Temporary HW: Difference between revisions
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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: