Homework Set 1

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:

${\hat{\rho }}_1=\frac{|\uparrow ⟩⟨\uparrow |+|\downarrow ⟩⟨\downarrow |}{2}$,

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

${\hat{\rho }}_2=|u⟩⟨u|$ , where $|u⟩=\frac{e^{i\alpha }|\uparrow ⟩+e^{i\beta }|\downarrow ⟩}{\sqrt{2}}$.

Here $|\uparrow ⟩$ and $|\downarrow ⟩$ are the eigenstates of the Pauli spin matrix ${\hat{\sigma }}_z$:

${\hat{\sigma }}_z|\uparrow ⟩=+1|\uparrow ⟩,{\hat{\sigma }}_z|\downarrow ⟩=-1|\downarrow ⟩$.

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 $P_{x,y,z}=⟨{\sigma }_{x,y,z}⟩=\mathrm{T}\mathrm{r}[\hat{\rho }{\hat{\sigma }}_{x,y,z}]$.

Problem 2

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

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

where $\overrightarrow{\sigma }=({\hat{\sigma }}_x,{\hat{\sigma }}_y,{\hat{\sigma }}_z)$ is the vector of the Pauli matrices. Show that the equation of motion

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

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

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

can be recast into the equation of motion for the spin-polarization (or Bloch) vector

$\frac{d\mathbf{𝐏}}{dt}=-\mathbf{𝐁}\times \mathbf{𝐏}$

since $\mathit{\rho }$ and $\mathbf{𝐏}$ are in one-to-one correspondence. Remember that

${\hat{\sigma }}_{\alpha }{\hat{\sigma }}_{\beta }-{\hat{\sigma }}_{\beta }{\hat{\sigma }}_{\alpha }=2i{ϵ}_{\alpha \beta \gamma }{\hat{\sigma }}_{\gamma }$.

Problem 3: Does entropy increase in quantum systems?

In classical Hamiltonian systems the nonequilibrium entropy

$S=-k_B\int \rho \ln \rho$

is constant. In this problem we want to demonstrate that in microscopic evolution of an isolated quantum system, the entropy is also time independent, even for general time-dependent density matrix $\hat{\rho }$. That is, using the equation of motion:

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

prove that von Neumann entropy

$S=-k_B\mathrm{T}\mathrm{r}(\hat{\rho }\ln \hat{\rho })$

is time independent for arbitrary density matrix $\hat{\rho }$.

HINT: Use $\mathrm{T}\mathrm{r}(\hat{A}\hat{B}\hat{C})=\mathrm{T}\mathrm{r}(\hat{C}\hat{A}\hat{B})$ for any operators $\hat{A}$, $\hat{B}$, $\hat{C}$, as well as that an operator $\hat{M}$ commutes with any function $f(\hat{M})$:

$[\hat{M},f(\hat{M})]=0$.