On Multi-Energy Square Wells

G.R.Dixon, 3/14/2007

In this article the wave functions of multiple-energy particles in an ensemble of infinite square wells are computed and summed to obtain a resultant Y. As in a previous article, the spinning spiral model for the individual Yi is used. The probability density (YY*) is plotted vs. x within the well.

Fig. 1 depicts the well. In the present case only energies whose wavelengths satisfy

(1)

are allowed. The 6 lowest energies are modeled.

Figure 1

The Infinite Square Well

Here again it is assumed that all of the Yi are Real and in phase at x=0. As in the multi-energy free particle case, we might accordingly expect a wave packet that, at t=0, has its maximum located at the left side of the well. Conventional wisdom in the present, bound case might lead us to expect the probability density not to vary in time. More specifically, conventional wisdom might lead us to expect the wave packet to remain stationary in the left part of the well. Such turns out not to be the case. As Figs. 2-4 show, in integral time multiples of (p/wo) the wave packet propagates to the right half of the well, is reflected back to the left half of the well, etc.

Figs. 2-4 plot YY* vs. x at the epochs t=n(p/wo), n=0,1,2 where

. (2)

Figure 2

YY* vs. x, t=0

Figure 3

YY* vs. x, t=p/wo

Figure 4

YY* vs. x, t=2p/wo

An obvious question is: what form does YY* have at times t=.5p/wo, t=1.5p/wo, etc. It might be expected that the wave packet will have its maximum in the middle of the well. However, here again this also turns out not to be the case. Figs. 5-6 plot YY* vs. x at these two epochs.

Figure 5

YY* vs. x, t=.5p/wo

Figure 6

YY* vs. x, t=1.5p/wo

If more energies are added, then the wave packet becomes more condensed. Fig. 7 depicts the case where 12 energies are modeled at t=0.

Figure 7

YY* vs. x, 12 Energies Modeled, t=0

Similar to the double slit experiment, the practice of summing the Yi prior to computing YY* amounts to stating that a priori a given well could contain a particle at energy Eo OR at E1 OR … Of course if we interact with the well then the sum shown in Fig. 2 (for example) is altered. Fig. 8 shows YY* vs. x when it is known that the energy is Eo. In such single-energy cases there are no wave packets, but only the curves derived in most elementary wave mechanics texts. The plot is unchanged at t=.5p/wo, etc. Fig. 9 shows the situation at this epoch.

Figure 8

YY* vs. x, E=Eo, t=0

Figure 9

YY* vs. x, E=Eo, t=.5p/wo