Einstein’s Paper on Light Quanta

ReferenceA Logical Approach to Theoretical Physics

In his very first paper published in 1905 Einstein establishes the concept of “energy quantum” or “light quantum”. The energy of light quantum (photon) is proportional to frequency that becomes more pronounced as one moves up the electromagnetic spectrum. Here is a summary of Einstein’s 1905 paper on Light Quantum followed by some comments.

SECTION 0: Introduction

Einstein says,

The energy of a ponderable body cannot be split into arbitrarily many, arbitrarily small parts, while the energy of a light ray, emitted by a point source of light is according to Maxwell’s theory (or in general according to any wave theory) of light distributed continuously over an ever increasing volume.

In other words, Einstein observes that atoms are not infinitely divisible, but the electromagnetic radiation is treated as a continuum and infinitely divisible.

Einstein says,

“In fact, it seems to me that the observations on “black-body radiation”, photoluminescence, the production of cathode rays by ultraviolet light and other phenomena involving the emission or conversion of light can be better understood on the assumption that the energy of light is distributed discontinuously in space.”

In other words, Einstein proposes that the creation and conversion of light may not be continuous.

SECTION 1. On a Difficulty in the Theory of “Black-body Radiation’’

In this section Einstein sets up a thought experiment. He assumes electrons to be particles that are colliding like gas molecules. There are bound “resonator electrons” that emit and absorb electromagnetic waves with definite periods. The average kinetic energy of a resonator electron must equal the average kinetic energy corresponding to the translational motion of a gas molecule under dynamic equilibrium. Similarly, it should also equal the energy of interaction with radiation present in space.

Einstein says,

“This relation, which we found as the condition for dynamic equilibrium does not only lack agreement with experiment, but it also shows that in our picture there can be no question of a definite distribution of energy between aether and matter. The greater we choose the range of frequencies of the resonators, the greater becomes the radiation energy in space…”

This was famously known as the ultraviolet catastrophe.

SECTION 2. On Planck’s Determination of Elementary Quanta

In this section Einstein shows that “determination of elementary quanta given by Mr. Planck is, to a certain extent, independent of the theory of “black-body radiation” constructed by him.”

Using mathematics to back up his argument, Einstein concludes:

“The higher the energy density and the longer the wavelengths of radiation, the more usable is the theoretical basis used by us; for short wavelengths and low radiation densities, however, the basis fails completely.”

In other words, the radiation appears continuous per Maxwell’s theory at lower frequencies, but not at higher frequencies.

SECTION 3. On the Entropy of the Radiation

In this section Einstein presents Wien’s consideration that entropy of radiation may be determined completely from black body radiation law when the radiation energy is given for all frequencies.

SECTION 4. Limiting Law for the Entropy of Monochromatic Radiation for Low Radiation Density

In this section Einstein uses Wien’s approximation (valid for higher frequencies of black body radiation) to derive an equation for the entropy of radiation.

Einstein writes:

“This equation shows that the entropy of a monochromatic radiation of sufficiently small density varies with volume according to the same rules as the entropy of a perfect gas or of a dilute solution.”

Thus, Einstein proves that the energy distribution of radiation becomes particle-like at high frequencies. This is an ingenious way of arriving at this conclusion.

SECTION 5. Molecular-Theoretical Investigation of the Volume-dependence of the Entropy of Gases and Dilute Solutions

In this section Einstein shows that, when applied to a large number of discrete particles, the use of “statistical probability” is compatible with macroscopic laws of physics.

SECTION 6. Interpretation of the Expression for the Volume-dependence of the Entropy of Monochromatic Radiation according to Boltzmann’s Principle

In this section, Einstein uses mathematical arguments to conclude:

“Monochromatic radiation of low density behaves—as long as Wien’s radiation formula is valid—in a thermodynamic sense, as if it consisted of mutually independent energy quanta of magnitude Rßv/N.”

Each quantum is the energy of one interaction. Einstein mathematically determines the theoretical value of a quantum.

SECTION 7. On Stokes’ Rule

In this section Einstein uses the new idea of “energy quanta” to explain the Stokes’ Rule for photoluminescence and indicates new possibilities.

SECTION 8. On the Production of Cathode Rays by Illumination of Solids

In this section Einstein brilliantly verifies the calculated value of energy quanta from the experimental value obtained from the study of photoelectricity. Here we have the conclusive evidence that energy of light is made up of frequency (kinetic energy) and not amplitude (wave energy).

SECTION 9. On the Ionization of Gases by Ultraviolet Light

In this section Einstein tests his ideas to explain the existing experimental observations and further proves the viability of the idea of “energy quantum” or “light quantum”.

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Comments

Einstein’s concept of quantum is based on energy. From the phenomenon of photoelectricity, it can be seen that this energy is kinetic (based on frequency) and not that of a wave (based on amplitude). Light carries this energy with it. This means light is a fast-moving substance.

The kinetic energy depends on relative velocity between two things. Therefore, this energy is visible only when there is interaction between two things, such as, light and electron. Einstein is mathematically comparing this interaction to collisions among gas molecules in the kinetic theory of gases.

Therefore, quantum is tied to the energy of interaction. If there is no interaction, the energy, or the quantum, cannot be perceived. The very perception of quantum requires an interaction. Interactions are discrete. Therefore, quantum is discrete also, but only as energy of interaction. This is what Einstein is thinking about when he says, The energy of a ponderable body cannot be split into arbitrarily many, arbitrarily small parts…”

In the kinetic theory of gases, not only the interactions are discrete, but the interacting gas molecules are discrete also. The molecules are discrete because they have individual centers of mass. This is not the case with light because light has no centers of mass. Therefore, light “particles” cannot be distinguished from each other. Light forms a continuum in space even when its interactions are discrete.

Einstein disagreed with Maxwell treating energy of light as a continuous function across the spectrum; but he did agree with Maxwell for energy being continuous at the lower end of the spectrum. Maxwell treated light as a wave, which is not really the case. There is no stationary aether through which light is moving as a disturbance. Therefore, a continuous energy function at lower frequencies can only mean that energy interactions are so frequent that they appear continuous.

Thus, as the frequency reduces, light starts to act as a continuum in terms of energy interactions also. This can be used as an argument to support the observation that, fundamentally, light is infinitely divisible. As frequency increases, the energy interactions become increasingly differentiable.

Electrons also form a continuum in space similar to light, but their frequency is much higher. When light interacts with electrons in the photoelectric phenomenon, it is two continuums of very different frequencies interacting with each other, and not two particles. Any interaction shall only be in terms of partial resonance, and not as an impact between two billiard balls.

Much seems to be unknown about the nature of this interaction between light and electron.

The frequency of light may best be understood as the density of its continuum. The higher is the frequency, the greater is the density of light. There appears to be a high-density gradient from the electronic region to the nucleus within the atom. This is where the charge appears and the center of mass forms. This is an area of transition where radiation appears to be in equilibrium with matter.

Much seems to be unknown about this area of transition from electronic region to nucleus of atom.

In conclusion, let us look at the following assumption made by Einstein in this paper:

According to the assumption considered here, when a light ray starting from a point is propagated, the energy is not continuously distributed over an ever increasing volume, but it consists of a finite number of energy quanta, localized in space, which move without being divided and which can be absorbed or emitted only as a whole.

It appears that quanta are more particle-like only because the density of the continuum has increased.

From Faraday’s perspective, quanta can be represented by thicker lines of force, but those lines are still continuous.

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