Quantum Mechanics (1) · Problem 9

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Physics — Quantum Mechanics · quantum-mechanical-resonances-breit-wigner

Problem

Resonances II: Quantum mechanical resonances -- Earlier we investigated some features of a classical oscillator with a ``resonant'' behavior under a driving force. Let us begin now to develop a quantum mechanical analogue, of relevance also to scattering and particle decays. For concreteness, consider an atom with two energy levels, E0<E1E_0 < E_1, where the transition E0→E1E_0 \to E_1 may be effected by photon absorption, and the decay E1→E0E_1 \to E_0 via photon emission. Because the level E1E_1 has a finite lifetime -- we denote the mean lifetime of the E1E_1 state by τ\tau -- it does not have a precisely defined energy. In other words, it has a finite width, which (assuming that E0E_0 is the ground state) can be measured by measuring precisely the distribution of photon energies in the E1→E0E_1 \to E_0 decay. Call the mean of this distribution ω0\omega_0. (a) Assume that the amplitude for the atom to be in state E1E_1 is given by the damped oscillatory form: ψ(t)=ψ0 e−iω0t−t2τ\psi(t) = \psi_0\, e^{-i\omega_0 t - \frac{t}{2\tau}} Show that the mean lifetime is given by τ\tau, as desired. (b) Note that our amplitude above satisfies a ``Schrödinger equation'': idψ(t)dt=(ω0−i2τ)ψ(t)i\frac{d\psi(t)}{dt} = \left(\omega_0 - \frac{i}{2\tau}\right)\psi(t) Suppose we add a sinusoidal ``driving force'' Fe−iωtFe^{-i\omega t} on the right hand side, to describe the situation where we illuminate the atom with monochromatic light of frequency ω\omega. Solve the resulting inhomogeneous equation for its steady state solution. (c) Convince yourself (e.g., by ``conservation of probability'') that the intensity of the radiation emitted by the atom in this steady-state situation is just ∣ψ(t)∣2|\psi(t)|^2. Thus, the incident radiation is ``scattered'' by our atom, with the amount of scattering proportional to the emitted radiation intensity in the steady state. Give an expression for the amount of radiation scattered (per unit time, per unit amplitude of the incident radiation), as a function of ω\omega. For convenience, normalize your expression to the amount of scattering at ω=ω0\omega = \omega_0. Determine the full-width at half maximum (FWHM) of this function of ω\omega, and relate to the lifetime τ\tau. Note that the ``Breit-Wigner'' function is just the Cauchy distribution in probability.

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