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 Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \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:

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \hat{\rho}_\mathrm{eq}} is the same density matrix in canonical ensemble studied in Problem 1 and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \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:

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathbf{P} } :

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{\partial \mathbf{P}}{\partial t} = \hat{A} \mathbf{P} + \mathbf{C} } ,

where you should find explicit form of Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \hat{A} } as a 3 x 3 matrix and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathbf{C} \in {\rm I\!R}^3 } as a vector.

(b) Solve differential equation for Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathbf{P} } assuming the initial condition Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathbf{P}(t=0) \equiv (0,1,0) } . For typical situation Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathrm{TiCuCl_3} } , spins Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathbf{B}=(0,0,B) } is given by:

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle J>0 } is the exchange coupling constant and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle N } dimers, as well as find their entropy.

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

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \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:

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \hat{H}=\frac{\hat{p}^2}{2m} + \frac{m \omega^2 q^2}{2} } ,

where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \hat{p} = -i{\hbar} \frac{d}{dq} } .

(a) Find the partition function Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle Z } in the quantum canonical ensemble at temperature Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle T } .

(b) Using the result from (a), calculate the averge energy Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle E = \langle \hat{H} \rangle } .

(c) Write down the formal expression for the canonical density operator Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \hat{\rho} } in terms of the eigenstates Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle |n\rangle } of the Hamiltonian and the corresponding energy levels Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \varepsilon_n = \hbar \omega (n + 1/2) } .

(d) Using the result in (c), write down the density matrix in the coordinate representation Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \langle q' |\hat{\rho}|q\rangle } .

(e) In the coordinate representation, calculate explicitly Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \langle q' |\hat{\rho}|q\rangle } in the high temperature limit Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle T \rightarrow \infty } .

HINT: One approach is to utilize the following result

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 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:

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle e^{-\beta \hat{H}} = e^{ -\beta \frac{\hat{p}^2}{2m} - \beta \frac{m\omega^2 q^2}{2} } }

while neglecting terms of order Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \beta^2 } and higher since Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \beta } is very small in the high temperature limit.

(f) At low temperatures, Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \hat{\rho} } is dominated by low-energy states. Use the ground state wave function Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \langle q|0 \rangle } only, evaluate the limiting behavior of Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \langle q' |\hat{\rho}|q\rangle } as Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle T \rightarrow 0 } .