物理代写|PHYS451 Quantum mechanics

Brain-warmer: oscillator algebra. Convince yourself that an operator $\mathcal{O}$ made of creation and annihilation operators $\mathbf{a}_k$ and $\mathbf{a}_k^{\dagger}$ for various $k$ commutes with the number operator $\sum_k \mathbf{N}_k$ if and only if it has the same number of as as $\mathbf{a}^{\dagger} \mathrm{s}$.

Brain-warmer: time evolution. Recall the expression for $\mathbf{q}_n$ in terms of creation and annihilation operators given in the lecture notes. Check that the expression for $\mathbf{p}_n$ in terms of creation and annihilation operators is consistent with the Heisenberg equations of motion
$$
\mathbf{p}_n=m \dot{\mathbf{q}}_n=\frac{\mathbf{i} m}{\hbar}\left[\mathbf{H}, \mathbf{q}_n\right] .
$$
(That is, evaluate the right hand side of this expression using the algebra of $\mathbf{a}_k$ and $\mathbf{a}_k^{\dagger}$

Entropy and thermodynamics. Consider a quantum system with hamiltonian $\mathbf{H}$ and Hilbert space $\mathcal{H}$ Its behavior in thermal equilibrium at temperature $T$ can be described using the thermal density matrix
$$
\boldsymbol{\rho}\beta \equiv \frac{1}{Z} e^{-\beta \mathbf{H}} $$ where $\frac{\beta \equiv 1}{T}$ specifies the temperature and $Z$ is a normalization factor. (We can think about this as the density matrix resulting from coupling the system to a heat bath and tracing out the Hilbert space of the heat bath.) (a) Find a formal expression for $Z$ by demanding that $\boldsymbol{\rho}\beta$ is normalized appropriately. This is called the partition function.
(b) Recall that the von Neumann entropy of a density matrix is defined as
$$
S[\rho]=-\operatorname{tr} \rho \log \rho
$$
Show that the von Neumann entropy of $\boldsymbol{\rho}\beta$ can be written as $$ S\beta=E / T+\log Z
$$
where $E \equiv\langle\mathbf{H}\rangle$ is the expectation value for the energy. Convince yourself that this is same as the thermal entropy.
(c) Evaluate $Z$ and $E$ and the heat capacity $C=\partial_T E$ for the case where the system is a simple harmonic oscillator
$$
\mathcal{H}=\operatorname{span}{|n\rangle, n=0,1,2 \ldots}, \quad \mathbf{H}=\hbar \omega\left(\mathbf{n}+\frac{1}{2}\right)
$$
with $\mathbf{n}|n\rangle=n|n\rangle$.
(d) Now evaluate the low-temperature equilibrium heat capacity for a harmonic mattress (the $d$-dimensional version of the harmonic chain). That is, find the heat capacity for a collection of harmonic oscillators labelled by wavenumber $\vec{k}$ in $d$ dimensions,
$$
\mathbf{H}=\sum_k \hbar \omega\left(a_k^{\dagger} a_k+\frac{1}{2}\right)
$$
with dispersion relation $\omega_k=v_s|k|$.

Momentum. In this problem we consider a scalar field theory in $d$ spatial dimensions. Consider the operator
$$
\overrightarrow{\mathbf{P}} \equiv \int \mathrm{d}^d k \hbar \vec{k} a_k^{\dagger} a_k
$$
where $\int d^d k \cdots \equiv \int \frac{d^d k}{(2 \pi)^d} \ldots$
(a) Find $\left[\overrightarrow{\mathbf{P}}, a_k^{\dagger}\right]$, and $\left[\overrightarrow{\mathbf{P}}, a_k\right]$
(b) Show using $4 \mathrm{a}$ and the mode expansion of a scalar field that
$$
[\overrightarrow{\mathbf{P}}, \phi(x)]=\mathbf{i} \hbar \vec{\nabla} \phi(x)
$$
(c) Conclude (using Taylor’s theorem) that
$$
e^{-\mathbf{i} \vec{a} \cdot \overrightarrow{\mathbf{P}} / \hbar} \phi(x) e^{\mathbf{i} \vec{a} \cdot \overrightarrow{\mathbf{P}} / \hbar}=\phi(x+a)
$$
and that therefore $\overrightarrow{\mathbf{P}}$ generates translations. Therefore $\overrightarrow{\mathbf{P}}$ is the operator representing the momentum carried by the field (like the Poynting vector for the electromagnetic field).
(d) Find $\overrightarrow{\mathbf{P}}\left|\vec{k}_1, \vec{k}_2 \ldots \vec{k}_n\right\rangle$, the action of this operator on a state of $n$ phonons. Conclude that $\hbar \vec{k}$ is the momentum of the phonon labelled by wavenumber $\vec{k}$.

Gaussian identity. Show that for a gaussian quantum system
$$
\left\langle e^{\mathbf{i} K \mathbf{q}}\right\rangle=e^{-A(K)\left\langle\mathbf{q}^2\right\rangle}
$$
and determine $A(K)$. Here $\langle\ldots\rangle \equiv\langle 0|\ldots| 0\rangle$. Here by ‘gaussian’ I mean that $\mathbf{H}$ contains only quadratic and linear terms in both $\mathbf{q}$ and its conjugate variable $\mathbf{p}$ but for the formula to be exactly correct as stated you must assume $\mathbf{H}$ contains only terms quadratic in $\mathbf{q}$ and $\mathbf{p}$; for further entertainment fix the formula for the case with linear terms in $\mathbf{H}$ .

I recommend using the path integral representation (with hints from the previous problem). Alternatively, you can use the harmonic oscillator operator algebra. Or, better, do it both ways.