Homework Set 1

Problem 1: Pure vs. mixed quantum states

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 expectation value $P_{x,y,z}=\langle \sigma _{x,y,z}\rangle =\mathrm {Tr} \,[{\hat {\rho }}{\hat {\sigma }}_{x,y,z}]$. The colloquial "spin-polarization" discussed in spintronics literature is $|\mathbf {P} |$ in precise formal language.

Problem 2: Dynamics of Bloch vector

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

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

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

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

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

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

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 $\mathbf {\rho }$ and $\mathbf {P}$ are in one-to-one correspondence.

HINT: Use the following property of the Pauli matrices:

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

Problem 3: Does entropy increase in closed 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:

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

prove that von Neumann entropy

$S(t)=-k_{B}\mathrm {Tr} [{\hat {\rho }}(t)\ln {\hat {\rho }}(t)]$

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

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

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