Temporary HW: Difference between revisions

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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 \mathbf{\sigma} \right) </math>  


can be recast into the equation of motion for the Bloch polarization vector
can be recast into the equation of motion for the spin-polarization (or Bloch) vector


<math> \frac{d \mathbf{P}}{dt} </math>
<math> \frac{d \mathbf{P}}{dt} = -\mathbf{B} \times \mathbf{P}  </math>


since <math> \mathbf{\rho} </math> and <math> \mathbf{P} </math> are in one-to-one correspondence. Remember that  
since <math> \mathbf{\rho} </math> and <math> \mathbf{P} </math> are in one-to-one correspondence. Remember that  

Revision as of 17:48, 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