Phys 451 — Fall 2015 — Assignment 3

  1. The Pauli matrices, defined as

    \[\sigma_x = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix},\ \sigma_y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix},\ \sigma_z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix},\]

    play an important role in describing the intrinsic angular momentum—or spin—of a quantum particle. Here, we’ll use them in a different context and take advantage of the fact that, together with the identity matrix, they span the space of \(2\times 2\) matrices. Prove the following identities.

    (a) \(\sigma_x^2 = \sigma_y^2 = \sigma_z^2 = 1\).

    (b) \(\sigma_x \sigma_y = -\sigma_y \sigma_x = i\sigma_z\), along with all cyclic permutations.

    (c) \(\exp(\lambda\sigma_x) = I\cosh \lambda + \sigma_x \sinh \lambda\). (Hint: proceed by power series.)

    (d) \(\exp(i\phi\sigma_z) = I\cos \phi + i\sigma_z \sin \phi\).

  2. We now consider the problem of barrier tunnelling in one spatial dimension using the transfer matrix approach introduced in lecture. Suppose that a quantum particle with energy \(E\) is described by plane wave states, with wave vector \(k_1\) in the region \(x < 0\) and with wave vector \(k_2\) in the region \(x > 0\):

    \[\begin{align} \psi(x < 0) &= u_1 e^{ik_1x} + v_1 e^{-ik_1 x},\\ \psi(x > 0) &= u_2 e^{ik_2x} + v_2 e^{-ik_2 x}. \end{align}\]

    The sudden change from \(k_1\) to \(k_2\) is due to a jump discontinuity of height \(V_0\) in the otherwise flat potential \(V(x)\).

    (a) Show that

    \[\begin{pmatrix} u_1 \\ v_1 \end{pmatrix} = \Theta(k_1,k_2) \begin{pmatrix} u_2 \\ v_2\end{pmatrix},\]

    where \(2\Theta(k_1,k_2) = (1+k_2/k_1)I + (1-k_2/k_1)\sigma_x\). This is just a matter of imposing continuity and smoothness at \(x=0\).

    (b) Re-express the matrix as \(\Theta(k_1,k_2) = \sqrt{k_1/k_2}\exp(w\sigma_x)\), where

    \[w = \tanh^{-1} \biggl(\frac{k_1-k_2}{k_1+k_2}\biggr).\]

    You can do this by working backwards from your result in 1(c).

  3. In the transfer-matrix language, free propagation of a particle of momentum \(\hbar k\) over a distance \(a\) is represented by the matrix \(\Gamma(k;a) = \exp(-ika\sigma_z)\).

    (a) Argue that the effect of a rectangular barrier is described by the matrix product

    \[\begin{split}M(k_1,k_2;a) &\equiv \Theta(k_1,k_2)\Gamma(k_2;a)\Theta(k_2,k_1)\\ &= \exp(w\sigma_x)(I\cos k_2a - i\sigma_z \sin k_2a)\exp(-w\sigma_x).\end{split}\]

    (b) Taking \(A = w\sigma_x\) and \(B = I\cos \phi - i\sigma_z \sin \phi\), compute the nested commutators \([A,B]\), \([A,[A,B]]\), \([A,[A,[A,B]]]\), and so on. Show that

    \[\underbrace{[A,[A,\ldots,[A}_{\text{$n$ times}},B]]\cdots] = -2^n w^n \sin \phi \times \begin{cases} \sigma_y & \text{$n$ odd}\\ i\sigma_z & \text{$n$ even}\end{cases}\]

    (c) Apply the identity

    \[e^ABe^{-A} = B + [A,B] + \frac{1}{2!}[A,[A,B]] + \frac{1}{3!}[A,[A,[A,B]]] + \cdots\]

    (known as the “Hadamard Lemma”) to show that

    \[M = I\cos k_2a - \frac{\sin k_2a}{2k_1k_2}\bigl[ (k_1^2-k_2^2)\sigma_y + (k_1^2+k_2^2)i\sigma_z \bigr].\]

    To eliminate the \(w\) in favour of \(k_1\) and \(k_2\), you’ll have to make use of the fact that

    \[\tanh^{-1}(x) = \frac{1}{2}\ln\Bigl(\frac{1+x}{1-x}\Bigr).\]

    (d) Show explicitly that the tunnelling and reflection amplitudes (which can be read off from the first column of the transfer matrix via \(M_{1,1} = 1/t\) and \(M_{2,1} = r/t\)) sum to unit probability; i.e., \(\lvert t \rvert^2 + \lvert r \rvert^2 = 1\).

  4. If the height of the step potential is large enough with respect to \(E\), then there is no propagating wave inside the rectangular barrier; rather, one finds exponential decay.

    (a) Convince yourself that the expression you’ve derived for \(M\) still holds, provided that you make the replacement \(k_2 \to i\lambda_2\). Give the correct expression for the positive real number \(\lambda_2\).

    (b) Justify the claim that the limit \(\lambda_2 a \gg 1\) is associated with large barrier height and/or large barrier width.

    (c) Demonstrate that under the assumption of \(\lambda_2 a\) large, the transfer matrix simplifies to

    \[M \to \tilde{M} = \frac{1}{2}e^{\lambda_2 a}\begin{pmatrix} 1 - \frac{i(k_1^2-\lambda_2^2)}{2k_1\lambda_2} & \frac{i(k_1^2+\lambda_2^2)}{2k_1\lambda_2} \\ -\frac{i(k_1^2+\lambda_2^2)}{2k_1\lambda_2} & 1 + \frac{i(k_1^2-\lambda_2^2)}{2k_1\lambda_2} \end{pmatrix}\]

    (d) Tell me what kind of barrier is described by the transfer matrix \(\tilde{M}(k_1,\lambda_2;a)\Gamma(k_1;b)\tilde{M}(k_1,\lambda_2;a)\). Identify the tunnelling amplitude \(t\) for this system. Show that there is now the possibility of resonant tunnelling, in which the system achieves perfect transmission at discrete values of the incoming kinetic energy \(E\). Try to offer an explanation for this phenomenon

    (e) Explain how you would perform (but don’t actually carry out) a calculation of the barrier tunnelling probability (again, for a particle of energy \(E\) incidient from the left) in the case of a linear ramp potential:

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

    (Hint: You’ll want to stick with the transfer matrix approach. Think about the Simpon’s rule limiting process that you use to compute area intergrals in calculus.)