Introduction to Quantum Mechanics
Phys 451 — Fall 2014 — Assignment 3
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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.
\[H = \begin{pmatrix} H_{1,1} & ? & ? & ? & ? & \cdots \\ ? & H_{2,2} & ? & ? & ? \\ ? & ? & H_{3,3} & ? & ? \\ ? & ? & ? & H_{4,4} & ? \\ ? & ? & ? & ? & \ddots \\ \vdots \end{pmatrix}\](c) 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 \(\ell \gg 1/\sqrt{\alpha}\).