Temporary HW: Difference between revisions

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<math> \hat{H} = -\frac{\hbar}{2} \mathbf{B} \cdot \mathbf{\sigma} </math>
<math> \hat{H} = -\frac{\hbar}{2} \mathbf{B} \cdot \mathbf{\sigma} </math>


where <math> \mathbf{\sigma}=(\sigma_x,\sigma_y,\sigma_z) </math> is the vector of the Pauli matrices. Show that the equation of motion for the density matrix of  
where <math> \mathbf{\sigma}=(\sigma_x,\sigma_y,\sigma_z) </math> is the vector of the Pauli matrices. Show that the equation of motion  
spin-<math> \frac{1}{2} </math> discussed in the class
 
<math> \frac{i \hbar \partial \mathbf{\rho}}{\partial t} = [\hat{H},\mathbf{\rho}] </math>
 
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 \mathbf{\sigma} \right) </math>  
<math> \mathbf{\rho} = \frac{1}{2} \left( 1 + \mathbf{P} \cdot \mathbf{\sigma} \right) </math>  


is can be written as:
is can be written as:
<math> \mathbf{\rho} = \frac{1}{2} \left( 1 + \mathbf{P} \cdot \mathbf{\sigma} \right) </math>


<math> \frac{dP}{dt} </math>
<math> \frac{dP}{dt} </math>


== Problem 3 ==
== Problem 3 ==

Revision as of 15:55, 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

is can be written as:

Problem 3