Temporary HW

Problem 1

The Hamiltonian for an electron spin degree of freedom in the external magnetic field $\mathbf{𝐁}$ is given by:

$\hat{H}=-{\mu }_B\overrightarrow{\sigma }\cdot \mathbf{𝐁}$

where ${\mu }_B$ is the Bohr magneton </math> and $\overrightarrow{\sigma }=({\hat{\sigma }}_x,{\hat{\sigma }}_y,{\hat{\sigma }}_z)$ is the vector of the Pauli matrices:

${\hat{\sigma }}_x=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right),$

${\hat{\sigma }}_x=\left(\begin{array}{cc}0& -i\\ i& 0\end{array}\right),$

${\hat{\sigma }}_x=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right).$

(a) In the quantum canonical ensemble, evaluate the density matrix if $\mathbf{𝐁}$ is along the z axis.

(b) Repeat the calculation from (a) assuming that $\mathbf{𝐁}$ points along the x axis.

(c) Calculate the average energy in each of the above cases.