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 Y_{i} 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 ____Y _{i} are Real and in phase at x=0.__ As in the multi-energy

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

. (2)

Figure 2

YY* vs. x, t=0

Figure 3

**YY* vs. x, t=**p/w_{o}

Figure 4

**YY* vs. x, t=**2p/w_{o}

An obvious question is: what form does YY* have at times t=.5p/w_{o}, t=1.5p/w_{o}, 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/w_{o}

Figure 6

YY* vs. x, t=1.5p/w_{o}

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 Y_{i} prior to computing **YY* **amounts to stating that *a priori* a given well could contain a particle at energy E_{o} __OR__ at E_{1} __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 E_{o}. 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/w_{o}, etc. Fig. 9 shows the situation at this epoch.

Figure 8

YY* vs. x, E=E_{o}, t=0

Figure 9

YY* vs. x, E=E_{o}, t=.5p/w_{o}