Historical research guide · 0773 · 1994
Computer modeling of resonance scattering in the time domain
R. C. Greenhow
In one minute
Quantum scattering is usually taught as a still photograph: send in a single-energy wave, read off how much comes out at each angle, plot a cross-section. R. C. Greenhow’s teaching paper does the opposite. He integrates Schrödinger’s time-dependent equation on a microcomputer and turns the answers into a motion picture, so that a student watches a Gaussian wave packet — a lump carrying a spread of energies, which is what a real particle actually is — travel towards a spherical step-potential well and meet it. The film shows the thing the still photograph hides. At certain energies the packet neither passes straight through nor bounces straight off. Part of it is captured, sits inside the well ringing for a while, and then leaks back out. Those are transient resonance states, and Greenhow measures how long each one lasts. The decay times line up with the widths of the resonances: the narrower the resonance, the longer the trapped state lives. A resonance, watched in time, is a delay.
Connection
Chapter 2 rests on the idea that the vacuum is a medium with modes and resonances, and this paper teaches the one fact that makes those resonances physical rather than bookkeeping — a resonance width is a lifetime, and the state really does sit there for that long before letting go. Chapter 4 needs the same habit of mind: the interesting quantity is not only where a wave ends up but how long it is held on the way, which is a thing the geometry can change.
From the source
Solutions of Schrödinger’s time-dependent equation can be generated numerically on a microcomputer and rendered as motion pictures — a quantum process shown unfolding in time rather than summarised as a stationary cross-section. Greenhow describes the technique as the paper’s first contribution, and the point of it is pedagogical: what a student sees is the process, not its summary statistic.Abstract, first sentence
The technique is applied to the resonance scattering of a Gaussian wave packet by a spherical step-potential well. The choice of a packet rather than a plane wave is the load-bearing one: a packet carries a spread of energies at once, so a single run contains every energy in the band, and the well selects from among them by itself instead of the calculation being repeated energy by energy.Abstract, second sentence
Transient resonance states are observed to form. At the resonant energies the incoming amplitude is not simply transmitted or reflected; part of it is held inside the well as a quasi-bound state that persists for a time and then leaks away. This is what a resonance is when you watch it rather than plot it — a state that exists, briefly, at an energy where no bound state is allowed to exist permanently.Abstract, third sentence
The measured decay times of those transient states correlate with the widths of the resonances. This is the time-and-energy relation made visible: a narrow resonance is a long-lived state and a broad one is a short-lived state, and the simulation recovers that correspondence by direct measurement of the decay rather than by assuming it. It is the same relation that ties the width of a spectral line to the lifetime of the level that produced it.Abstract, third sentence
What to watch: the method Greenhow taught for a spherical well is now the working method for the vacuum itself. Casimir forces are computed in the time domain by propagating fields and reading off the response rather than by summing static modes; the dynamical Casimir effect is by definition a time-domain phenomenon; and the newest work asks what an indistinguishability-based, time-domain view of quantum fields changes about how vacuum and source contributions are counted. The question worth following is how far a resonance-and-lifetime picture of the vacuum’s modes can be pushed before it needs replacing.Written from the site’s own reading; see the cross-links below
The way in
https://doi.org/10.1119/1.17604How to cite it
R. C. Greenhow (1994) Computer modeling of resonance scattering in the time domain doi:10.1119/1.17604