Homework Set 2

Problem 1: Canonical partition function for a single spin-1/2

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 canonical ensemble, find matrix representation of the density operator ${\hat {\rho }}_{\mathrm {eq} }=e^{-\beta {\hat {H}}}/Z$ for $\mathbf {B}$ 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: Bloch equation for thermal relaxation and decoherence of a two-level system

The Bloch equation is widely used phenomenological tool to understand relaxation toward thermal equilibrium and decoherence of two-level systems like qubits in quantum computing, spin-1/2 of electrons in spintronics and nuclear spins in medical physics. For example, in MRI imaging of brain and spine tissue is characterized by two different relaxation times, T₁ and T₂. T₁ is the so-called longitudinal relaxation time which determines the rate at which excited proton spins return to equilibrium orientation. It is a measure of the time taken for spinning protons to realign with the external magnetic field. T₂ is the so-called transverse relaxation time (or decoherence time in modern terminology) which determines the rate at which excited protons spins go out of phase with each other. It is a measure of the time taken for spinning protons to lose phase coherence among the nuclei spinning perpendicular to the main field.

Using the language of spin-1/2 in an external magnetic field, the Bloch equation can be written as:

${\frac {\partial {\hat {\rho }}}{\partial t}}=-{\frac {i}{\hbar }}[{\hat {H}},{\hat {\rho }}]-{\frac {1}{T_{1}}}({\hat {\rho }}-{\hat {\rho }}_{\mathrm {eq} })-{\frac {1}{T_{2}}}{\begin{pmatrix}0&\rho _{\uparrow \downarrow }\\\rho _{\downarrow \uparrow }&0\end{pmatrix}},$

where ${\hat {\rho }}_{\mathrm {eq} }$ is the same density matrix in canonical ensemble studied in Problem 1 and ${\hat {H}}=-g\mu _{B}{\hat {\sigma }}_{z}B$ is the Hamiltonian for the z-axis chosen along the magnetic field. Note that although Bloch did not specify the physical mechanisms giving rise to T₁ and T₂, he did correctly conclude that T₁ must result from thermal agitation and T₂ from interaction of single spin with neighboring spins.

(a) Using the fact that arbitrary (pure or mixed) quantum state of spin-1/2 can always be written as:

${\hat {\rho }}={\frac {1}{2}}\left({\hat {I}}+\mathbf {P} \cdot {\boldsymbol {\sigma }}\right)$,

show that the Bloch equation can be re-written as differential equation for the Bloch vector $\mathbf {P}$:

${\frac {\partial \mathbf {P} }{\partial t}}={\hat {A}}\mathbf {P} +\mathbf {C}$,

where you should find explicit form of ${\hat {A}}$ as a 3 x 3 matrix and $\mathbf {C} \in {\rm {I\!R}}^{3}$ as a vector.

(b) Solve differential equation for $\mathbf {P}$ assuming the initial condition $\mathbf {P} (t=0)\equiv (0,1,0)$. For typical situation $T_{1}\gg T_{2}\gg \hbar /(g\mu _{B}B)$, plot your solution using Mathematica and comment on its physical meaning. You should obtain a spiral curve resembling a "tent" show below:

Problem 3: 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) 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.

HINT: An equivalent pedagogical expression for the Hamiltonian above is given by:

${\hat {H}}=J({\hat {S}}_{1}^{x}\otimes {\hat {S}}_{2}^{x}+{\hat {S}}_{1}^{y}\otimes {\hat {S}}_{2}^{y}+{\hat {S}}_{1}^{z}\otimes {\hat {S}}_{2}^{z})+2\mu _{B}B({\hat {S}}_{1}^{z}\otimes {\hat {I}}+{\hat {I}}\otimes {\hat {S}}_{2}^{z})$.

Problem 4: 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 ${\hat {p}}=-i{\hbar }{\frac {d}{dq}}$.

(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 $T\rightarrow 0$.