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:

which is called the

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

Note that if , this reduces to the famous rest-energy expression . However, photons are massless particles that always have a finite momentum . In this case, Einstein's formula becomes . From Planck's hypothesis, one quantum of electromagnetic radiation has energy . Thus, equating these two expressions for the kinetic energy of a photon, we have

Solving for the wavelength gives

Now, this relation pertains to photons not massive particles. However, de Broglie argued that if particles can behave as waves, then a relationship like this, which pertains particularly to waves, should also apply to particles. Hence, we associate a wavelength to a particle that has momentum , which says that as the momentum becomes larger and large, the wavelength becomes shorter and shorter. In both cases, this means the energy becomes larger. That is, short wavelength

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

where denotes the complex conjugate of . The function is positive-definite, and if is appropriately chosen, will satisfy the normalization condition

where and denote the endpoints of the screen. is an example of a probability density or probability distribution function. Because is continuous, we cannot define a discrete probability since the probability to be at any particular value of is zero (there are an infinite number of values for ). What tells us is the probability to measure the particle's position on the screen within a small interval . In particular, the probability that a measurement of position yields a value in an interval centered on the point is . The amplitude is called a

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

If the two possibilities were completely independent, then their corresponding probabilities would simply add, and we we just have the first two terms in the last expression. However, there is a cross term that is generally nonzero, and it is this cross term that gives rise to the inteference between the two possibilites, which leads to the observed interference pattern. Let us look at this term in greater detail. Recall that a complex number can be expressed as

where is the magnitude of the number

and is the phase of the number

Letting

so that

and substituting into the probability distribution expression, we obtain

Recognizing that

the probability finally becomes

The oscillatory nature of the cosine is suggests of the oscillating pattern of bright and dark fringes in the interference pattern.