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:
,
while the spins comprising the current in the other device are described by the density matrix
, where .
Here and are the eigenstates of the Pauli spin matrix :
.
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 . The colloquial "spin-polarization" discussed in spintronics literature is in precise formal language.
Problem 2: Dynamics of Bloch vector
The Hamiltonian of a single spin of an electron in external magnetic field is given by (assuming that gyromagnetic ration is unity):
where is the vector of the Pauli matrices. Show that the equation of motion
for the density matrix of spin- discussed in the class
can be recast into the equation of motion for the spin-polarization (or Bloch) vector
since and are in one-to-one correspondence.
HINT: Use the following property of the Pauli matrices:
.
Problem 3: Does entropy increase in closed quantum systems?
In classical Hamiltonian systems the nonequilibrium entropy
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 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 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} } . That is, using the equation of motion:
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 i \hbar \frac{\partial \hat{\rho}}{\partial t} = [\hat{H},\hat{\rho}] }
prove that von Neumann entropy
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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 } .