Hendrik A. Lorentz

(1853-1928)

Hendrik A. Lorentz was giant in theoretical physics who bridged the gap between classical electromagnetic field theory and modern relativity theories.

He put forward a theory of the electron and he developed the famous Lorentz transformations that describe how objects appear contracted in the direction of their motion to observers in a frame at rest. Lorentz's equations provided the basis for Albert Einstein's theory of special relativity.

Lorentz had many unpublished conversations with Einstein, some of which provide insight into Einstein's thoughts on the mysterious relationship between discrete light quanta (particles) and the continuous waves of classical electromagnetic theory. It shows that Einstein had a statistical view of the quanta. The probability of finding quanta is determined by the continuous wave, which controls the interference even for one quantum at a time.

Lorentz also describes the two-slit experiment.

Excerpt from *Problems of Modern Physics* (1922 Lectures at Cal Tech)

**50. Interference and the Quantum Theory.** I tried to explain to
you how the production of light by quantum jumps can perhaps
be reconciled with our old views concerning radiation, so that
these would hold as to the constitution of the emitted radiation.
But the question arises, Can this constitution be really just what
we have thought; that is, can there be a propagation according
to Maxwell's laws, with a tendency to spread out in all directions
and the impossibility of a lasting concentration of energy?

You know that phenomena like those of photo-electricity
have led Einstein to his hypothesis of light-quanta. According to
this, quantities of energy equal to *hν* would be concentrated in
small spaces, moving with the speed of light; they would even
be light and would produce all optical effects. In this way we
can understand that even very feeble light can give to an electron
the amount of energy *hv*, for the smallness of the intensity
would be due to the small number of quanta which it contains,
the magnitude of each remaining the same.

Einstein described this difficulty in 1905.

So we should
escape the difficulty which, in the case of wave-motion, arises
from the continual spreading out and weakening of the energy.

The hypothesis of light-quanta, however, is in contradiction
with the phenomena of interference. Can the two views be
reconciled? I should like to put forward some considerations
about this question, but I must first say that Einstein is to be
given credit for whatever in them may be sound. As I know
his ideas concerning the points to be discussed only by verbal
communication, however, and even by hearsay, I have to take
the responsibility for all that remains unsatisfactory.

Let us suppose that in the emission and propagation of light
there is something that conforms wholly to Maxwell's equations,
but that it has practically no energy at all, the electric
and magnetic forces being infinitely small.

Today this Fresnel (interference) radiation is the probability amplitude wave function *ψ*

Then in this, let
us say, Fresnel radiation we shall have the ordinary laws of
reflection, interference, and refraction, but we shall see nothing
of it. On a screen you will have something like an undeveloped
photographic image.

We can now imagine that in the production of light this
Fresnel radiation is accompanied by the emission of certain
quanta of energy that are of a different nature. Although their
precise nature is unknown, we may suppose that energy is concentrated
in small spaces and remains so. These quanta move
in such a way in our "pattern" that they can never come to a
place where in this pattern there is darkness. In thus traveling
from the source outward each quantum has a choice between
many paths.

The intensity of the radiation gives the probability of finding light quanta, just as Born's rule (1926) says the probability of finding material particles is proportional to the square of the wave function

The probability of following different paths is
proportional to the intensity of the radiation along these paths
in Fresnel's radiation.

Now in all real cases the act of emission is repeated a great
many times. Suppose it is repeated *N* times, and let the Fresnel
radiation be the same in these different cases. Then we shall
have *N* quanta moving in this pattern, and if their number is
very great and the probability of following different paths as
stated, the number of quanta coming on different parts of a
screen on which we observe an interference phenomenon will be
proportional to the intensity which we have in Fresnel's pattern.
These considerations can easily be extended. Take, for
instance, polarization. The polarization will be in the Fresnel
pattern, not in the quanta, but the quanta will illuminate a
screen or a photographic plate or our retina to exactly the
degree determined by the classical theory.

Or consider light passing through

two slits, one particle at a time

When light falls on the surface of a piece of glass, there is
a partition between the reflected and refracted parts. The
probability of the quantum's following one path or another is
determined by the well-known formulae of Fresnel for the intensities
of the reflected and the refracted light.

Suppose that in an elementary act of radiation there are a
million waves; these exist in Fresnel's pattern; but the quantum
of energy can have any place in the train of waves, either
near the front or near the rear of these waves.

If we have an ordinary beam of light consisting of the superposition
of a great number of elementary beams, we have quanta
in great number distributed all through the space occupied by
the beam.

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