Temporary HW: Difference between revisions
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<math> \hat{H}=\frac{\hat{p}^2}{2m} + \frac{m \omega^2 q^2}{2} </math>, | <math> \hat{H}=\frac{\hat{p}^2}{2m} + \frac{m \omega^2 q^2}{2} </math>, | ||
where <math> p = \frac{\hbar}{i} \frac{d}{dq} </math>. | where <math> \hat{p} = \frac{\hbar}{i} \frac{d}{dq} </math>. | ||
(a) Find the partition function <math> Z </math> in quantum canonical ensemble at temperature T. | (a) Find the partition function <math> Z </math> in quantum canonical ensemble at temperature T. |
Revision as of 15:30, 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
Failed to parse (syntax error): {\displaystyle e^{\beta \hat{A}) e^{\beta \hat{B}} = exp[\beta(\hat{A} + \hat{B}) + \beta^2[A,B]/2 + O(\beta^3)] }
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.