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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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