Homework Set 2

Problem 1

The Hamiltonian for an electron spin degree of freedom in the external magnetic field $\mathbf {B}$ is given by:

${\hat {H}}=-\mu _{B}{\vec {\sigma }}\cdot \mathbf {B}$

where $\mu _{B}$ is the Bohr magneton and ${\vec {\sigma }}=({\hat {\sigma }}_{x},{\hat {\sigma }}_{y},{\hat {\sigma }}_{z})$ is the vector of the Pauli matrices:

${\hat {\sigma }}_{x}={\begin{pmatrix}0&1\\1&0\end{pmatrix}},$

${\hat {\sigma }}_{y}={\begin{pmatrix}0&-i\\i&0\end{pmatrix}},$

${\hat {\sigma }}_{z}={\begin{pmatrix}1&0\\0&-1\end{pmatrix}}.$

(a) In the quantum canonical ensemble, evaluate the density matrix if $\mathbf {B}$ is along the z axis.

(b) Repeat the calculation from (a) assuming that $\mathbf {B}$ 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:

${\hat {H}}={\frac {{\hat {p}}^{2}}{2m}}+{\frac {m\omega ^{2}q^{2}}{2}}$,

where ${\hat {p}}=-i{\hbar }{\frac {d}{dq}}$.

(a) Find the partition function $Z$ in the quantum canonical ensemble at temperature $T$.

(b) Using the result from (a), calculate the averge energy $E=\langle {\hat {H}}\rangle$.

(c) Write down the formal expression for the canonical density operator ${\hat {\rho }}$ in terms of the eigenstates $|n\rangle$ of the Hamiltonian and the corresponding energy levels $\varepsilon _{n}=\hbar \omega (n+1/2)$.

(d) Using the result in (c), write down the density matrix in the coordinate representation $\langle q'|{\hat {\rho }}|q\rangle$.

(e) In the coordinate representation, calculate explicitly $\langle q'|{\hat {\rho }}|q\rangle$ in the high temperature limit $T\rightarrow \infty$.

HINT: One approach is to utilize the following result

$e^{\beta {\hat {A}}}e^{\beta {\hat {B}}}=e^{\beta ({\hat {A}}+{\hat {B}})+\beta ^{2}[{\hat {A}},{\hat {B}}]/2+O(\beta ^{3})}$

which you can apply to the Boltzmann operator:

$e^{-\beta {\hat {H}}}=e^{-\beta {\frac {{\hat {p}}^{2}}{2m}}-\beta {\frac {m\omega ^{2}q^{2}}{2}}}$

while neglecting terms of order $\beta ^{2}$ and higher since $\beta$ is very small in the high temperature limit.

(f) At low temperatures, ${\hat {\rho }}$ is dominated by low-energy states. Use the ground state wave function $\langle q|0\rangle$ only, evaluate the limiting behavior of $\langle q'|{\hat {\rho }}|q\rangle$ as $T\rightarrow 0$.