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 \mathbf{\rho}}{\partial t} = [\hat{H},\mathbf{\rho}] </math>  
<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> \mathbf{\rho} = \frac{1}{2} \left( 1 + \mathbf{P} \cdot \vec{\sigma} \right) </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 (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 \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: