Mechanical WKB Barrier Approximation
The WKB (Wentzel-Kramers-Brillouin) applet shows how the transmission coefficient
through a barrier of variable potential, shown in blue, can be approximated using a succession of square barriers.
The transmission by each square element is shown. The number of square approximations can be altered. Energies
in eV and dimensions in Å are relevant to electrons. The same qualitative effect with suitable scaling applies
to alpha particles at energies MeV.
- What you see is the value of the natural logarithm (ln) of the transmission coefficient
at the end of each square barrier (purple)
calculated by the wkb method. This is graphed as lnT(app). Also drawn is a schematic representation of the shape
of the incident and transmitted wavefunction (green). Below the diagram is shown the total transmission factor calculated by numerical integration
compared with the wkb approximation for the number of square barriers chosen.
- The main diagram shows the potential drawn as a blue
line, the square barriers used in the approximation (shaded yellow).
- To illustrate that the transmission coefficient drops after each barrier the
shading of the boxes gets progressively lighter.
- The energy of the incident electrons is shown by a red
line whose height can be adjusted up to the top of the potential.
- The height and width of the barrier can be adjusted by clicking and dragging
on the orange markers
in the centre and to the right of the potential.
- The box labelled number of steps can be set at any number from 1 to 100. The default value is 6.
- Scales are provided to show how wide the barrier is and also to show the energy
- Boxes at the top of the screen display the maximum value of the potential, the
excess energy of the electrom from this maximum (shown as a negative number, in
red) and the width of the barrier at the electron energy.
- Left click on the University Logo to decrease the scale by a factor of 10. The
scale marker changes to 0.2 Å.
- Quantum Mechanics Concepts and Applications
N Zettili. John Wiley and Sons 2001
- Quantum Physics S Gasiorowicz. John
Wiley and Sons 1974
The applet and web pages were
designed by Gary Skinner, as part of a project sponsored by LSTN