Phys 451 — Fall 2014 — Assignment 5

  1. Consider an infinite square well of width \(L\), modified so that it has an finite interior step of height \(V_0\) jutting out a distance \(a\) from the left edge:

    \[V(x) = \begin{cases} \infty & \text{if $x<0$ or $x>L$}\\ V_0 & \text{if $0 < x < a$}\\ 0 & \text{if $a < x < L$} \end{cases}\]

    (a) Sketch the potential.

    (b) Explain why

    \[\psi(x) = \begin{cases} 0 & \text{if $x<0$ or $x>L$}\\ \sin k'x & \text{if $0 < x < a$}\\ b\cos kx + c\sin kx & \text{if $a < x < L$} \end{cases}\]

    is an appropriate functional form for the (unnormalized) wavefunction.

    (c) Suppose that \(\psi\) is an eigenstate satisfying \(H\psi = E \psi\). Provide an expression for \(k\) in terms of \(E\) and an expression for \(k'\) in terms of \(k\) and \(V_0\).

    (d) Impose matching conditions on \(\psi(a)\) and \(\psi'(a)\) to find expressions for \(b\) and \(c\).

    (e) Impose the correct boundary condition at the right edge of the well to derive the quantization condition.

    (f) Make two plots of the energy spectrum (using at least the first few energy eigenvalues): one with \(V_0\) fixed and \(a\) varying from \(0\) to \(L\) and another with \(a\) fixed and \(V_0\) varying from zero to infinity. Explain the behaviour.

  2. The quantum harmonic oscillator Hamiltonian is

    \[\hat{H} = -\frac{\hbar^2}{2m}\frac{d^2}{dx^2} + \frac{1}{2}m\omega^2 x^2.\]

    (a) Confirm that \(\psi_0(x) = \exp(-\alpha x^2)\) is the ground state wavefunction with energy \(E_0 = \hbar \omega / 2\). Determine the correct value of \(\alpha\).

    (b) Treat the wavefunctions of the higher modes as products of the form \(\psi_n(x) = f_n(x)\psi_0(x)\). Show that

    \[-\frac{\hbar^2}{2m}f_n''(x) + \hbar\omega x f_n'(x) = (E_n - E_0) f_n(x).\]

    (c) Use the result \(E_n - E_0 = \hbar\omega n\) and transform to a dimensionless variable \(\xi = (m\omega/\hbar)^{1/2}x\) in order to prove that the differential equation in 2(b) can be recast as

    \[-\frac{1}{2}g_n''(\xi) + \xi g_n'(\xi) - ng_n(\xi) = 0.\]

    (d) We suspect that \(g_n(\xi)\) is a polynomial of order \(n\). Let’s write it as

    \[g_n(\xi) = \sum_{j=0}^n a_n^{(j)} \xi^j.\]

    Derive the following recursion relation for the polynomial coefficients:

    \[a_n^{(j+2)} = \frac{2(j-n)}{(j+1)(j+2)}a_n^{(j)}.\]

    (e) The \(g_n\) functions show alternating parity and thus can be seeded with \(a_n^{(0)} = 1\) and \(a_n^{(1)} = 0\) for \(n\) even or \(a_n^{(0)} = 0\) and \(a_n^{(1)} = 1\) for \(n\) odd.
    Compute the coefficients for \(g_1(x)\), \(g_2(x)\), \(g_3(x)\), and \(g_4(x)\). Their normalization is arbitrary. Report results that follow the convention \(a_n^{(n)} = 2^n\).

  3. The quantum harmonic oscillator Hamiltonian can also be written

    \[\hat{H} = \hbar \omega \Bigl( a^\dagger a + \frac{1}{2}\Bigr),\]

    in terms of ladder operators.

    (a) Compute the commutators \([a,\hat{H}]\) and \([a^\dagger,\hat{H}]\).

    (b) Prove the Ehrenfest relation,

    \[\frac{d}{dt}\langle \hat{x} \rangle = \langle [\hat{x},\hat{H}] \rangle = \frac{1}{m}\langle \hat{p} \rangle,\]

    by expressing \(\hat{x}\) in terms of \(a\) and \(a^\dagger\), applying the results of 3(a), and then converting the \(a\) and \(a^\dagger\) difference to \(\hat{p}\).

    (c) Find an expression for the time evolution of the expectation value of \(\hat{p}\) in the case of a system that is prepared in the state

    \[\lvert \psi \rangle = \frac{1}{2}\lvert 0 \rangle + \frac{\sqrt{3}}{2}\lvert 1 \rangle\]

    at time \(t=0\).