On the Infinite Square Well Solution for Y

G.R.Dixon, 2/16/2007

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:

. (1)

The magnitude of Y_{R} and Y_{L} is constant, say Y_{0}.
Looking down the positive x axis from x=0, Y_{R} rotates CW in time and CCW in x:

. (2)

Similarly, Y_{L} also rotates CW in time, but it also rotates CW in x:

. (3)

(Note the -Y_{0} in Eq. 3; Y_{L} and Y_{R} must sum to zero at x=0 and L).

Using

, (4a)

, (4b)

, (4c)

, (4d)

, (4e)

it is readily shown that

. (5)

And

. (6)

Y_{0} can be found from the normalization condition that

. (7)

The model of Y_{R} and Y_{L} spiraling through a stack of complex planes can be used to demonstrate how, in any given plane, the first partial derivative of Y_{R} or Y_{L} with respect to time, and the second with respect to x, point in opposite "directions." This must be true if, for example,

(8)

is to be satisfied at all x. For example, we might say that Y_{0} is positive real, in which case points down (is negative imaginary) in the following complex plane.

Figure 1

Direction of

Multiplication of by i rotates it CCW so that i points to the right (is real positive).

Since Y_{R} 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

(9)

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

. (10)

It is also insightful to consider the product of Y_{R} (or of Y_{L}) with its complex conjugate. From Eq. 2

. (11)

Thus Y_{R}Y_{R}* 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 Y_{R} and Y_{L}.

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) … a statement that does not seem to preclude the electron from ever *being* at x=0 or x=L. What Born’s 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 *any*where 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 atom’s 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 of YY*, however, belies this assertion. Until we intervene and determine otherwise, the electron *is* simultaneously traveling in both directions, with the concomitant interference of Y_{R} and Y_{L.}