Homework Set 2

Problem 1: Canonical partition function for a single non-interacting spin

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

$\hat{H}=-g{\mu }_B\hat{\mathit{\sigma }}\cdot \mathbf{𝐁}$

where ${\mu }_B$ is the Bohr magneton, $g$ is the gyromagnetic ratio, and $\hat{\mathit{\sigma }}=({\hat{\sigma }}_x,{\hat{\sigma }}_y,{\hat{\sigma }}_z)$ is the vector of the Pauli matrices:

${\hat{\sigma }}_x=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right),$

${\hat{\sigma }}_y=\left(\begin{array}{cc}0& -i\\ i& 0\end{array}\right),$

${\hat{\sigma }}_z=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right).$

(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: Canonical partition function for two-interacting spins

In some antiferromagnetic materials, such as $\mathrm{T}\mathrm{i}\mathrm{C}\mathrm{u}\mathrm{C}{\mathrm{l}}_{\mathrm{3}}$, spins $S=1/2$ are arranged in pairs. To first approximation, such dimers can be considered independently of each other. The Hamiltonian of a single dimer in the external magnetic field $\mathbf{𝐁}=(0,0,B)$ is given by:

$\hat{H}=J{\widehat{\mathbf{𝐒}}}_1\cdot {\widehat{\mathbf{𝐒}}}_2+2{\mu }_{B}B({\hat{S}}_1^z+{\hat{S}}_2^z)$

where $J>0$ is the exchange coupling constant and ${\mu }_B$ is the Bohr magneton.

(a) How many energy eigenlevels does this Hamiltonian have? List all eigenergies explicitly.

(b) Using your result in (a), compute the canonical partition function and free energy of $N$ dimers, as well as find their entropy.

NOTE: An equivalent and more pedagogical expression for the Hamiltonian above is given by:

$\hat{H}=J({\widehat{\mathbf{𝐒}}}_1\cdot {\widehat{\mathbf{𝐒}}}_2$ is given by ${\hat{S}}_1^x\otimes {\hat{S}}_2^x+{\hat{S}}_1^y\otimes {\hat{S}}_2^y+{\hat{S}}_1^z\otimes {\hat{S}}_2^z)+2{\mu }_{B}B({\hat{S}}_1^z\otimes \hat{I}+\hat{I}\otimes {\hat{S}}_2^z)$.

Problem 3: Density matrix and canonical partition function for one-dimensional harmonic oscillator

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

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

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

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

(d) Using the result in (c), write down the density matrix in the coordinate representation $⟨q^{\prime }|\hat{\rho }|q⟩$.

(e) In the coordinate representation, calculate explicitly $⟨q^{\prime }|\hat{\rho }|q⟩$ in the high temperature limit $T\to \mathrm{\infty }$.

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 $⟨q|0⟩$ only, evaluate the limiting behavior of $⟨q^{\prime }|\hat{\rho }|q⟩$ as $T\to 0$.