General principles

The quantum theory of line-spectra rests on the following fundamental assumptions:

1. An atomic system can only exist permanently in a certain series of states corresponding to a discontinuous series of values for its energy.

Consequently, any change of the energy of the system, including emission and absorption of electromagnetic radiation, must take place by a complete transition between 2 such states.

These states will be denoted as the “stationary states” of the system.*

2. The radiation absorbed or emitted during a transition between 2 stationary states is “unifrequentic” and possesses a frequency ν, given by the relation:

E' - E'' = hv (1)

where:

• h is Planck’s constant
• E' and E'' are the values of the energy in the 2 states under consideration.

These explain the combination of spectral lines deduced from the frequency-measurements of the series spectra of the elements.

According to the laws discovered by Balmer, Rydberg and Ritz, the frequencies of the lines of the series spectrum of an element is expressed by:

ν = fτ''(n'')-fτ'(n') (2)

where:

• n' and n'' are whole numbers
• fτ(n) is one among a set of functions of n, characteristic for the element under consideration.

This means that the stationary states of an atom form a set of series. The energy in the nth state of the τth series, omitting an arbitrary constant, is given by:

Eτ (n) = −hfτ (n). (3)

Thus, the values for the energy in an atom’s stationary states may be obtained directly from the measurements of the spectrum by means of relation (1).

To explain these, we are forced to assume that an atom or molecule consists of a number of electrified particles in motion.

• This implies that no emission of radiation takes place in the stationary states.

Thus, the ordinary laws of electrodynamics must be radically changed.

In many cases, however, the electrodynamical forces that emit radiation will be very small compared to simple electrostatic attractions or repulsions of the charged particles corresponding to Coulomb’s law.

We assume that it is possible to describe the motion in the stationary states, by retaining only the latter forces.

We shall calculate the motions of the particles in the stationary states as the motions of mass-points according to ordinary mechanics including the modifications claimed by Relativity.*

A transition between 2 stationary states is obvious from the essential discontinuity involved in Assumptions I and II.

In general, it is impossible even approximately to describe this phenomenon by means of ordinary mechanics or to calculate the frequency of the radiation absorbed or emitted by such a process by means of ordinary electrodynamics.

On the other hand, it has been possible by means of ordinary mechanics and electrodynamics to account for the phenomenon of temperature-radiation in the limiting region of slow vibrations.

Thus, any theory can describe this phenomenon in accordance with observations to form some sort of natural generalisation of the ordinary theory of radiation.

The Theory of Temperature-radiation

Planck gave the theory of temperature-radiation.

It lacked internal consistency, since, in the deduction of his radiation formula, assumptions of similar character as I and II were used in connection with assumptions which were in obvious contrast to them.

Einstein has succeeded, on the basis of the assumptions I and II, to give a consistent deduction of Planck’s formula by introducing supplementary assumptions on:

• the probability of transition of a system between 2 stationary states
• how this probability depends on the density of radiation of the corresponding frequency in the surrounding space
  • This is suggested from analogy with the ordinary theory of radiation.

Einstein compares the emission or absorption of radiation of frequency ν corresponding to a transition between 2 stationary states with the emission or absorption to be expected on ordinary electrodynamics for a system consisting of a particle executing harmonic vibrations of this frequency.

Einstein assumes that on the quantum theory there will be a certain probability An'n''dt that the system in the stationary state of greater energy, characterised by the letter n' in the time interval dt will start spontaneously to pass to the stationary state of smaller energy, characterised by the letter n''

Moreover, on ordinary electrodynamics, the harmonic vibrator will:

• make an independent emission.
• emit or absorb radiation-energy in the presence of a radiation of frequency ν in the surrounding space, dependent on the accidental phasedifference between this radiation and the vibrator

Einstein assumes secondly that in the presence of a radiation in the surrounding space, the system will on the quantum theory, possess a certain probability of:

• passing in the time dt from the state n' to the state n'', as well as from the state n00 to the state n'depending on this radiation
• spontaneous transition from the state n' to the state n''

These latter probabilities are assumed to be proportional to the intensity of the surrounding radiation. They are denoted by ρνBn 0 n00 dt and ρνBn 00 n0 dt

where:

• ρν dν denotes the amount of radiation in unit volume of the surrounding space distributed on frequencies between ν and ν + dν
• Bn 0 n00 and Bn 00 n0 are constants which, like An 0 n00, depend only on the stationary states under consideration.

Einstein does not give details on the values of these constants, no more than to the conditions by which the different stationary states of a given system are determined or to the “a-priori probability” of these states on which their relative occurrence in a distribution of statistical equilibrium depends.

He shows, however, how it is possible to deduce a formula for the temperature radiation.

• This formula is based on:
  • Boltzmann’s principle on the relation between entropy and probability
  • Wien’s well-known displacement-law
• This formula, apart from an undetermined constant factor, coincides with Planck’s, if we only assume that the frequency corresponding to the transition between the 2 states is determined by (1) [complete state change].

By reversing the line of argument, Einstein’s theory may be considered as a very direct support of the latter relation.

The motions in successive stationary states differ very little from each other. I shall show, however, that the frequencies in these states coincides with the frequencies from the motion of the system in the stationary states using the ordinary theory of radiation.