Temporary HW: Difference between revisions

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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 ==
== Problem 3 ==

Revision as of 17:50, 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