Phys 451 — Fall 2014 — Assignment 2

  1. In class, we proved that \([\hat{x},\hat{p}] = i\hbar\). We did this by showing that \([\hat{x},\hat{p}]\psi(x) = i\hbar\psi(x)\) for any differentiable function \(\psi(x)\). Apply that same technique to demonstrate each of the following equalities. [Keep in mind the position representation correspondences \(\hat{x} = x\) and \(\hat{p} = (\hbar/i)(d/dx)\).]

    (a) \([\hat{x}^n,\hat{p}] = i\hbar\, n\hat{x}^{n-1}\)

    (b) \([f(\hat{x}),\hat{p}] = i\hbar \, df(x)/dx\)

    (c) \([\hat{x},\hat{x}\hat{p}] = [\hat{x},\hat{p}\hat{x}] = i\hbar\,\hat{x}\)

    (d) \([\hat{x}^2,\hat{p}^2] = 4i\hbar\,\hat{x}\hat{p} + 2\hbar^2\)

  2. A single quantum particle of mass \(m\) is subject to a square-well potential of finite depth \(V_0\). Specifically, \(V(x) = V_0\) when \(x < 0\) or \(x > L\) and \(V(x) = 0\) otherwise.

    (a) Demonstrate that any such square well of nonzero depth results in at least one bound state, whose energy \(E_1\) tends to \(mV_0^2L^2/2\hbar^2\) as \(V_0 \rightarrow 0^+\).

    (b) What is the shallowest the potential can be and still support at least two symmetric bound states?

  3. \(N\) identical potential wells are arranged around a ring of circumference \(L\) (i.e., a one-dimensional line segment with periodic boundary conditions). The wells are regularly spaced a distance \(a = L/N\). The ket \(\lvert j \rangle\) denotes the state in which a particle resides in the lowest energy bound state of the \(j\)th well. Adjacent wells are tunnel coupled with a (real-valued) strength \(-t\). The resulting Hamiltonian is

    \[\hat{H} = \sum_{j} \bigl[ \varepsilon\lvert j \rangle\langle j \rvert - t \lvert j \rangle\langle j+1 \rvert - t\lvert j\rangle\langle j-1 \rvert \bigr]\]

    (a) Convince yourself that \(\hat{H}\) is hermitian.

    (b) The \(N=6\) system has the following matrix represention. What is the correct value of \(b\)?

    \[H = \begin{pmatrix} \varepsilon & -t & 0 & 0 & 0 & b\\ -t & \varepsilon & -t & 0 & 0 & 0\\ 0 & -t & \varepsilon & -t & 0 & 0 \\ 0 & 0 & -t & \varepsilon & -t & 0\\ 0 & 0 & 0 & -t & \varepsilon & -t\\ b & 0 & 0 & 0 & -t & \varepsilon \end{pmatrix}\]

    (c) Show that states \(\lvert k \rangle \sim \sum_{j=1}^N \exp(ijka) \lvert j \rangle\) of definite “crystal momentum” are energy eigenstates. Determine their proper normalization. What are the relevant values of \(k\) such that \(\{ \lvert k \rangle \}\) forms an orthonormal basis set?