General principles

The quantum theory of line-spectra rests upon 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 two such states.

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

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

E' - E'' = hv (1)

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

These assumptions offer an immediate interpretation of the fundamental principle of combination of spectral lines deduced from the measurements of the frequencies 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 can be expressed by a formula of the type:

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. On the above assumptions this formula may obviously be interpreted by assuming that the stationary states of an atom of an element form a set of series, and that the energy in the nth state of the τ th series, omitting an arbitrary constant, is given by

Eτ (n) = −hfτ (n). (3)

We thus see that the values for the energy in the stationary states of an atom may be obtained directly from the measurements of the spectrum by means of relation (1).

In order, however, to obtain a theoretical connection between these values and the experimental evidence about the constitution of the atom obtained from other sources, it is necessary to introduce further assumptions about the laws which govern the stationary states of a given atomic system and the transitions between these states.

Now on the basis of a vast amount of experimental evidence, we are forced to assume that an atom or molecule consists of a number of electrified particles in motion, and, since the above fundamental assumptions imply that no emission of radiation takes place in the stationary states, we must consequently assume that the ordinary laws of electrodynamics cannot be applied to these states without radical alterations.

In many cases, however, the effect of that part of the electrodynamical forces which is connected with the emission of radiation will at any moment be very small in comparison with the effect of the simple electrostatic attractions or repulsions of the charged particles corresponding to Coulomb’s law.

Even if the theory of radiation must be completely altered, it is therefore a natural assumption that it is possible in such cases to obtain a close approximation in the description of the motion in the stationary states, by retaining only the latter forces.

In the following we shall therefore, as in all the papers mentioned in the introduction, for the present 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 the theory of relativity, and we shall later in the discussion of the special applications come back to the question of the degree of approximation which may be obtained in this way.

If next we consider a transition between two stationary states, it is obvious at once from the essential discontinuity, involved in the assumptions I and II, that 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, from the fact that 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, we may expect that any theory capable of describing this phenomenon in accordance with observations will form some sort of natural generalisation of the ordinary theory of radiation.

Now the theory of temperature-radiation in the form originally given by Planck confessedly 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.

Quite recently, however, Einstein1) has succeeded, on the basis of the assumptions I and II, to give a consistent and instructive deduction of Planck’s formula by introducing certain supplementary assumptions about the probability of transition of a system between two stationary states and about the manner in which this probability depends on the density of radiation of the corresponding frequency in the surrounding space, 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.

In analogy with the fact that on the latter theory such a system will without external excitation emit a radiation of frequency ν, Einstein assumes in the first place 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, in addition to the above mentioned independent emission, in the presence of a radiation of frequency ν in the surrounding space, and dependent on the accidental phasedifference between this radiation and the vibrator, emit or absorb radiation-energy.

In analogy with this, Einstein assumes secondly that in the presence of a radiation in the surrounding space, the system will on the quantum theory, in addition to the above mentioned probability of spontaneous transition from the state n’ to the state n’’, possess a certain probability, depending on this radiation, of passing in the time dt from the state n’ to the state n’’, as well as from the state n 00 to the state n'.

These latter probabilities are assumed to be proportional to the intensity of the surrounding radiation and are denoted by ρνBn 0 n00 dt and ρνBn 00 n0 dt respectively, where ρν dν denotes the amount of radiation in unit volume of the surrounding space distributed on frequencies between ν and ν + dν, while 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 introduce any detailed assumption as to 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 from the above general assumptions, by means of Boltzmann’s principle on the relation between entropy and probability and Wien’s well known displacement-law, to deduce a formula for the temperature radiation which apart from an undetermined constant factor coincides with Planck’s, if we only assume that the frequency corresponding to the transition between the two states is determined by (1).

It will therefore be seen that by reversing the line of argument, Einstein’s theory may be considered as a very direct support of the latter relation.

In the following discussion of the application of the quantum theory to determine the line-spectrum of a given system, it will, just as in the theory of temperature-radiation, not be necessary to introduce detailed assumptions as to the mechanism of transition between two stationary states.

We shall show, however, that the conditions which will be used to determine the values of the energy in the stationary states are of such a type that the frequencies calculated by (1), in the limit where the motions in successive stationary states comparatively differ very little from each other, will tend to coincide with the frequencies to be expected on the ordinary theory of radiation from the motion of the system in the stationary states.

