Temporary HW: Difference between revisions

From phys813
Jump to navigationJump to search
No edit summary
Line 22: Line 22:


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 <math> P_{x,y,z} = \langle \sigma_{x,y,z}\rangle =\mathrm{Tr}\, [\hat{\rho} \hat{\sigma}_{x,y,z}] </math>.
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 <math> P_{x,y,z} = \langle \sigma_{x,y,z}\rangle =\mathrm{Tr}\, [\hat{\rho} \hat{\sigma}_{x,y,z}] </math>.


== Problem 2 ==
== Problem 2 ==
The Hamiltonian of a single spin in external magnetic field <math> \mathbf{B} </math> is given by (assuming that gyromagnetic ration is unity):
<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
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>
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>


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

Revision as of 15:54, 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

is can be written as:


Problem 3