File:BarrierWave.gif

English: Animation showing rightward-traveling 1D free-particle (zero potential V) waves of energy U encountering a variable-V/U barrier/trough between 0<x<1, with help from a piece-wise continuous-potential Schrödinger-equation calculation. Wavefunction amplitude (brightness-saturation) and deBroglie phase (hue) are rendered in logarithmic complex-color.

In each of these panels, the horizontal axis runs from -6≤x≤+6; if we set particle mass and Planck's constant to 1, while the vertical axis shows step-potentials ranging from -6≤V/U≤+6. In the rightmost panel, the dotted horizontal-band is the region of positive barrier still lower than the particle energy (transmission still classically allowed), while the dotted vertical-band marks the barrier/trough region itself.

The top of the figure (esp. when V/U > 1) illustrates the exponential quantum-tunneling falloff in the transmitted wave (for 1≤x≤6) when the barrier's potential-energy V is above that of the incoming particle energy U. For x<0, it also illustrates the standing-wave formed by almost total reflection of the incident wave.

The bottom of the figure (when V/U < 0) illustrates the phase-lag proportional to projected-potential associated with electron deBroglie-phase contrast imaging, an important tool for the characterization of atom positions in modern-day nanoscale science and technology.