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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 \vec{\sigma} </math>
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| where <math> \vec{\sigma}=(\hat{\sigma}_x,\hat{\sigma}_y,\hat{\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 \hat{\rho}}{\partial t} = [\hat{H},\hat{\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> \hat{\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
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| <math> \frac{d \mathbf{P}}{dt} = -\mathbf{B} \times \mathbf{P} </math>
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| since <math> \mathbf{\rho} </math> and <math> \mathbf{P} </math> are in one-to-one correspondence. Remember that
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| <math> \hat{\sigma}_i \hat{\sigma}_j - \hat{\sigma}_j \hat{\sigma}_i = 2 i \epsilon_{ijk} \hat{\sigma}_k </math>.
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| == Problem 3: Does entropy increase in quantum systems? ==
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| In classical Hamiltonian systems the nonequilibrium entropy
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| <math> S = -k_B \int \rho \ln \rho </math>
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| is constant. In this problem we want to demonstrate that in microscopic evolution of an isolated quantum system, the entropy is also time independent, even
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| for general time-dependent density matrix <math> \hat{\rho} </math>. That is, using the equation of motion:
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| <math> i \hbar \frac{\partial \hat{\rho}}{\partial t} = [\hat{H},\hat{\rho}] </math>
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| prove that von Neumann entropy
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| <math> S=-k_B \mathrm{Tr}(\hat{\rho} \ln \hat{\rho}) </math>
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| is time independent for arbitrary density matrix <math> \hat{\rho} </math>.
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| HINT: Use <math> \mathrm{Tr}(\hat{A}\hat{B}\hat{C})=\mathrm{Tr}(\hat{C}\hat{A}\hat{B}) </math> for any operators <math> \hat{A} </math>, <math> \hat{B} </math>, <math> \hat{C} </math>, as well as that an operator <math> \hat{M} </math> commutes with any function <math> f(\hat{M}) </math>:
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| <math> [\hat{M},f(\hat{M})]=0 </math>.
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