Introduction to Quantum Mechanics
Phys 451 — Fall 2015 — Assignment 2
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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) \([\hat{x}^2,\hat{p}^2] = 4i\hbar\,\hat{x}\hat{p} + 2\hbar^2\)
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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?
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We’ll use the notation \(\{ \lvert n \rangle \} = \{ \lvert 1 \rangle, \lvert 2 \rangle, \lvert 3 \rangle, \ldots \}\) to denote the set of energy eigenstates of a one-dimensional, infinite square well potential that confines a particle of mass \(m\) between \(x=0\) and \(x=L\).
(a) Suppose that we are given a potential
\[V(x) = \begin{cases} V_0 \sin(\pi x/L) & \text{if}\ 0 < x < L\\ \infty & \text{otherwise.} \end{cases}\]Write down—both don’t yet compute—the integral that corresponds to the general matrix element \(H_{m,n} = \langle m \rvert \hat{H} \lvert n \rangle\), evaluated in the infinite square well basis.
(b) Show that the diagonal matrix elements satisfy
\[H_{n,n} = \langle n \rvert \hat{H} \lvert n \rangle = \frac{n^2h^2}{8mL^2} + \frac{V_08n^2}{\pi(4n^2-1)}.\](c) For each element marked with a question mark, indicate with 0,+,– whether the entry is zero, postive, or negative. No explicit calculation is required.
\[H = \begin{pmatrix} H_{1,1} & ? & ? & ? & ? & \cdots \\ ? & H_{2,2} & ? & ? & ? \\ ? & ? & H_{3,3} & ? & ? \\ ? & ? & ? & H_{4,4} & ? \\ ? & ? & ? & ? & \ddots \\ \vdots \end{pmatrix}\](d) Compute H explicitly in the restricted basis consisting of the only lowest three states, \(\{ \lvert 1 \rangle, \lvert 2 \rangle, \lvert 3 \rangle \}\). Find its energy eigenvalues.
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Consider an electron living in a double well potential. \(\lvert L \rangle\) and \(\lvert R \rangle\) denote occupation in the left and right positions. The shape of the wavefunction in each state is identical but translated by a distance \(\ell\). In other words, \(\psi_L(x) = \langle x | L \rangle = \phi(x)\) and \(\psi_R(x) = \langle x | R \rangle = \phi(x-\ell)\).
(a) Write down an integral expression for the overlap \(\langle L \vert R \rangle\).
(b) In the case where \(\phi(x) = Ce^{-(\alpha/2)x^2}\), show that the overlap is equal to \(e^{-(\alpha/4)\ell^2}\).
(c) The Hamiltonian of the system is
\[\hat{H} = E_L\lvert L \rangle\langle L \rvert + E_R\lvert R \rangle\langle R \rvert\]Formulate the generalized eigenvalue problem that follows from the time-independent Schrödinger equation.
(d) Solve for the eigenvalues and eigensates.
(e) Comment on the asymptotic behaviour of the system in the limit \(\ell \gg 1/\sqrt{\alpha}\).