物理代写|PHYS3034 Quantum mechanics

Brain-warmer: fluctuations of the EM field. Let us focus on a single mode, and a single polarization, of the EM field, at a point in space:
$$
\mathbf{E}(t) \equiv \mathbf{E}(\vec{r}=0, t)=\mathbf{i} \sqrt{\frac{\hbar \omega}{2 \varepsilon_0 V}}\left(\mathbf{a} e^{-\mathbf{i} \omega t}-\mathbf{a}^{\dagger} e^{\mathbf{i} \omega t}\right)
$$
where $V$ is the volume of the cavity in which this mode lives. Define the variance of an operator in a state $\psi$ as
$$
\Delta_\psi E \equiv \sqrt{\left\langle\psi\left|\mathbf{E}^2(t)\right| \psi\right\rangle-(\langle\psi|\mathbf{E}(t)| \psi\rangle)^2}
$$
(a) Find the variance of $\mathbf{E}$ in the vacuum, $\Delta_0 E$.
(b) Find the variance of $\mathbf{E}$ in the state of exactly $n$ photons (all in this mode).
(c) A more realistic state, for both a single-mode laser, and for a classical source of light, is a coherent state. (In a classical source of light, different modes have random phases relative to each other.) In this state, what is the expected number of photons? What is its variance?
Find the variance of $\mathbf{E}$ in a coherent state $|z\rangle$, where $\mathbf{a}|z\rangle=z|z\rangle$.

Maxwell’s equations, quantumly.
(a) Check that the oscillator algebra for the photon creation and annihilation operators
$$
\left[\mathbf{a}{k s}, \mathbf{a}{k^{\prime} s}^{\dagger}\right]=\delta_{k k^{\prime}} \delta_{s s^{\prime}}
$$
implies (using the mode expansion for $\mathbf{A}$ ) that
$$
\left[\mathbf{A}i(\vec{r}), \mathbf{E}_j\left(\vec{r}^{\prime}\right)\right]=-\mathbf{i} \hbar \int \mathrm{d}^3 k e^{\mathrm{i} \vec{k} \cdot\left(\vec{r}-\vec{r}^{\prime}\right)}\left(\delta{i j}-\hat{k}_i \hat{k}_j\right) / \varepsilon_0
$$
(and also $\left[\mathbf{A}_i(\vec{r}), \mathbf{A}_j\left(\vec{r}^{\prime}\right)\right]=0$ and $\left[\mathbf{E}_i(\vec{r}), \mathbf{E}_j\left(\vec{r}^{\prime}\right)\right]=0$ ).
Conclude that it’s not possible to simultaneously measure $E_x(\vec{r})$ and $B_y(\vec{r})$.
(b) Using the result of the previous part, check that the wave equation for $\mathbf{A}_i(x)$ follows from the Heisenberg equations of motion
$$
-\partial_t \overrightarrow{\mathbf{E}}=\frac{\mathbf{i}}{\hbar}[\mathbf{H}, \overrightarrow{\mathbf{E}}]
$$

A charged particle, classically. This problem is an exercise in calculus of variations, as well as an important ingredient in our discussion of particles in electromagnetic fields.
Consider the following action functional for a particle in three dimensions:
$$
S[x]=\int d t\left(\frac{m}{2} \dot{\vec{x}}^2-e \Phi(\vec{x})+\frac{e}{c} \dot{\vec{x}} \cdot \vec{A}(x)\right) .
$$
(a) Show that the extremization of this functional gives the equation of motion:
$$
\frac{\delta S[x]}{\delta x^i(t)}=-m \ddot{x}^i(t)-e \partial_{x^i} \Phi(x(t))+\frac{e}{c} \dot{x}^j F_{i j}(x(t))
$$
where $F_{i j} \equiv \partial_{x^i} A_j-\partial_{x^j} A_i$. Show that this is the same as the usual CoulombLorentz force law
$$
\vec{F}=e\left(\vec{E}+\frac{\vec{v}}{c} \times \vec{B}\right)
$$
with $B_i \equiv \frac{1}{2} \epsilon_{i j k} F_{j k}$.
(b) Show that the canonical momenta are
$$
\Pi_i \equiv \frac{\partial L}{\partial \dot{x}^i}=m \dot{x}^i+\frac{e}{c} A_i(x)
$$
Here $S=\int d t L$. (I call them I rather than $p$ to emphasize the difference from the ‘mechanical momentum’ $m \dot{x}$.) Show that the resulting Hamiltonian is
$$
H \equiv \sum_i \dot{x}^i \Pi^i-L=\frac{1}{2 m}\left(\Pi_i-\frac{e}{c} A_i(x(t))\right)^2+e \Phi
$$

Phonons in salt. Consider a model of a more complex solid, where there are two kinds of atoms, of masses $m$ and $M$, connected by springs of strength $\kappa_1$ and $\kappa_2$, as in the figure. (This is a cartoon of an ionic solid, like $\mathrm{NaCl}$.)

The unit cell, i.e. the pattern that is repeated, contains two atoms. Let $q_n$ be the deviation from equilibrium position of the $n$th light atom and $Q_n$ be the deviation from equilibrium position of the $n$th heavy atom. Assume periodic boundary conditions. You may wish to rescale the variables $q, Q$ to simplify the dependence on the parameters. (If you prefer, study the special case where $m=M$, but $\kappa_1 \neq \kappa_2$.)
By making the Fourier ansatz
$$
\left(\begin{array}{c}
q_n \
Q_n
\end{array}\right)=\frac{1}{\sqrt{N}} \sum_k e^{\mathrm{i} k n a}\left(\begin{array}{c}
q_k \
Q_k
\end{array}\right)
$$
find the spectrum of normal modes. Introduce creation and annihilation operators and reduce the problem to a collection of decoupled oscillators.

In addition to the acoustic phonons, which have dispersion $\omega_k \sim v_s|k|$ near $k=0$, and represent the quantum avatar of the sound mode, you should find a branch of optical phonons which have $\omega_k \stackrel{k \rightarrow 0}{\rightarrow}$ a finite number. Interpret the optical phonon mode in terms of the motion of the atoms.