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| == Problem 1 ==
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| The Hamiltonian for an electron spin degree of freedom in the external magnetic field <math> \mathbf{B} </math> is given by:
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| <math> \hat{H} = - \mu_B \vec{\sigma} \cdot \mathbf{B} </math>
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| where <math> \mu_B </math> is the Bohr magneton </math> and <math> \vec{\sigma}=(\hat{\sigma}_x,\hat{\sigma}_y,\hat{\sigma}_z) </math> is the vector of the Pauli matrices:
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| <math> \hat{\sigma}_x =
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| \begin{pmatrix}
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| 0 & 1 \\
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| 1 & 0
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| \end{pmatrix},
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| </math>
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| <math> \hat{\sigma}_x =
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| \begin{pmatrix}
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| 0 & -i \\
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| i & 0
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| \end{pmatrix},
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| </math>
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| <math> \hat{\sigma}_x =
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| \begin{pmatrix}
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| 1 & 0 \\
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| 0 & -1
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| \end{pmatrix}.
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| </math>
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| (a) In the quantum canonical ensemble, evaluate the density matrix if <math> \mathbf{B} </math> is along the ''z'' axis.
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| (b) Repeat the calculation from (a) assuming that <math> \mathbf{B} </math> points along the ''x'' axis.
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| (c) Calculate the average energy in each of the above cases.
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| == Problem 2 ==
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| Consider a single one-dimensional quantum harmonic oscillator described by the Hamiltonian:
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| <math> \hat{H}=\frac{\hat{p}^2}{2m} + \frac{m \omega^2 q^2}{2} </math>,
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| where <math> \hat{p} = \frac{\hbar}{i} \frac{d}{dq} </math>.
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| (a) Find the partition function <math> Z </math> in quantum canonical ensemble at temperature T.
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| (b) Using result from (a), calculate the averge energy <math> E = \langle \hat{H} \rangle </math>.
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| (c) Write down the formal expression for the canonical density operator <math> \hat{\rho} </math> in terms of eigenstates <math> |n\rangle </math> and
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| energy levels <math> \varepsilon_n = \hbar \omega (n + 1/2) </math>.
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| (d) Using result in (c), write down the density matrix in a coordinate representation <math> \langle q' |\hat{\rho}|q\rangle </math>.
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| (e) In the coordinate representation, calculate explicitly <math> \langle q' |\hat{\rho}|q\rangle </math> in the high temperature limit <math> T \rightarrow \infty </math>. HINT: One approach is to apply the following result
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| <math> e^{\beta \hat{A}} e^{\beta \hat{B}} = e^{\beta(\hat{A} + \hat{B}) + \beta^2 [\hat{A},\hat{B}]/2 + O(\beta^3)} </math>
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| which you can apply to the Boltzmann operator:
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| <math> e^{-\beta \hat{H}} = e^{ -\beta \frac{\hat{p}^2}{2m} - \beta \frac{m\omega^2 q^2}{2} } </math>
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| while neglecting terms of order <math> \beta^2 </math> and higher since <math> \beta </math> is very small in this limit.
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| (f) At low temperatures, <math> \hat{\rho} <\math> is dominated by low-energy states. Use the ground state wave function to evaluate the limiting behavior of <math> \langle q' |\hat{\rho}|q\rangle </math> as <math> T \rightarrow 0 </math>.
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