Temporary HW: Difference between revisions
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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 | (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 | ||
<math> e^{\beta \hat{A} | <math> e^{\beta \hat{A}} e^{\beta \hat{B}} = exp[\beta(\hat{A} + \hat{B}) + \beta^2 [\hat{A},\hat{B}]/2 + O(\beta^3)] </math> | ||
which you can apply to the Boltzmann operator: | which you can apply to the Boltzmann operator: |
Revision as of 15:31, 23 February 2011
Problem 1
The Hamiltonian for an electron spin degree of freedom in the external magnetic field is given by:
where is the Bohr magneton </math> 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:
Failed to parse (unknown function "\q"): {\displaystyle exp(-beta\hat{H}) = exp(-\beta \frac{\hat{p}^2}{2m} - \beta \frac{m\omega^2\q^2}{2}) <math> while neglecting terms of order <math> \beta^2 } and higher since is very small in this limit.