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3-6: The Hypothesis of Light Quanta
and the Photoelectric Effect

[Einstein's Hypothesis of Light Quanta]
As learned on the preceding page, it was clarified that the energy of the radiation (light) in a cavity is integral multiples of the elementary unit . This can never be explained with the classical theory which consist of Newtonian mechanics and Maxwellian electromagnetism. The reason is as follows.
A cavity in a thermal equilibrium is exchanging energies with the surrounding wall every moment. In exchanging energies, the amounts of energies being given and taken are equal in average, and they are balanced. However, they are always fluctuating from moment to moment. Strictly speaking, the energies being given and taken are not necessarily equal, but always fluctuating. Accordingly, the total energy of the cavity radiation is always fluctuating. The energy of the cavity radiation must be a collection of the elementary quanta . The energy fluctuation must therefore be an integral multiple of the unit . Hence a finite amount of energy is coming in or going out of the cavity. Such a finite amount of instantaneous energy transfer is not allowed in the classical theory. Therefore Planck's idea of energy quanta can never be explained within the classical theory.
In 1905, Einstein proposed a hypothesis that the light (radiation) exists in a form of grains (or corpuscles or particles) in a space. This is the Einstein's hypothesis of light quanta.

[The Miraculous Year]
The year 1905 was a miraculous year (annus mirabilis) in the history of sciences. In 1905, A. Einstein published three great papers, each of which may be worth being awarded a Nobel Prize independently. The first was on the discovery of light quanta, which contains the theory of the photoelectric effect as discussed below. The second was on the theory of Brownian motion, which shows the real existence of atoms and molecules. The third contained the special theory of relativity.
An office worker of the Berne Patent Office published three great works just in one year, the amazing year of 1905. This was nothing other than a "miracle".

[The Photoelectric Effect]
A metal surface that is illuminated by light emits charged particles. This phenomenon was discovered by H. R. Hertz (Germany, 1857 - 94) while he was doing experiments concerning the electromagnetic waves. One of his students, P. E. A. von Lenard (Germany, 1862 - 1947) measured the charge-to-mass ratio of these particles and confirmed that the particles are electrons (1900). Hence this phenomenon is called the photoelectric effect.
A schematic drawing of the apparatus detecting the photoelectric effect is shown in the following figure.

The apparatus detecting the photoelectric effect
Two plates (electrodes) are put in a vacuum glass tube. When ultraviolet rays are illuminated on the one of the plates, an electric current flows if the illuminated plate is electrically negative as Case (A), but no current if it is positive as Case (B).

[Einstein's Theory on the Photoelectric Effect]
On the basis of the hypothesis of light quanta, Einstein proposed the following theory about the photoelectric effect (1905). This was contained in the same paper as the idea of light quanta was proposed.
Einstein thought that the light of a frequency is existing in a form of grains (corpuscles or particles) of the energy of and absorbed as whole units by electrons in a metal. These whole units are light quanta. If the energy gained by an electron is greater than the energy W necessary to carry the electron from inside to outside of the metal, then the electron would be emitted as a photoelectron.

Here W is called the work function, which has already been known by Richardson's research on the thermal electrons. ( Richardson: See the following figure (C).)

[Millikan's Experiment on the Photoelectric Effect]
The above-mentioned Einstein's theory on the photoelectric effect was proved by Millikan's experiment carried out in 1916. The schematic drawing of the apparatus for Millikan's experiment is shown in the above figure (D).
The experimental results are summarized as follows:
(1) If all the electrons emitted by the photoelectric effect are collected on the anode by setting the voltage V rather high, the electric current flowing through the ammeter would be proportional to the intensity of the light illuminated on the cathode.
(2) There is a threshold frequency of the light illuminated on the cathode for photoelectric emission. If the frequency of the light is less than this threshold value, no photoelectrons are released no matter how intense the illumination is.
(3) The maximum kinetic energy of the photoelectron is independent of the intensity of the light illuminated.
(4) The maximum kinetic energy of the emitted electrons is a linear function of the frequency, which is exactly the same as Einstein's hypothesis, i.e.,

Millikan obtained the constant h from this experiment as

which is well fit to the value obtained by Planck's analysis about the cavity radiation.

[Difficulty of the Classical Theory]
It is impossible to explain the above-mentioned results of Millikan's experiment within the classical theory consisting of Newtonian mechanics and Maxwellian electromagnetism.
We can easily imagine the followings within the classical theory: When a light is illuminated on the surface of a metal, electrons in the metal are violently shaked and oscillated by the electromagnetic field of the light. If the oscillation is too hard to keep the electrons inside the metal, they jump out of the metal surface. According to the classical theory, the energy given to the electrons in this case must be proportional to the square of the strength of the electromagnetic field. Hence the maximum energy of the photoelectron must be dependent on the intensity of the light illuminated. This completely contradicts Millikan's experimental results summarized above, in which (2), (3) and (4) can never be explained with the classical theory.
Moreover, we cannot explain the time to produce the photoelectric effect within the classical theory. The emission of the photoelectrons appears practically instantaneous. This cannot be explained with the classical theory. Let us discuss this below.
The value of the work function W of standard metals is Here let us discuss the photoelectric effect for a metal with The energy of light passing through an area of in 1 s (second) at a distance of 1 m from a point source of light at a rate of 1 W (watt) is When we illuminate this light on the metal, the energy of the light hit on an atom is since the cross section of an atom is nearly Suppose that the atom would absorb all this energy and it would be given to one electron in the atom, and thereby a photoelectric effect would occur. More energy than the value of the work function must then be accumulated on the electron before the emission. It takes about 100 s for this. This means that the photoelectron comes out about 100 s after switching on the light source. However the photoelectric effect occurs instantaneously in practice. Thus the classical theory cannot explain the photoelectric effect.
We could understand all without any contradiction, if we think that light is instantaneously absorbed as a whole unit of the energy by the electron in the metal. This is Einstein's hypothesis of energy quanta.
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