Temporary HW

Problem 1

A researcher in spintronics is investigated two devices in order to generate spin-polarized currents. One of those devices has spins comprising the current described by the density matrix:

${\hat {\rho }}_{1}={\frac {|\uparrow \rangle \langle \uparrow |+|\downarrow \rangle \langle \downarrow |}{2}}$,

while the spins comprising the current in the other device are described by the density matrix

${\hat {\rho }}_{2}=|u\rangle \langle u|$ , where $\ |u\rangle ={\frac {e^{i\alpha }|\uparrow \rangle +e^{i\beta }|\downarrow \rangle }{\sqrt {2}}}$.

Here $|\uparrow \rangle$ and $|\downarrow \rangle$ are the eigenstates of the Pauli spin matrix ${\hat {\sigma }}_{z}$:

${\hat {\sigma }}_{z}|\uparrow \rangle =+1|\uparrow \rangle ,\ {\hat {\sigma }}_{z}|\downarrow \rangle =-1|\downarrow \rangle$.

What is the spin polarization of these two currents? Comment on the physical meaning of the difference between the spin state transported by two currents.

HINT: Compute the x, y, and z components of the spin polarization vector using both of these density matrices following the quantum-mechanical definition of an average value $P_{x,y,z}=\langle \sigma _{x,y,z}\rangle =\mathrm {Tr} \,[{\hat {\rho }}{\hat {\sigma }}_{x,y,z}]$.

Problem 2

The Hamiltonian of a single spin in external magnetic field $\mathbf {B}$ is given by (assuming that gyromagnetic ration is unity):

${\hat {H}}=-{\frac {\hbar }{2}}\mathbf {B} \cdot \mathbf {\sigma }$

where $\mathbf {\sigma } =(\sigma _{x},\sigma _{y},\sigma _{z})$ is the vector of the Pauli matrices. Show that the equation of motion

$i\hbar {\frac {\partial \mathbf {\rho } }{\partial t}}=[{\hat {H}},\mathbf {\rho } ]$

for the density matrix of spin-${\frac {1}{2}}$ discussed in the class

$\mathbf {\rho } ={\frac {1}{2}}\left(1+\mathbf {P} \cdot \mathbf {\sigma } \right)$

can be written as:

${\frac {dP}{dt}}$