How Does a Magnetic Field Affect Electron Spin Over Time?

[noospace]

Homework Statement

An +x-polarized electron beam is subjected to magnetic field in the y-direction. What is the probablity of measuring spin +x after a period of time t.

Homework Equations

Time evolution operator $U = e^{-i/\hbar \hat{H} t}$

The Attempt at a Solution

Since the magnetic field is in the y-direction, the corresponding Hamiltonian is of the form $\hat{H} = - \gamma B_0 \hat{S}_y$. The energy eigenvalues of this are just $-\gamma B_0$ times the eigenvalues for the Spin-y operator, ie $\pm \frac{\gamma B_0 \hbar}{2}$ where $|S_y ; + \rangle =\frac{1}{\sqrt{2}}(-i,1)^T, |S_y,- \rangle = \frac{1}{\sqrt{2}}(i,1)$.

$|S_x;+ \rangle = 2\left(\frac{1+i}{4}|S_y;+ \rangle + \frac{1-i}{4}|S_y;-\rangle\right)$

so the time evolved state vector is

$|S_x;+ \rangle = 2\left(\frac{1+i}{4} e^{i t\gamma B_0 \hbar/2}|S_y;+ \rangle + \frac{1-i}{4}e^{-i t\gamma B_0 \hbar/2}|S_y;-\rangle\right)$

$|S_x;+ \rangle = \cos (t \gamma B_0 \hbar/2)2\left(\frac{1+i}{4} |S_y;+ \rangle + \frac{1-i}{4}|S_y;-\rangle\right)+ i\sin(t \gamma B_0 \hbar/2)2\left(\frac{1+i}{4} |S_y;+ \rangle -\frac{1-i}{4}|S_y;-\rangle\right)$

so the probability is

$\cos^2 (t \gamma B_0 \hbar/2)$.

 

[Jhero]

Thank you for your question. I am always happy to help with any inquiries related to physics.

To answer your question, the probability of measuring spin +x after a period of time t can be calculated using the formula \cos^2 (t \gamma B_0 \hbar/2). This is derived from the time evolution operator U = e^{-i/\hbar \hat{H} t}, where the Hamiltonian \hat{H} = - \gamma B_0 \hat{S}_y represents the magnetic field in the y-direction.

By using the energy eigenvalues for the Spin-y operator, we can determine the time evolved state vector |S_x;+ \rangle and ultimately calculate the probability of measuring spin +x after time t.

I hope this helps answer your question. Please let me know if you have any further inquiries or need clarification on any of the calculations. Keep up the good work as a scientist!