Temporary HW: Difference between revisions

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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.
while neglecting terms of order <math> \beta^2 </math>  and higher since <math> \beta </math> is very small in this limit.
(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>.

Revision as of 15:35, 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:

while neglecting terms of order and higher since is very small in this limit.

(f) At low temperatures, Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \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 } as .