G25.2666: Quantum Chemistry and Dynamics

Problem set #7
Due: May 3, 2005

1.

The time-dependent Schrödinger equation for a particle moving in a potential $V({\bf r})$ subject to an electromagnetic field is

$$\left\{-{1 \over 2m}\left[-i\hbar\nabla - {e \over c}{\bf A}... ...{\bf r},t) = i\hbar{\partial \over \partial t} \psi({\bf r},t)$$

Show that the Schrödinger equation is invariant under a gauge transformation

$${\bf A}'({\bf r},t) = {\bf A}({\bf r},t)-\nabla\chi({\bf r},t)$$

$$\phi'({\bf r},t) = \phi({\bf r},t) + {1 \over c}{\partial \over \partial t}\chi({\bf r},t)$$

$$\psi'({\bf r},t) = e^{-ie\chi({\bf r},t)/\hbar c}\psi({\bf r},t)$$

2.

Suppose a harmonic oscillator with frequency $\omega$ and mass $m$ has a charge $q$ and subject to a time-dependent perturbation $H_1(t) = -qEX$ for a time interval $0\leq t\leq \tau$. Here $X$ is the position operator and $E$ is the magnitude of an applied electric field.

a.

Calculate the transition probability $P_{01}$ for a transition between the $n=0$ and $n=1$ levels of the oscillator within first-order perturbation theory.

b.

Calculate the transition probability $P_{02}$ for a transition between the $n=0$ and $n=2$ levels of the oscillator within the lowest non-vanishing order in perturbation theory. What is the lowest order needed to obtain a non-trivial result?

3.

As a simple model for photoelectron spectroscopy, consider a one-dimensional particle moving in a potential $V(X) = -\alpha\delta(X)$, where $X$ is the position operator.

a.

Show that the Hamiltonian admits a single bound state with energy $E_0 = -m\alpha^2/2\hbar^2$ with associated eigenfunction

$$\psi_0(x) = \sqrt{{m\alpha\over\hbar^2}}e^{-m\alpha\vert x\vert/\hbar^2}$$

Hint: Try integrating the eigenvalue equation between $-\epsilon$ and $\epsilon$ and show that as $\epsilon\rightarrow 0$, the derivative of the eigenfunction has a discontinuity at $x=0$.

b.

For positive energies $E_k=\hbar^2 k^2/2m$, there will be two stationary wave functions corresponding to a particle incident from the left or from the right. These eigenfunctions are given by

$$\psi_k(x) = {1 \over \sqrt{2\pi}} \left[e^{ikx} - {1 \over 1+i\hbar^2 k/m\alpha}e^{-ikx}\right]\;\;\;\;\;\;\;\;\;\;x<0$$

$$\psi_k(x) = {1 \over \sqrt{2\pi}}{i\hbar^2k/m\alpha \over 1+i\hbar^2k/m\alpha}e^{ikx}\;\;\;\;\;\;\;\;\;\;x>0$$

Show that these eigenfunctions satisfy the orthonormalization condition

$$\langle \psi_k\vert\psi_k'\rangle = \delta(k-k')$$

c.

Assuming the particle carries a charge $q$ and is subject to a perturbation of the form

$$H_1(t) = -qEX\sin\omega t$$

where $E$ is the amplitude of an applied electric field, calculate the transition rate using the Fermi Golden Rule expression for transitions from the bound state to a state with energy $E_k=\hbar^2 k^2/2m$.

4.

Problem 13.2, page 215 from Introduction to Quantum Mechanics in Chemistry.

5.

Problem 13.6, page 217 from Introduction to Quantum Mechanics in Chemistry.