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.
The basical hamiltonian is on the form: [tex] H = \vec{B} \cdot \vec{S} = B_x \cdot S_x + B_y \cdot S_y + B_z \cdot S_z [/tex] And the [tex] S_x = \frac{1}{2} \sigma _x [/tex] pauli matrix, etc (I use natural units, so dont bother)ยจ I hope my hint helped you anyway to solve a)
So with this I get: [tex] H = \frac{1}{\sqrt{2}} B_0 \frac{\hbar}{2} \[ \left( \begin{array}{cc} 0 & 1 \\ 1 & 0 \\ \end{array} \right)\] [/tex] [tex]+ \frac{1}{\sqrt{2}} B_0 \frac{\hbar}{2} \[ \left( \begin{array}{cc} 1 & 0 \\ 0 & -1 \\ \end{array} \right)\] [/tex] [tex] H = B_0 \hbar \[ \left( \begin{array}{cc} \frac{1}{2\sqrt{2}} & \frac{1}{2\sqrt{2}} \\ \frac{1}{2\sqrt{2}} & \frac{-1}{2\sqrt{2}} \\ \end{array} \right)\] [/tex] Right?
For the eigenvalue: [tex] (\frac{B_0 \hbar}{2\sqrt2})^2 (1-\lambda)(1-\lambda) - (\frac{B_0 \hbar}{2\sqrt2})^2 = 0 [/tex] [tex](1-\lambda)(1-\lambda) = 0[/tex] [tex] \lambda = 1 [/tex] And the eigenvector: [tex]\frac{B_0 \hbar} {2\sqrt2} \[ \left( \begin{array}{cc} 0 & 1 \\ 1 & -2 \\ \end{array} \right)\] \times \[ \left(\begin{array}{c} c_1 \\ c_2 \\ \end{array} \right)\] = 0 [/tex] This gives me [tex] c_2 = 0 [/tex] and [tex] c_1 = \frac{B_0 \hbar} {2\sqrt2} [/tex]. I think I did something wrong.
for matrix: [tex] A = \[ \left( \begin{array}{cc} a & a\\ a & -a \\ \end{array} \right)\] [/tex] The secular eq is [tex] (a- \lambda )(-a- \lambda ) - a^2 = 0 [/tex] if lambda is the eigenvalue.
Ok so I get the following system of equations for the case of the eigenvalue [tex]+\sqrt2[/tex] [tex](1-\sqrt2)c_1 + c_2 = 0[/tex] [tex]c_1 + (-1-\sqrt2)c_2 = 0 [/tex] which according to myself and my calculator has no solution....
If you can get eigenvalues to a matrix, then there exists corresponding eigenvectors. Eigen vectors are, by using Matlab: for sqrt2 = (-0.92388,-0.38268) for -sqrt2 = (0.38268,-0.92388)
Ok I found the two eigenvectors using Matlab. I'm not sure how to write them when applied to the system so that it makes sense. [tex] |\psi(t)> = 0.92388 | + > + 0.382683 | - > [/tex] [tex] |\psi(t)> = 0.382683 | + > - 0.92388 | - > [/tex] Is this correct? And for part c), which one do I use to find the probability?
But in order to get them analytically, just do substituion [tex]c_1 = (1+\sqrt2)c_2[/tex] From your second equation and put in into the first one and solve for c_2 Now how does a state evolve with time? Ever heard of "Time evolution operator" or similar? Time evolution of a ket is [tex] |a(t) \rangle = \exp (-i E_a t/\hbar)|a(0) \rangle [/tex] where E_a is the energy eigenvalue of that ket. So your egeinvectors are: [tex] |\psi +> = 0.92388 | + > + 0.382683 | - > [/tex] [tex] |\psi -> = 0.382683 | + > - 0.92388 | - > [/tex] Dont use time, as you did, it is not correct. The psi + has eigenvalue +sqrt2 etc. Now I have helped you very much.
no, just for the hamiltonian. at time = 0; the state is in |-> Then you must find out what just |-> is in superposition of the eigenvectors to the hamiltonian, in order to get the time evolution. i.e you should first write |-> = a|phi + > + b|phi ->