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) Compute the free energy of $N$ dimers in zero magnetic field and find their entropy.

(b) Using your result in (a) and the thermodynamic formula for average magnetization $M=-\partial F/\partial B$, compute the zero-field magnetic susceptibility $\chi =(\partial M/\partial B){|}_{B=0}$.

Problem 3: Canonical partition function and density matrix of 1D 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 $T\to 0$.