[QUE/QFT-04011 QFT-PROBLEM

$\newcommand{\matrixelement}[3]{\langle#1|#2|#3\rangle}\newcommand{\dd}[2][]{\frac{d#1}{d#2}}$

Expand \(\psi(x)\) in terms of plane waves as \[ \psi(\mathbf x) = \frac{1}{(2\pi)^{3/2}}\int d\mathbf k \exp(i\mathbf k\mathbf x) a(\mathbf k)\]

1. Working in in one dimension, show that \[[a(\mathbf k), \psi^\dagger(\mathbf x)] = \frac{e^{-i\mathbf k\mathbf x}}{(2\pi)^{3/2}}\qquad \text{and} \qquad [a^\dagger(\mathbf k), \psi(\mathbf x)]= -\frac{e^{i\mathbf k\mathbf x}}{(2\pi)^{3/2}}\]
2. Express the free Hamiltonian \begin{equation} H = \frac{\hbar^2}{2m}\int d\mathbf x (\nabla \psi^\dagger(\mathbf x) )(\nabla \psi(\mathbf x)) \end{equation} in terms of number operators and verify that \begin{equation} H = \int d\mathbf k \Big(\frac{\hbar^2\mathbf k^2}{2m}\Big) a(\mathbf k)^\dagger a(\mathbf k) \end{equation}
3. Given interaction Hamiltonian \[H{'} = \int d\mathbf x \psi^\dagger(\mathbf x) V(\mathbf x) \psi(\mathbf x)\]. Compute the matrix element \(\matrixelement{\mathbf k_f}{H{'}}{\mathbf k_i}\) and verify that \[ \matrixelement{\mathbf k_f}{H{'}}{\mathbf k_i} = \frac{1}{(2\pi)^{3/2}} \int\,d\mathbf q \exp(i\mathbf q\mathbf x) V(\mathbf x) \] where \(\mathbf q=\mathbf k_f-\mathbf k_i\).