Temporary HW: Difference between revisions
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where <math> \vec{\sigma}=(\hat{\sigma}_x,\hat{\sigma}_y,\hat{\sigma}_z) </math> is the vector of the Pauli matrices. Show that the equation of motion | where <math> \vec{\sigma}=(\hat{\sigma}_x,\hat{\sigma}_y,\hat{\sigma}_z) </math> is the vector of the Pauli matrices. Show that the equation of motion | ||
<math> i \hbar \frac{\partial \ | <math> i \hbar \frac{\partial \hat{\rho}}{\partial t} = [\hat{H},\hat{\rho}] </math> | ||
for the density matrix of spin-<math> \frac{1}{2} </math> discussed in the class | for the density matrix of spin-<math> \frac{1}{2} </math> discussed in the class | ||
<math> \ | <math> \hat{\rho} = \frac{1}{2} \left( 1 + \mathbf{P} \cdot \vec{\sigma} \right) </math> | ||
can be recast into the equation of motion for the spin-polarization (or Bloch) vector | can be recast into the equation of motion for the spin-polarization (or Bloch) vector | ||
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<math> \hat{\sigma}_i \hat{\sigma}_j - \hat{\sigma}_j \hat{\sigma}_i = 2 i \epsilon_{ijk} \hat{\sigma}_k </math>. | <math> \hat{\sigma}_i \hat{\sigma}_j - \hat{\sigma}_j \hat{\sigma}_i = 2 i \epsilon_{ijk} \hat{\sigma}_k </math>. | ||
== Problem 3: Does entropy increase in quantum systems? == | == Problem 3: Does entropy increase in quantum systems? == |
Revision as of 17:54, 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: