Homework Set 2: Difference between revisions

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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>


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:


<math> \hat{\sigma}_x =  
<math> \hat{\sigma}_x =  

Revision as of 15:42, 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 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 .