Phys 451 — Fall 2015 — Assignment 4

  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.

    (g) Compute the approximate energy of the first few energy eigenstates using first-order perturbation theory. Use the states of the infinite square well and treat the step (of height \(V_0\) and width \(a\)) as the small perturbation. Compare these results to the plots in part (f).

  2. A quantum particle confined to the \(x\)–\(y\) plane is subject to a central potential \(V(r) = V(\sqrt{x^2 + y^2})\). In polar coordinates (defined by \(x = r\cos\theta\) and \(y = r\sin\theta\)), the Hamiltonian

    \[\hat{H} = -\frac{\hbar^2 \nabla^2}{2m} + V(r)\]

    looks like

    \[\hat{H} = -\frac{\hbar^2}{2m}\biggl[ \frac{1}{r}\frac{\partial}{\partial r} \biggl( r \frac{\partial}{\partial r}\biggr) + \frac{1}{r^2}\frac{\partial^2}{\partial \theta^2}\biggr] + V(r).\]

    (a) Express the transformed wave function \(\psi(x,y) \to \psi(r,\theta)\) as the product \(\psi(r,\theta) = R(r)\Theta(\theta)\). Show that doing so leads to two independent eigen-equations for \(R(r)\) and \(\Theta(\theta)\).

    (b) The angular momentum operator (a scalar rather than a vector in 2D) is

    \[\hat{L} = \mathbf{e}_z \cdot (\mathbf{r} \times \mathbf{p}) = - i \hbar \mathbf{e}_z \cdot (\mathbf{r} \times \nabla).\]

    Show that both of the following representations are correct:

    \[\hat{L} = \frac{\hbar}{i}\biggl( x\frac{\partial}{\partial y} - y\frac{\partial}{\partial x} \biggr) = \frac{\hbar}{i} \frac{\partial}{\partial \theta}.\]

    (c) Prove that \([\hat{H},\hat{L}] = 0\).

    (d) By virtue of the result in question 3, we know that the wavefunction can be a simultaneous eigenfunction of the Hamiltonian and the angular momentum. Solve for the states of definite angular momentum,

    \[\hat{L}\Theta_m(\theta) = \hbar m \Theta_m(\theta).\]

    What is the functional form of \(\Theta_m(\theta)\), and what values can \(m\) take?

    (e) Consider the specific example of a quantum harmonic oscillator with

    \[V(r) = \frac{1}{2}m\omega^2r^2 = \frac{1}{2}m\omega^2(x^2+y^2).\]

    Argue that the Hamiltonian can be expressed as \(\hat{H} = \hbar\omega(a^\dagger a + b^\dagger b + 1)\), where \(a\) and \(b\) are lowering operators in each of the \(x\) and \(y\) directions.

    (f) Use the usual definitions,

    \[\begin{align} \hat{x} &= \sqrt{\frac{\hbar}{2m\omega}} (a^\dagger+a)\\ \hat{y} &= \sqrt{\frac{\hbar}{2m\omega}} (b^\dagger+b)\\ \hat{p}_x &= i\sqrt{\frac{m\hbar\omega}{2}} (a^\dagger-a)\\ \hat{p}_y &= i\sqrt{\frac{m\hbar\omega}{2}} (b^\dagger-b), \end{align}\]

    to show that

    \[\hat{L} = i\hbar(a b^\dagger - a^\dagger b).\]

    (g) Again, prove that \([\hat{H},\hat{L}] = 0\), but this time do so by manipulating the raising and lowering operators.

  3. (Optional bonus question) Consider a basis \(\{ \lvert 0,0 \rangle, \lvert 1,0 \rangle, \lvert 0,1 \rangle, \lvert 2,0 \rangle, \lvert 1,1 \rangle, \lvert 0,2 \rangle, \ldots \}\) consisting of states \(\lvert n_x,n_y \rangle\) of definite number:

    \[\begin{align} a^\dagger a \lvert n_x,n_y \rangle = n_x \lvert n_x,n_y \rangle,\\ b^\dagger b \lvert n_x,n_y \rangle = n_y \lvert n_x,n_y \rangle. \end{align}\]

    Expressed in this basis, the Hamiltonian is diagonal.

    \[H = \begin{pmatrix} \hbar \omega \\ & 2\hbar \omega & 0\\ & 0 & 2\hbar \omega \\ &&& 3\hbar \omega & 0 & 0 \\ &&& 0 & 3\hbar \omega & 0 \\ &&& 0 & 0 & 3\hbar \omega \\ &&&&&& \ddots \\ \end{pmatrix}.\]

    The angular momentum, however, is not.

    \[L = \hbar \begin{pmatrix} 0 \\ & 0 & 1\\ & 1 & 0 \\ &&& ? & ? & ? \\ &&& ? & ? & ? \\ &&& ? & ? & ? \\ &&&&&& \ddots \\ \end{pmatrix}.\]

    (a) Compute matrix elements to fill in the nine missing entries.

    (b) Construct states with energy \(\hbar\omega\), \(2\hbar\omega\), and \(3\hbar\omega\) that are also states of definite angular momentum. Proceed by diagonalizing the \(1\times 1\), \(2 \times 2\), and \(3 \times 3\) blocks that appear in the matrix representation of \(L\).