Homework Set 2

Problem 1: Canonical partition function for a single spin in magnetic field

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

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

where $\mu _{B}$ is the Bohr magneton, $g$ is the gyromagnetic ratio, and ${\hat {\boldsymbol {\sigma }}}=({\hat {\sigma }}_{x},{\hat {\sigma }}_{y},{\hat {\sigma }}_{z})$ 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: Canonical partition function for two-interacting spins

In some antiferromagnetic materials, such as $\mathrm {TiCuCl_{3}}$, 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:

${\hat {H}}=J{\hat {\mathbf {S} }}_{1}\cdot {\hat {\mathbf {S} }}_{2}+2\mu _{B}B({\hat {S}}_{1}^{z}+{\hat {S}}_{2}^{z})$

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

(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 $T\rightarrow 0$.