The Origin of the Quantum Theory

The Rayleigh-Jeans Law

1. Blackbody Radiation in Equilibrium. Historically, the quantum theory began with the attempt to account for the equilibrium distribution of electromagnetic radiation in a hollow cavity. We shall, therefore, begin with a brief description of the characteristics of this distribution of radiation. The radiant energy originates in the walls of the cavity, which continually emit waves of every possible frequency and direction, at a rate which increases very rapidly with the temperature. The amount of radiant energy in the cavity does not, however, continue to increase indefinitely with time, because the process of emission is opposed by the process of absorption that takes place at a rate proportional to the intensity of radiation already present in the cavity. In the state of thermodynamic equilibrium, the amount of energy U(v)dv, in the frequency range between v and v + dv, will be determined by the condition that the rate at which the walls emit this frequency shall be balanced by the rate at which they absorb this frequency. It has been demonstrated both experimentally and theoretically, that after equilibrium has been reached, U(v) depends only on the temperature of the walls, and not on the material of which the walls are made nor on their structure.

To observe this radiation, we make a hole in the wall. If the hole is very small compared with the size of the cavity, it produces a negligible change in the distribution of radiant energy inside the cavity. The intensity of radiation per unit solid angle coming through the hole is then readily shown to be I(v) = c/4π U(v), where c is the velocity of light.

Measurements disclose that, at a particular temperature, the function U(v) follows a curve resembling the solid curve of Fig. 1. At low frequencies the energy is proportional to v2, while at high frequencies it drops off exponentially. As the temperature is raised, the maximum is shifted in the direction of higher frequencies; this accounts for the change in the color of the radiation emitted by a body as it gets hotter.

By thermodynamic arguments Wien showed that the distribution must be of the form U(v) = v3f(v/T). The function f, however, cannot be determined from thermodynamics alone. Wien obtained a fairly good, but not perfect, fit to the empirical curve with the formula

U(v) dv ~ v3e-hv/κT dv (Wien's law) (1)

Here κ is Boltzmann's constant, and h is an experimentally determined constant (which later turned out to be the famous quantum of action).

Classical electrodynamics, on the other hand, leads to a perfectly definite and quite incorrect form for U(v). This theoretical distribution, which will be derived in subsequent sections, is given by

U(v) dv ~ κTv2 dv (Rayleigh-Jeans law) (2)

Reference to Fig. 1 shows that the Rayleigh-Jeans law is in agreement with experiment at low frequencies, but gives too much radiation for high frequencies. In fact, if we attempt to integrate over all frequencies to find the total energy, the result diverges, and we are led to the absurd conclusion that the cavity contains an infinite amount of energy. Experimentally, the correct curve begins to deviate appreciably from the Rayleigh-Jeans law where hv becomes of the order of κT. Hence, we must try to develop a theory that leads to the classical results for hv< κT, but which deviates from classical theory at higher frequencies.

Before we proceed to discuss the way in which the classical theory must be modified, however, we shall find it instructive to examine in some detail the derivation of the Rayleigh-Jeans law. In the course of this deviation we shall not only gain insight into the ways in which classical physics fails, but we shall also be led to introduce certain classical physical concepts that are very helpful in the understanding of the quantum theory. In addition, the introduction of Fourier analysis to deal with this classical problem will also constitute some preparation for its later use in the problems of quantum theory.

2. Electromagnetic Energy. According to classical electrodynamics, empty space containing electromagnetic radiation possesses energy. In fact, this radiant energy is responsible for the ability of a hollow cavity to absorb heat. In terms of the electric field, ε(x, y, z, t), and the magnetic field, H(x, y, z, t), the energy can be shown to be

E = 1/8π [integrity] (E2 + H2) dτ (3)

where dτ signifies integration over all the space available to the fields.

Our problem, then, is to determine the way in which this energy is distributed among the various frequencies present in the cavity when the walls are at a given temperature. The first step will be to use Fourier analysis for the fields and to express the energy as a sum of contributions from each frequency. In so doing, we shall see that the radiation field behaves, in every respect, like a collection of simple harmonic oscillators, the so-called "radiation oscillators." We shall then apply statistical mechanics to these oscillators and determine the mean energy of each oscillator when it is in equilibrium with the walls at the temperature T. Finally, we shall determine the number of oscillators in a given frequency range and, by multiplying this number by the mean energy of an oscillator, we shall obtain the equilibrium energy corresponding to this frequency, i.e., the Rayleigh-Jeans law.

3. Electromagnetic Potentials. We begin with a brief review of electrodynamics. The...