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| == Problem 1 ==
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| 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:
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| <math> \hat{\rho}_1 = \frac{|\uparrow \rangle \langle \uparrow| + |\downarrow \rangle \langle \downarrow|}{2} </math>,
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| while the spins comprising the current in the other device are described by the density matrix
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| <math> \hat{\rho}_2 = |u \rangle \langle u|</math> , where <math> \ |u\rangle = \frac{e^{i\alpha} |\uparrow\rangle + e^{i\beta}|\downarrow\rangle}{\sqrt{2}}</math>.
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| Here <math> |\uparrow\rangle </math> and <math> |\downarrow\rangle </math> are the eigenstates of the Pauli spin matrix <math> \hat{\sigma}_z </math>:
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| <math> \hat{\sigma}_z |\uparrow \rangle = +1 |\uparrow \rangle, \ \hat{\sigma}_z |\downarrow \rangle = -1 |\downarrow \rangle </math>.
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| 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.
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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>.
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| == Problem 2 ==
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| The Hamiltonian of a single spin in external magnetic field <math> \mathbf{B} </math> is given by (assuming that gyromagnetic ration is unity):
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| <math> \hat{H} = -\frac{\hbar}{2} \mathbf{B} \cdot \mathbf{\sigma} </math>
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| where <math> \mathbf{\sigma}=(\sigma_x,\sigma_y,\sigma_z) </math> is the vector of the Pauli matrices. Show that the equation of motion
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| <math> i \hbar \frac{\partial \mathbf{\rho}}{\partial t} = [\hat{H},\mathbf{\rho}] </math>
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| 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>
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| can be written as:
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| <math> \frac{dP}{dt} </math>
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| == Problem 3 ==
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