Homework Set 1

Problem 1: Three-dimensional isotropic harmonic oscillator

The Hamiltonian of a three-dimensional (3D) harmonic oscillator is given by ${\hat {H}}={\frac {{\hat {p}}_{x}^{2}}{2m}}+{\frac {{\hat {p}}_{y}^{2}}{2m}}+{\frac {{\hat {p}}_{z}^{2}}{2m}}+{\frac {1}{2}}m\omega ^{2}({\hat {x}}^{2}+{\hat {y}}^{2}+{\hat {z}}^{2})$ which is isotropic since constant $\omega ^{2}$ is the same in all three directions $x,y,z$. Using second quantization formalism find energy levels of this Hamiltonian and their degeneracy (in contrast to one-dimensional harmonic oscillator whose all energy levels are non-degenerate). NOTE: This problem is often used to analyze Bose-Einstein condensation of trapped ultracold atomic gases, see, e.g, J. Low Temp. Phys. 106, 615 (1997).

Problem 2: Autocorrelation function of harmonic oscillator at finite temperature

For one-dimensional harmonic oscillator, the expectation value of the number operator at finite temperature $T$ is given by

$n(\omega )=\langle {\hat {a}}^{\dagger }{\hat {a}}\rangle ={\frac {1}{e^{\beta \hbar \omega }-1}}$

where $\beta =1/(k_{B}T)$, whereas $\langle {\hat {a}}{\hat {a}}^{\dagger }\rangle =n(\omega )+1$.

(a) Compute the autocorrelation function of the position operator

${\frac {1}{2}}\langle \{{\hat {x}}(t),{\hat {x}}(0)\}\rangle$

and give physical interpretation of its first and second term. Here $\{{\hat {A}},{\hat {B}}\}$ is antucommutator of two operators.

(b) What happens to the autocorrelation function in the high temperature limit?

Problem 3: Coordinate representation of coherent state

The coherent state can be generated from vacuum state as $|\alpha \rangle ={\hat {D}}(\alpha )|0\rangle$ using the unitary operator ${\hat {D}}(\alpha )=\exp(\alpha {\hat {a}}^{\dagger }-\alpha ^{*}{\hat {a}})$.

(a) Show that ${\hat {D}}(\alpha )$ acts as a displacement operator, ${\hat {D}}^{\dagger }(\alpha ){\hat {a}}{\hat {D}}(\alpha )={\hat {a}}+\alpha$.

(b) Find coordinate representation of the coherent state, $\langle x|{\hat {D}}(\alpha )|0\rangle (t)$ at time $t=0$ and later times $t>0$.

HINT: You will find useful the so-called Glauber formula $\exp({\hat {A}})\exp({\hat {B}})=\exp({\hat {A}}+{\hat {B}})\exp([{\hat {A}},{\hat {B}}]/2)$ which is valid on the proviso that both operators ${\hat {A}}$ and ${\hat {B}}$ commute with their commutator $[{\hat {A}},[{\hat {A}},{\hat {B}}]]=0=[{\hat {B}},[{\hat {A}},{\hat {B}}]]$.

Problem 4: Acoustic and optical phonons in one-dimensional chain with two different atoms per unit cell

Consider an infinite one-dimensional chain described by classical Hamiltonian:

$H=\sum _{j}\left[{\frac {p_{j}^{2}}{2m_{j}}}+{\frac {k}{2}}(x_{j}-x_{j-1})^{2}\right]$

whose even sites host atoms of mass $m_{2j}=m$ and odd sites host atoms of mass $m_{2j+1}=M$.

(a) Find classical normal mode frequencies and sketch their dispersion curves $\omega (k)$. There will be two dispersion curves called acoustic and optical branch in solid state physics.

(b) What is the energy gap between the acoustic and optical branch?

(c) Write the diagonalized Hamiltonian in second quantized form and discuss how you might arrive at your final answer. You will now need two types of creation operators.