What is an Atomic System?

To get the necessary relation to the ordinary theory of radiation in the limit of slow vibrations, we need a conclusion on the probability of transition between 2 stationary states in this limit. We thus connect:

• the probability of a transition between any 2 stationary states
• the motion of the system in these states.

This will explain the polarisation and intensity of the different lines of the spectrum of a given system.

I define an atomic system as a number of electrified particles which move in a force field that has a potential depending only on the position of the particles.

• This is a system under constant external conditions.

The stationary states have variations caused by a variation of the external conditions, such as when the atom is under some external force field.

• This variation cannot be calculated by ordinary mechanics

If the external variations are very slow, the system’s motion at any given moment during the variation will differ very little from the motion in a stationary state corresponding to the instantaneous external conditions.

If the external variations are constant or change very slowly, the atom’s particles will be exposed to forces that will not differ at any moment from those to which they would be exposed if the external forces are caused by slowly moving additional particles which form a stationary atom, together with the original atom.

Thus, the motion of a stationary atom can be calculated by direct application of ordinary mechanics:

• under constant external conditions
• during a slow and uniform variation of these conditions.

I call this assumption as the principle of the “mechanical transformability” of the stationary states.

• This has been introduced in the quantum theory by Ehrenfest1
• This is important in discussing the conditions to be used to fix the stationary states of an atomic system among the continuous multitude of mechanically possible motions.

This principle allows us to overcome a fundamental difficulty involved in the definition of the energy difference between 2 stationary states ) P. Ehrenfest.

• He calls it the “adiabatic hypothesis” in accordance with his logic in which considerations of thermodynamical problems play an important part.

I assumed that the direct transition between 2 such states cannot be described by ordinary mechanics. On the other hand, we cannot define an energy difference between 2 states if there exists no possibility for a continuous mechanical connection between them.

Such a connection is afforded by Ehrenfest’s principle.

• It allows us to transform mechanically the stationary states of a given system into those of another
• In 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.

It has the problem of the statistical distribution of the different stationary states between many atoms of the same kind in temperature equilibrium.

• The number of atoms in the different states may be deduced via Boltzmann’s fundamental relation between entropy and probability
  • This can be done 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, the quantum theory has no direct means of determining these apriori probabilities.

This is 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 an atom, 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.

Ehrenfest1 examined the necessary conditions to explain the second law of thermodynamics.

• He has deduced a certain general condition regarding the variation of the a-priori probability corresponding to a small change of the external conditions
• It follows that the a-priori probability of a given stationary state of a atom remains unaltered during a continuous transformation.
  • This is except in special cases where the values of the energy in some of the stationary states will tend to coincide during the transformation.

This gives us a rational basis to determine the a-priori probability of the different stationary states of a given atomic system.