Eqtn 4.3.1
Eqtn 4.3.1
Although the dependence on barrier height is quite complicated through the kappa dependence, the variation with barrier thickness for fixed height is a simple exponential, at least to first order.In the following sequence, we show movies for the standard wavepacket approaching a rectangular potential barrier which has a height equal to 150% of the mean incident kinetic energy.
[The transmitted wavepacket is relatively small, and as the barrier thickness increases it becomes difficult to display the incident and transmitted packets on the same linear scale. For example, the transmitted packet peak amplitude psi psi* for a barrier thickness of .05 distance units is less than 10^-4. Therefore, in the following movies we use a logarithmic scale for the probability amplitude. The wavepackets now take on a parabolic shape:]
As a useful exercise, the amplitudes of the transmitted packets can be plotted as a function of barrier height, making use of a ruler to measure amplitudes directly off the screen. The analytical prediction of an exponential dependence, Eqtn. 4.3.1, can then be checked. (The actual exponent will differ slightly from the prediction of Eqtn. 4.3.1, since we are not dealing here with a simple plane wave)
Before leaving the simple barrier, there is one interesting feature which has been recently discussed in the popular scientific press ("Faster than the speed of light", Julian Brown, New Scientist, 1 April 1995, pp 26-30), and in the scientific journals (eg. "Barrier Interaction Time in Tunnelling", R. Landauer and Th. Martin, Revs. Mod. Phys 66, 217-228, 1994). Tunnelling effects can also occur for electromagnetic waves in some situations, and it has been proposed that both types of waves can result in the transfer of information with a speed greater than that of light. The movies shown here can be used to illustrate a speed enhancement on tunnelling, although the effect is too small to measure directly from the screen. Using data from the Schroedinger program for the last movie (.025 thick barrier), the following graph shows the distance versus time relations for the incident, reflected and transmitted wavepackets. (Note that it is impossible to determine these relations close to the barrier because of distortion of the envelope shapes. The dashed lines represent extrapolations through to the barrier region). The incident wavepacket is the standard one with k0=100*pi.
Fig 4.3.1. Wavepacket Incident on a Potential Barrier of height 1.5(k0)^2, Width .025 Distance Units.
The transmitted wavepacket has a slightly larger mean wavevector, and hence a greater velocity, than the incident one because of the non-linearity of the transmission coefficient with wavevector (components with larger wave vector have a greater tunnelling probability). However, by comparing the extrapolated x vs t curves at the barrier centre it can be seen that there is indeed a measurable time advance for the transmitted wavepacket compared to a packet which did not have to tunnel through the barrier.
Also of interest is the time delay for the reflected wave. (Time delays
on reflection are discussed by J. Lekner, "Theory of Reflection"
(Martin Nijhoff, Dordrecht, 1987).) The time delay is evident in the figure
in that the point of intersection of the (extrapolated) incident and reflected
wavepacket trajectories lies inside the barrier; ie the effective reflecting
surface is below the barrier surface.
Reference: Reference: "Quantum
Mechanics", E. Merzbacher, Wiley International, New York, 1970, 2nd edition,
Exercise 8.12