On the Infinite Square Well Solution forY
In an infinite square well, with walls at x=0 and L, Y is the sum of a wave traveling to the right and one traveling to the left:
The magnitude of YR and YL is constant, say Y0. Looking down the positive x axis from x=0, YR rotates CW in time and CCW in x:
Similarly, YL also rotates CW in time, but it also rotates CW in x:
(Note the -Y0 in Eq. 3; YL and YR must sum to zero at x=0 and L).
it is readily shown that
Y0 can be found from the normalization condition that
The model of YR and YL spiraling through a stack of complex planes can be used to demonstrate how, in any given plane, the first partial derivative of YR or YL with respect to time, and the second with respect to x, point in opposite "directions." This must be true if, for example,
is to be satisfied at all x. For example, we might say that Y0 is positive real, in which case points down (is negative imaginary) in the following complex plane.
Multiplication of by i rotates it CCW so that i points to the right (is real positive).
Since YR spirals CCW with increasing x, points to the left at x=0. The two terms thus point in opposite directions, and the directional requirement for summing to zero is satisfied.
It is instructive to recast Eq. 8 as
For an electron (m=9.11E-31 kg) the magnitude of is ~5.8E5 that of . Nevertheless it is readily shown (using the deBroglie relations) that Eq. 9 is equivalent to
It is also insightful to consider the product of YR (or of YL) with its complex conjugate. From Eq. 2
Thus YRYR* is the classically expected constant across the entire breadth of 0<x<L. The "quantum" behavior (Eq. 6) is a consequence of the interference (or superposition) of YR and YL.
Since YY*=0 at x=0 and L, an interesting question is whether the electron is never at x=0 or x=L. Born was careful to suggest that YY* is the probability the electron will be found in a given (x,dx) Ö a statement that does not seem to preclude the electron from ever being at x=0 or x=L. What Bornís interpretation of YY*dx seems to imply is that the electron will never be found when it is at x=0 or x=L. (In other words, the implication is that the electron will never interact with its environment when it is at x=0 or x=L.) But if the electron is occasionally at x=0 or x=L when we look for it, then there might be a small fraction of times when we fail to locate the electron anywhere in the potential well. The fraction is presumably so small (or even zero for a finite number of looks) that it need not concern us in normalizing Y.
The interference pattern in any given atom suggests that all possible modes (or electron orbitals), consistent with the atomís energy, are simultaneously occupied. Indeed without explicit knowledge of which (of the ensemble of possible modes) an electron occupies, it seems illogical to insist that it really is in one mode or another, notwithstanding macroscopic experience that such is the case. In the instance of the simpler square well, an analogous assertion would be that, at any given moment, the electron is actually traveling to the right or to the left, even though we have no a priori knowledge of which direction might be momentarily true. The sinusoidal form ofYY*, however, belies this assertion. Until we intervene and determine otherwise, the electron is simultaneously traveling in both directions, with the concomitant interference of YR and YL.