Stationary states may be confined (waveform vanishes at infinity) or not. The latter describe particle scattering; QMTools is able to resolve such scattering states into their incident, reflected, and transmitted components, thereby permitting reflection and transmission probabilities to be found at any energy.

Displaying Components of a Scattering Wave

1. Begin by creating a stationary state wave in the applet or the Applet Editor, as described in QM Waves; graph this wave (right-click in its formula field and select Plot Function from the popup menu).
2. For a stationary state to be unconfined, its energy must exceed the maximum potential energy; adjust the energy of the state accordingly to yield a scattering waveform. The computational algorithm employed by QMTools assumes particles incident from the left, so the wave at the far right of the display is the transmitted portion of the scattering waveform! To show the incident and/or reflected component(s), select the relevant entry from the popup menu that appears when you right-click on the waveform.
   The popup menu options for displaying incident and reflected components are available only for scattering waves; these menu entries are greyed (inactive) once the component is plotted.

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Computing Scattering Probabilities

1. The transmission characteristics follow directly from the transmitted portion of the scattering wave. The amplitude of the transmitted wave is easily found by right- clicking in the graph background and activating the Trace feature (see Plot Markup).
2. The waveform to the far left of the display is a superposition of incoming and reflected components. Display either one of these by right-clicking on the waveform and selecting the desired entry from the popup menu that appears. The Trace feature can then be used to find the incident and/or reflected component amplitudes.
   Amplitude measurements are especially easy to make if the component waves are plotted using the Color-4-phase plotting style (the default for unconfined waves).
3. The reflection probability at this energy is calculated by finding the ratio of reflection amplitude to incident amplitude, and squaring the result. A similar calculation (adjusted by the ratio of wave numbers on the transmitted and incident sides of the potential, if these are different) yields the probability of transmission.

Scattering probabilities also can be found at any energy without displaying the associated stationary state. For that purpose, QMTools furnishes the [built-in] reflectance and transmittance functions, "refl(...)" and "trns(...)", respectively. These functions – which require only that a suitable mass and potential energy function be assigned – are especially useful for exploring the transmission characteristics of a system (device?) as a function of particle energy. See QM Math for usage details.

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