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 an average value $P_{x,y,z}=\langle \sigma _{x,y,z}\rangle =\mathrm {Tr} \,[{\hat {\rho }}{\hat {\sigma }}_{x,y,z}]$.

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 Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \hat{\rho}(t) } .

HINT: Use Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathrm{Tr}(\hat{A}\hat{B}\hat{C})=\mathrm{Tr}(\hat{C}\hat{A}\hat{B}) } for any operators Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \hat{A} } , Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \hat{B} } , Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \hat{C} } , as well as that an operator Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \hat{M} } commutes with any function Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f(\hat{M}) } :

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle [\hat{M},f(\hat{M})]=0 } .

Problem 4: Successive measurements on subsystems of composite bipartite quantum system

Consider a composite quantum system composed of two spins, labeled as subsystem A and B. The quantum state of the composite system is described by the following density matrix:

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \hat{\rho} = \frac{1}{8} \hat{I} + \frac{1}{2} |\Psi\rangle \langle \Psi| }

where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \hat{I} } denotes the Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 4 \times 4 } unit matrix and

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle |\Psi \rangle = \frac{1}{\sqrt{2}}\left( |\uparrow\rangle |\downarrow \rangle - |\downarrow \rangle |\uparrow \rangle \right) }

is entangled (in the context of spins also called "singlet") pure state of two spins.

Suppose we measure the first spin (subsystem A) along the axis described by the unit vector Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathbf{n} } , and the second spin (subsystem B) along the axis described by the unit vector Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathbf{m} } , where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathbf{n} \cdot \mathbf{m} = \cos \theta } . What is the probability that both spins are "spin-up" along their respective axes?

HINT: In general, the probability to measure eigenvalue Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \lambda } of a physical quantity in the quantum state described by the density matrix Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \hat{\rho} } is given by Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle p = \mathrm{Tr}[ \hat{\rho} \hat{P}_\lambda]} . Here Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \hat{P}_\lambda } is the projection operator on the eigensubspace corresponding to eigenvalue Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \lambda } . To find the probability of measurement on the subsystem, one should use the density matrix of that subsystem, obtained by partial trace over the states of the second subsystem. This means that the probability asked in the problem is formally obtained from:

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle p = \mathrm{Tr}_B \left[ \hat{P}_\mathbf{m}^B \hat{\rho}^B \right] = \mathrm{Tr}_B \left[ \hat{P}_\mathbf{m}^B \mathrm{Tr}_A \left[(\hat{P}_\mathbf{n}^A \otimes \hat{I}^B) \hat{\rho} \right] \right] }

The eigenprojector for the "spin-up" (i.e., +1) eigenvalue along the Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathbf{n} } -axis is simply:

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \hat{P}_\mathbf{n}^A = |\uparrow_\mathbf{n} \rangle \langle \uparrow_\mathbf{n}| = \frac{1}{2} \left( \hat{I}^A + \mathbf{n} \cdot \hat{\boldsymbol{\sigma}}^A \right) } .

We also use the fact that resulting state of the composite system after the selective measurement on subsystem A is described by Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \hat{P}_\mathbf{n}^A \hat{\rho} \hat{P}_\mathbf{n}^A } , so that subsystem B after the measurement on system A is "collapsed" onto the state described by the density matrix Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \hat{\rho}^B = \mathrm{Tr}_A [\hat{P}_\mathbf{n}^A \hat{\rho} \hat{P}_\mathbf{n}^A]= \mathrm{Tr}_A [\hat{P}_\mathbf{n}^A \hat{\rho}] } .