Phys 712 — Spring 2015 — Assignment 2

  1. In manipulating quantum operators, it is crucial to preserve operator ordering. In Question 2 of Assignment 1, a number of students gave a proof that hinged on the following being an equality:

    \[\frac{\partial}{\partial t}\bigl(f(t)\hat{A} + g(t)\hat{B}\bigr)^n \neq n\bigl(f(t)\hat{A} + g(t)\hat{B}\bigr)^{n-1}\bigl(f'(t)\hat{A} + g'(t)\hat{B}\bigr).\]

    Consider the specific case of \(f(t) = 1\), \(g(t) = t\), and \(n=3\). Show explicitly that

    \[\frac{\partial}{\partial t}\bigl(\hat{A} + \hat{B}t\bigr)^3 \neq 3\bigl(\hat{A} + \hat{B}t\bigr)^{2}\hat{B},\]

    unless \(\hat{A}\) commutes with \(\hat{B}\).

  2. The Hamiltonian \(\hat{H} = \hat{H}_0 + \hat{V}(t)\) describes a two-state system

    \[\hat{H}_0 = \sum_{n=1,2}E_n \lvert n \rangle\langle n \rvert\]

    coupled to an oscillating external field

    \[\hat{V}(t) = \gamma e^{i\omega t}\lvert 1 \rangle\langle 2 \rvert + \gamma e^{-i\omega t}\lvert 2 \rangle\langle 1 \rvert.\]

    Here, \(\gamma\) is a real-valued quantity representing the strength of the field, and \(\omega\) is the angular freqency of oscillation. The wavefunction can be written in the form

    \[\lvert \psi(t) \rangle = \sum_{n=1,2} c_n(t)e^{-iE_nt/\hbar}\lvert n \rangle,\]

    where the time-dependent coefficients \(c_1(t)\) and \(c_2(t)\) encode all the effects of \(\hat{V}(t)\). The states \(\lvert 1 \rangle\) and \(\lvert 2 \rangle\) are eigenstates of \(\hat{H}_0\) with energy eigenvalues \(E_1\) and \(E_2\).

    (a) Suppose that the system is in its ground state at time zero; i.e., \(\lvert \psi(0) \rangle = \lvert 1 \rangle\), and hence \(c_1(0) = 1\) and \(c_2(0) = 0\). In class, we worked through the details and found that

    \[c_2(t) = \frac{-2i\gamma}{\sqrt{(\hbar\omega-E_2 + E_1)^2 + 4\gamma^2}} \sin\biggl(\frac{t}{2\hbar}\sqrt{(\hbar\omega - E_2+E_1)^2+4\gamma^2} \biggr)e^{i(\hbar\omega - E_2 + E_1)t/2\hbar}.\]

    The modulus squared of this expression appears as Eq. (5.5.21a) of Sakurai. Find the comparable expression for \(c_1(t)\), and verify that \(\lvert c_1(t) \rvert^2 + \lvert c_2(t) \rvert^2 = 1\).

    (b) Expand the expression for \(c_2(t)\) as a power series in \(\gamma\). Compute the terms up to (and including) second order.

    (c) For a Hamiltonian \(\hat{H} = \hat{H}_0 + \hat{V}(t)\) that has all its time dependence in the \(\hat{V}\) term, it is convenient to switch to the interaction picture. One advantage is that we can easily write the evolution operator \(\hat{U}_I\) as a Dyson series in \(\hat{V}_I\). See Eq. (5.7.6) of Sakurai. Use this approach to compute

    \[c_2(t) = \langle 2 \vert \psi_I(t) \rangle = \langle 2 \rvert \hat{U}_I(t,0) \lvert 1 \rangle.\]

    Compare order by order with your results from part (b). Are they the same or different? Explain why.

    N.B. If you find this question overwhelming, simplify things by carrying out the calculations in parts (b) and (c) on resonance.

  3. The Hamiltonian

    \[\hat{H} = \frac{\hat{p}^2}{2m} + \frac{1}{2}m\omega^2\bigl(\hat{x} - q(t)\bigr)^2\]

    describes a particle in a quadradic trap whose position changes in time.

    (a) Apply a transformation to put the Hamiltonian in the form \(\hat{H} = \hbar \omega (a^\dagger a + 1/2) + \hat{V}(t)\), where \(a\) and \(a^{\dagger}\) are the creation and annihilation operators of the quantum harmonic oscillator.

    (b) Suppose that

    \[q(t) = \begin{cases} 0 & \text{for $t\le 0$} \\ q_0 \sin(\Omega t) & \text{for $t > 0$} \end{cases}\]

    and that the system is in its ground state before the trap starts shaking. Write down the system of equations governing the time evolution of the coefficents \(\{ c_n(t) \}\).

    (c) Produce a plot of the expectation values of \(\hat{x}\) and \(\hat{H}_0\) as a function of time for values of \(\Omega \ll \omega\), \(\Omega \approx \omega\), and \(\Omega \gg \omega\).

    (d) Suppose instead that

    \[q(t) = \frac{q_0}{2}\bigl(1 + \tanh(\nu_0 t)\bigr).\]

    Explain in words (no calculation necessary) how you would solve the problem in the limits \(\nu_0 \ll 1\) and \(\nu_0 \gg 1\).