A wave function changing its form according to external conditions seems at first look to be a promising viewpoint to resolve the wave particle dilemma, but there are some problems associated with it.

The most important problem is the indeterminacy in the collapse of the wave function during the position measurement. Let’s formulate the problem of indeterminacy for the double slit experiment.

If each electron is a wave passing through both slits simultaneously , the wave corresponding to a single electron must be spread over the whole screen and must have the same form (the observed interference pattern) for each electron when it arrives the screen. If so, then why does the bright dot corresponding to a single electron appear each time at a very different location on the scintillating screen namely (expressed in wave-function only terms) why does the "condensatiton" of the wave function into a sharp peak occurs each time at a different position?

If the wave function’s behavior during the interaction with the screen is determined by a deterministic equation like the Schrodinger equation, the same initial condition should lead to same final condition.

There are two possible answers to this question:

1. There is a fundamental indeterminacy so that the process of collapse (the transition from an extended wave to a localized peak ) cannot be described by the Schrodinger equation, or more generally by any deterministic equation.

2. As it is always the case in a real experiment, we don’t have EXACTLY the same initial conditions but only nearly the same initial conditions because the wave length may differ slightly, the position of the atoms in the screen may differ slightly because of the thermal motion of the atoms, and it could be the case that these small initial differences lead to very distinct final states. After all there are many known such cases where susceptibility to initial conditions lead to a quasi-indeterminacy although the whole process is described by deterministic laws.

Lets consider a small sphere placed on a convex surface as in figure 8

Figure 8 - a small sphere on top of a convex surface .

The sphere falls each time on a different side, although it may be placed as exactly as possible at the center of the top. The same initial condition apparently leads to different final conditions.

The reason is that the motion of the sphere is very sensitive to initial conditions. In reality we can never place it exactly at the equilibrium position. Each time it is placed on a slightly different position. The forces acting upon the system are in such a delicate balance at the equilibrium so that if the system is slightly off the equilibrium position it is "forced" to move even further away from the equilibrium. Such a delicate balance of forces is called instability. If there is instability there is also a sensitivity to initial conditions .

Can the time evolution according to the Schrodinger equation lead to such an amplification of the initially slight differences? The Schrodinger equation is however a linear equation and linear equations cannot lead to such a behavior. Should then the Schrodinger equation be modified? But it was so successful in explaining the frequencies we encounter in line spectra. Would a modification not destroy this agreement between the theory and the experiment?

The wave function of a system of n electrons is not a superposition of n distinct clouds in our ordinary 3 dimensional space.

The mathematical description is consistent only if one assumes that the system of n electrons is a single "cloud " in a 3n dimensional space(For example for 5 electrons 3x5 = 15 dimensions).

This is not a real space. For 5 electrons it only means that the wave-function depends on 15 independent parameters, namely 3 coordinates for each particle. One cannot speak of finding electron-1 at a certain point. But one can speak of the probability of finding them in a certain positional correlation, namely for a two electron system the wave function gives the probability of finding electron-1 at the position x1 and electron-2 at the position x2 simultaneously. Because of this correlation of electrons it is not possible to visualize a system of 2 electrons as simply a larger cloud in 3 dimensional space that is the result of the superposition of 2 clouds.

This was one of the most important reasons beside the indeterminacy why, for example, Einstein claimed that the wave function alone is an inadequate representation of physical reality¹. The fact that an n particle system is described by a single function of 3n variables rather than as n separate functions each of 3 variables is called the entanglement of particles. The particles don’t behave as separate entities but the whole system behaves as a single entity. The Schrodinger equation describes the time development of this single wave function as a whole. As we will later see this has consequences that lead to an apparent conflict with the special relativistic speed limit if two distant measurements are conducted .

The wave function is represented by a complex number. A complex number has a real part and an imaginary part. (See appendix-1) The imaginary part contains the imaginary number i = sqrt of –1. Complex numbers are used in many areas of physics but only because it makes the calculations simpler. In all these cases the calculations could be done without using complex numbers.

In the case of Schrodinger equation however the situation is a little different. The equation itself cannot be written down without using the imaginary number i .

Many physicists thought that imaginary numbers cannot represent a physically real entity.

During the measurement of position, the wave function of a single electron collapses in a "very short time" (suddenly?) from a very spread form to a very localized form. During the collapse process the amplitude increases in the close neighborhood of the capturing atom rapidly while it decreases rapidly in even very far regions simultanously. Such a process implies unlimited wave propagation velocities . Propagation velocities that exceeds the light velocity are forbidden by the theory of special relativity. This property is called the problem of nonlocality in quantum mechanics. The problem becomes more striking if we have a two particle system because of the entanglement of the particles. This is described in a paper of einstein Rosen and Padolsky and is known as EPR paradox. We will discuss it in chapter 8

Problem 5.

According to the wave-only viewpoint a free particle with a definite trajectory is represented by a localized wave packet, i.e. a peak propagating with constant velocity. However according to the Schrodinger equation a free wave packet spreads in time by itself. Einstein claimed that this shows that the localized wave-packet cannot represent a localized particle. One fact that seems to support Einstein’s objection is that there is no indication that macroscopic objects spread in space with elapsing time.

Because of these difficulties, the wave-only viewpoint was more and more considered as inadequate or insufficient so that physicists felt themselves forced to draw the following conclusions:

1. Wave particle duality is inevitable. The wave function alone cannot represent a physical system. In a position measurement the particle nature reveals itself.

2 On the one hand the wave function allows us only to make probabilistic predictions about the outcome of a particular measurement. Namely it describes only the statistical result one obtains if the same experiment is repeated with many electrons. This implies that the wave function is not something that describes fully the behavior of an individual electron. On the other hand two facts support the view that even a single electron is a wave. The first fact is that the interference pattern emerges after a while even if we let the electrons one by one through the slits. This shows that even a single electron is passing through both slits simultaneously as a wave. The second fact is, the agreement of the measured energy levels in atoms with the predictions of Schrodinger equation indirectly shows that it is not merely a statistical description of an ensemble of electrons but it has consequences for an individual electron, namely it determines its energy in an atom. Thus the interpretation of the wave function remains ambiguous .

3. The problems 2 and 3 rise serious doubts if the wave function could be a physically real field at all like electromagnetic fields were assumed to be in classical physics. There was a tendency to interprete the wave function only as a probability wave namely as a mathematical construction to predict the probability of a particular outcome in a measurement.

1. Belousek 1996 p. 450