Shape Resonance in the Continuum WaveFunction

Powis Group

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Bound and Continuum Wavefunctions: the Shape Resonance

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To Use

The main panel (white) shows a model potential and wavefunctions in familiar text-book style (i.e. origin of Ψ vertically offset to match against its eigenvalue). In this area:

  • Left-click to read coordinates
  • Right-click to save snapshot in a new window

The black column to the right is used to show the spectrum of energies of the discrete eigenvalues. In this area:

  • Left-click on a displayed energy level (green) to re-display its wavefunction
  • Right-click-drag to instantaneously solve and display the wave function at the corresponding energy.*

The control area (bottom) can be used to find solutions at particular energies or discrete bound levels specified by quantum number n. For the adventurous the model potential can also be changed. The three buttons in the right hand panel can be used as a convenience to locate Ψ around a shape resonance in the trial potential.

[*]Nb. While dragging in the bound region between the allowed discrete energy levels an attempted solution Ψ will be shown but which has to be disallowed as it fails to meet the boundary conditions — Ψ will be seen to diverge at the right.

Discussion

This applet uses a model potential with a well and a narrow barrier. This "shape" in the potential may "trap" a particle that otherwise has sufficient energy to escape the well and pass out to large positive X — in our understanding, to ionize. The so-called shape resonance resulting from the potential's shape will be explored.

Bound Levels

The bound state wavefunctions Ψn (in the well) show the expected pattern — discrete levels with characteristic spacing. Notice that Ψn penetrates into the slightly "soft" left hand potential wall.

Continuum Wavefunction

The continuum (free-particle) wavefunctions generally display a long-range sinusoidal form that doesn't decay as X goes towards infinity. Notice how the de Broglie wavelength changes as the energy (momentum) is varied and also how for fixed total energy E above the barrier the wavelength varies above different regions of the regions of the potential.

Shape Resonance

Dragging the energy through the region with the narrow potential barrier (E=600–1350), note how the amplitude of the wavefunction resides "outside" this barrier. But also observe how there seem to be discontinuities or jumps at certain energies. Examine one of these regions closely. Notice how at certain discrete energies, above the effective ionization limit (here E=603) but behind the barrier, the wavefunction acquires more amplitude in the inner region, while penetrating through the barrier to the outer region. This is a shape resonance — a quasi-bound state where the electron gets temporarily trapped behind the barrier.

Amplitude

The increased amplitude in the inner region provides greater overlap with a bound state wavefunction (which is necessarily restricted to the same region) and so provides an enhanced cross-section for photoionization into the continuum at the resonant energy — a characteristic of such resonances.

Phase Shift

Either by careful dragging through the shape resonance, or by using the buttons to examine just above-, below-, and on-resonance, notice how the phase of the wavefunction in the outer (right hand) region jumps rapidly by 180° as we pass through the resonance. This is characteristic "signature" of the shape resonance and is anticipated to be very evident in the photoelectron angular distribution patterns.

Resonance Width

Finally notice from the variation of inner-region amplitude with energy, that the shape resonance has a finite energy-"width" unlike the bound state solutions which are obtained for discrete energies. In very simple terms this width represents an uncertainty in the energy which, according to the uncertainty principle, reflects a shortened lifetime. In contrast for a bound state the lifetime is infinite, hence the energy certain. The resonant trapped photoelectron can tunnel through the barrier and eventually escape, unlike the fully bound electron which remains trapped forever.