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A Suggested Alternate Approach to the Normalization of Y

G.R.Dixon, 11/25/03

Fig. 1 depicts one of N infinite square potential energy (U) wells, where N is arbitrarily large. Trapped in each well is a single particle with mass m and speed v.

Figure 1

One of N Identical Square Wells

According to quantum theory a wave function, Y(x), is associated with this ensemble of wells and particles. Only integral half wavelengths can be accommodated:


DeBroglie suggested a connection between l and particle momentum magnitude:

. (2)

Thus only discrete momenta (and kinetic energies) can be accommodated.

Born interpreted the square of the wave function’s magnitude (multiplied by dx) to be the probability of finding a particle in a given interval dx:

. (3)

Fig. 2a shows an approximation to the lowest (ground state) P(x), and Fig. 2b shows an approximation to the first excited state’s P(x).

Figure 2a

P(x), Ground State

Figure 2b

P(x), First Excited State

But what is meant operationally by P(x)dx? Let us assume that the span {–L/2<x<L/2} is divided into M intervals of width dx, where M is very large:

. (4)

Our only requirement is that M be much less than N, the total number of identical systems in our ensemble (so that, at least classically, there is a reasonable chance of finding a particle one or more times in every interval). We now determine and tally the number of times the particle is found in each of the M intervals. Let n(x)dx be the number of times we find the particle in the interval dx centered on x. Then by definition

. (5)

Note that classically we would expect P(x) to be single-valued:

(classical). (6)

But as Figs. 2a and 2b indicate, the quantum reality is quite something else.

The amplitude of P(x) is determined by the process of normalization. The reasoning behind normalization is that the particle will presumably always be found somewhere, inside any given well, the first time we look for it. Thus the probability of finding it in some interval is unity:

. (7)

To the extent dx is infinitesimal, |Y(x)|2dx is sometimes referred to as the probability of finding the particle at x.

It is interesting that the assumption, that we will invariably find the particle the first time we check any of the N systems in our ensemble, logically leads to a paradox. Consider the first excited state (full wave) in Fig. 2b. The probability of finding the particle at x=0 is zero … a result often interpreted to mean that a particle is never at x=0! But if this is the case, then how can particles travel back and forth between x=-L/2 and x=L/2? (Indeed in all energy states the particles are presumably never at x=+L/2, the only points where their momenta can be classically reversed.)

Let us consider how the situation might change if we do not invariably find a particle the first time we check an arbitrary system? (Or more generally, what if a particle does not invariably interact with something between the walls when it encounters it?) There is ample evidence (e.g. the Franck-Hertz experiment) that particles do not always interact with other entities in their environment when an encounter occurs. In the present case, what if we must sometimes look at a system more than once in order to locate its particle?

If this latter possibility is the case, then the door is opened to interpret a zero (or infinitesimal) P(x)dx in a slightly different way. If YY*dx=0 then the particle will never interact with its environment when it is at that particular x. Note how this latter interpretation allows that a particle can be at every x between –L/2 and L/2. But (for mysterious reasons) it will never interact with its environment (including us) when it is at one of the YY* nodes.

Mathematically, let us say that we look at each system in our ensemble of N systems once. We actually detect a particle, at one x or another, Q times, where Q<N. And we fail to find the particle at any x (N-Q) times. The probability of finding the particle somewhere in the well is no longer 1 (a certainty); rather it is Q/N. The probability of finding a particle in (x,dx) is again

. (8)

But the normalization condition is now

. (9)

The new feature is how we interpret YY* at any given x. It is the number of times we find the particle at x, and not the number of times the particle is at x. There may be times the particle is at x but fails to interact with us.

Note that a solution of the Schroedinger equation (i.e. Y) does not provide information on what Q might be. Evidently it is only through experimentation that we can determine how often we are successful in finding a particle. Once we have determined what Q is, however, the theory (i.e. the wave equation, etc.) theoretically gives us the correct weighting function Y. And Eq. 9 (the modified normalization condition) determines the function’s amplitude.