form the expectations values with the psi(0) specified.

 

[BOAS]

Looks all ok, I think the only thing you are assumed still to do is to form the expectations values with the psi(0) specified.

I'm still having trouble solving this. I can't see how to find $\hat S_y (0)$

 

[DrDu]

At t=0, the operators in the Schroedinger and Heisenberg picture coincide.

 

[BOAS]

At t=0, the operators in the Schroedinger and Heisenberg picture coincide.

Thanks, I see.

Ok, so I know that $\psi (s_x = \hbar / 2) = \frac{1}{\sqrt{2}} \begin{pmatrix} 1\\ 1 \end{pmatrix}$. Does "solving" these equations mean taking the expectation value for my state?

 

[DrDu]

I think so, what do you get?

 

[BOAS]

I think so, what do you get?

I find that $\langle \hat S_y \rangle_{\psi} = - \frac{\hbar}{2} \sin (\gamma B_0 t)$ and $\langle \hat S_x \rangle_{\psi} = \frac{\hbar}{2} \cos (\gamma B_0 t)$

 

[DrDu]

I find that $\langle \hat S_y \rangle_{\psi} = - \frac{\hbar}{2} \sin (\gamma B_0 t)$ and $\langle \hat S_x \rangle_{\psi} = \frac{\hbar}{2} \cos (\gamma B_0 t)$

Looks good!

 

[BOAS]

Looks good!

Thank you for your help