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| <math> \hat{H} = - \mu_B \vec{\sigma} \cdot \mathbf{B} </math> | | <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: | | where <math> \mu_B </math> is the Bohr magneton 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 = | | <math> \hat{\sigma}_x = |
Problem 1
The Hamiltonian for an electron spin degree of freedom in the external magnetic field
is given by:
where
is the Bohr magneton and
is the vector of the Pauli matrices:
(a) In the quantum canonical ensemble, evaluate the density matrix if
is along the z axis.
(b) Repeat the calculation from (a) assuming that
points along the x axis.
(c) Calculate the average energy in each of the above cases.
Problem 2
Consider a single one-dimensional quantum harmonic oscillator described by the Hamiltonian:
,
where
.
(a) Find the partition function
in quantum canonical ensemble at temperature T.
(b) Using result from (a), calculate the averge energy
.
(c) Write down the formal expression for the canonical density operator
in terms of eigenstates
and
energy levels
.
(d) Using result in (c), write down the density matrix in a coordinate representation
.
(e) In the coordinate representation, calculate explicitly
in the high temperature limit
. HINT: One approach is to apply the following result
which you can apply to the Boltzmann operator:
while neglecting terms of order
and higher since
is very small in this limit.
(f) At low temperatures,
is dominated by low-energy states. Use the ground state wave function
only, evaluate the limiting behavior of
as
.