Temporary HW: Difference between revisions

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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  
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 <math> \hat{\rho} </math>. That is, using the equation of motion:
for general time-dependent density matrix <math> \hat{\rho} </math>. That is, using the equation of motion:
<math> i \hbar \frac{\partial \hat{\rho}}{\partial t} = [\hat{H},\hat{\rho}] </math>
prove that von Neumann entropy
<math> S=-k_B \mathrm{Tr}(\hat{\rho} \ln \hat{\rho}) </math>
is time independent for arbitrary density matrix <math> \hat{\rho} </math>.
HINT: Use <math> \mathrm{Tr}(\hat{A}\hat{B}\hat{C}})=\mathrm{Tr}(\hat{C}\hat{A}\hat{B}}) </math> for any operators <math> \hat{A} </math>, <math> \hat{B} </math>, <math> \hat{C} </math>, as well as that an operator <math> \hat{M} </math> commutes with any function <math> f(\hat{M}) </math>:
<math> [\hat{M},f(\hat{M})]=0 </math>.

Revision as of 18:00, 16 February 2011

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:


,


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 an average value .


Problem 2

The Hamiltonian of a single spin 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. Remember that

.

Problem 3: Does entropy increase in quantum systems?

In classical Hamiltonian systems the nonequilibrium entropy

Failed to parse (unknown function "\n"): {\displaystyle S = -k_B \int \rho \n \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 . That is, using the equation of motion:

prove that von Neumann entropy

is time independent for arbitrary density matrix .

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 , , , as well as that an operator commutes with any function :

.