In order to obtain the necessary relation to the ordinary theory of radiation in the limit of slow vibrations, we are therefore led directly to certain conclusions about the probability of transition between 2 stationary states in this limit.

This leads again to certain general considerations about the connection between the probability of a transition between any two stationary states and the motion of the system in these states, which will be shown to throw light on the question of the polarisation and intensity of the different lines of the spectrum of a given system.

In the above considerations we have by an atomic system tacitly understood a number of electrified particles which move in a field of force which, with the approximation mentioned, possesses a potential depending only on the position of the particles.

This may more accurately be denoted as a system under constant external conditions, and the question next arises about the variation in the stationary states which may be expected to take place during a variation of the external conditions, e. g. when exposing the atomic system to some variable external field of force. Now, in general, we must obviously assume that this variation cannot be calculated by ordinary mechanics, no more than the transition between 2 different stationary states corresponding to constant external conditions. If, however, the variation of the external conditions is very slow, we may from the necessary stability of the stationary states expect that the motion of the system at any given moment during the variation will differ only very little from the motion in a stationary state corresponding to the instantaneous external conditions. If now, moreover, the variation is performed at a constant or very slowly changing rate, the forces to which the particles of the system will be exposed will not differ at any moment 12 from those to which they would be exposed if we imagine that the external forces arise from a number of slowly moving additional particles which together with the original system form a system in a stationary state. From this point of view it seems therefore natural to assume that, with the approximation mentioned, the motion of an atomic system in the stationary states can be calculated by direct application of ordinary mechanics, not only under constant external conditions, but in general also during a slow and uniform variation of these conditions. This assumption, which may be denoted as the principle of the “mechanical transformability” of the stationary states, has been introduced in the quantum theory by Ehrenfest1 ) and is, as it will be seen in the following sections, of great importance in the discussion of the conditions to be used to fix the stationary states of an atomic system among the continuous multitude of mechanically possible motions. In this connection it may be pointed out that the principle of the mechanical transformability of the stationary states allows us to overcome a fundamental difficulty which at first sight would seem to be involved in the definition of the energy difference between two stationary states ) P. Ehrenfest, loc. cit. In these papers the principle in question is called the “adiabatic hypothesis” in accordance with the line of argumentation followed by Ehrenfest in which considerations of thermodynamical problems play an important part. From the point of view taken in the present paper, however, the above notation might in a more direct way indicate the content of the principle and the limits of its applicability.

which enters in relation (1). In fact we have assumed that the direct transition between two such states cannot be described by ordinary mechanics, while on the other hand we possess no means of defining an energy difference between 2 states if there exists no possibility for a continuous mechanical connection between them. It is clear, however, that such a connection is just afforded by Ehrenfest’s principle which allows us to transform mechanically the stationary states of a given system into those of another, because for the latter system we may take one in which the forces which act on the particles are very small and where we may assume that the values of the energy in all the stationary states will tend to coincide. As regards the problem of the statistical distribution of the different stationary states between a great number of atomic systems of the same kind in temperature equilibrium, the number of systems present in the different states may be deduced in the well known way from Boltzmann’s fundamental relation between entropy and probability, if we know the values of the energy in these states and the a-priori probability to be ascribed to each state in the calculation of the probability of the whole distribution. In contrast to considerations of ordinary statistical mechanics we possess on the quantum theory no direct means of determining these apriori probabilities, because we have no detailed information about the mechanism of transition between the different stationary states. If the a-priori probabilities are known for the states of a given atomic system, however, they may be deduced for any other system which can be formed from this by a continuous transformation without passing through one of the singular systems referred to below. In fact, in examining the necessary conditions for the explanation of the second law of thermodynamics Ehrenfest1 ) has deduced a certain general condition as regards the variation of the a-priori probability corresponding to a small change of the external conditions from which it follows, that the a-priori probability of a given stationary state of an atomic system must remain unaltered during a continuous transformation, except in special cases in which the values of the energy in some of the stationary states will tend to coincide during the transformation. In this result we possess, as we shall see, a rational basis for the determination of the a-priori probability of the different stationary states of a given atomic system.