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| 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 |
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| <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 \vec{\sigma} \right) </math> |
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| 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 |
Revision as of 17:49, 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