(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Consider a spin 1/2 particle placed in a magnetic field [tex]\vec{B_0}[/tex] with components:

[tex] B_x = \frac{1}{\sqrt{2}} B_0 [/tex]

[tex] B_y = 0 [/tex]

[tex] B_z = \frac{1}{\sqrt{2}} B_0 [/tex]

a) Calculate the matrix representing, in the {| + >, | - >} basis, the operator H, the Hamiltonian of the system.

b) Calculate the eigenvalues and the eigenvectors of H.

c) The system at time t = 0 is in the state | - >. What values can be found if the energy is measured, and with what probabilities?

2. Relevant equations

[tex] \omega_0 = - \gamma B_0 [/tex]

[tex] H = \omega_0 S_z [/tex]

[tex] S_z = \frac{\hbar}{2} \[ \left( \begin{array}{cc}

1 & 0 \\

0 & -1 \\ \end{array} \right)\] [/tex]

3. The attempt at a solution

I'm stuck on part a).

My initial instinct is to do this:

[tex] H = \omega_0 S_z [/tex]

[tex] H = - \gamma \vec{B_0} S_z [/tex]

[tex] H = - \gamma \vec{B_0} \frac{\hbar}{2} \[ \left( \begin{array}{cc}

1 & 0 \\

0 & -1 \\ \end{array} \right)\] [/tex]

But [tex]\vec{B_0}[/tex] is a 3D column vector, and I can't multiply that into a 2x2 matrix. And I have to somehow express that with | + > and | - >... I have a feeling I'm on the wrong track.

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# Spin 1/2 particle in B field

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