The first, proposed by Clinton Davisson and Lester
Germer in 1927, was based on a hypothesis put forth earlier by Louis de
Broglie in 1922. De Broglie suggested that if waves (photons) could
behave as particles, as demonstrated by the photoelectric effect,
then the converse, namely that particles could behave as waves,
should be true. He associated a wavelength to a
particle with momentum using Planck's constant as the
constant of proportionality:
In the following, we give a brief discussion of where the de Broglie hypothesis comes from. From the photoelectric effect, we have the first part of the particle-wave duality, namely, that electromagnetic waves can behave like particles. These particles are known as photons, and they move at the speed of light. Any particle that moves at or near the speed of light has kinetic energy given by Einstein's special theory of relatively. In general, a particle of mass and momentum has an energy
If particles can behave as waves, then we need to develop a theory for this type particle-wave. We will do this in detail when we study the Schrdinger wave equation. For now, suffice it to say that the theory of particle-waves has some aspects that are similar to the classical theory of waves, but by no means can a classical wave theory, like that used to describe waves on a string on the surface of a liquid, be used to formulate the theory of particle waves. To begin with, what is the very nature of a particle wave? Here, we will give only a brief conceptual answer.
A particle-wave is still described by some kind of amplitude function , but this amplitude must be consistent with the fact that we could, in principle, design an experiment capable of measuring the particle's spatial location or position. Thus, we seem to have arrived at a paradoxical situation: The electron diffraction experiment tells us that particle-waves can interfere with each other, yet it must also be possible to measure a particle-like property, the position, via some kind of actual experiment. The resolution of this dilemma is that the particle exhibits wave-like behavior until a measurement is performed on it that is capable of localizing it at a particular point in space. Making this leap, however, has a profound implication, namely, that the outcome of the position measurement will not be the same in each realization even if it is performed in the same way. The reason is that if it did yield the same result, then we could say that the particle was evolving in a particular way that would put it there when the measurement was made, and this negates the possibility of its ever having a wave-like character, since this is exactly how classical particles behave. Thus, if the result of a position measurement can yield different outcomes, then the only thing we can predict is the probability that a given measurement of the position yields a particular value. The quantum world is not deterministic but rather intrinsically probabilistic.
If we are only able to predict the probability that a measurement of position will yield a particular value, then how do we obtain this prediction. Recall that the particle-wave is described by an amplitude function . Let us suppose that the particle-wave reaches the screen at time when the amplitude is , which we denote as (when is fixed, the amplitude is a function of alone). Here, denotes the position along the screen. The screen, itself, acts as an apparatus for measuring the particle's position. The amplitude can be either positive or negative, and we could even choose it such that it is a complex number. Thus, is not a probability because it is not positive-definite, and it is not real. However, we can turn into a probability by taking the square magnitude of . We define a new function
So why does an interference pattern arise? Because we do not observe the particle's position until it reaches the screen, we have to consider two possibilities for the particle-wave: passage through the uppoer slit and passage through the lower slit, and we assign to one possibility a probability amplitude and other an amplitude . Just as with ordinary waves, we must add the amplitudes to obtain the total wave amplitude, to the total amplitude is . The probability that we observe something at a given point on the screen in an interval is