Systems of one degree of freedom

Let us now assume that the small variation of the motion is produced by a small external field established at a uniform rate during a time interval ϑ, long compared with σ, so that the comparative increase during a period is very small.

In this case δE is at any moment equal to the total work done by the external forces on the particles of the system since the beginning of the establishment of the field.

Let this moment be t = −ϑ

Let the potential of the external field at t ≥ 0 be given by Ω, expressed as a function of the q’s.

At any given moment t > 0 we have then

…

which gives by partial integration

where the values for the q’s to be introduced in Ω in the first term are those corresponding to the motion under the influence of the increasing external field, and the values to be introduced in the second term are those corresponding to the configuration at the time t.

Neglecting small quantities of the same order as the square of the external force.

However, we may in this expression for δE instead of the values for the q’s corresponding to the perturbed motion take those corresponding to the original motion of the system.

With this approximation the first term is equal to the mean value of the second taken over a period σ, and we have consequently

From (6) and (7) it follows that I will remain constant during the slow establishment of the small external field, if the motion corresponding to a constant value of the field is periodic.

If next the external field corresponding to Ω is considered as an inherent part of the system, it will be seen in the same way that I will remain unaltered during the establishment of a new small external field, and so on.

Consequently, I will be invariant for any finite transformation of the system which is sufficiently slowly performed, provided the motion at any moment during the process is periodic and the effect of the variation is calculated on ordinary mechanics.

This has a simple consequence of (6) for systems for which every orbit is periodic independent of the initial conditions.

In that case we may for the varied motion take an undisturbed motion of the system corresponding to slightly different initial conditions.

This gives δE constant, and from (6) we get therefore

is the frequency of the motion. This equation forms a simple relation between the variations in E and I for periodic systems, which will be often used in the following.

Returning now to systems of one degree of freedom, we shall take our starting point from Planck’s original theory of a linear harmonic vibrator.

According to this theory, the stationary states of a system, consisting of a particle executing linear harmonic vibrations with a constant frequency ω0 independent of the energy, are given by the well known relation

E = nhω0, (9)

where n is a positive entire number, h Planck’s constant, and E the total energy which is supposed to be zero if the particle is at rest.

From (8) it follows at once that (9) is equivalent to

…

where the latter integral is to be taken over a complete oscillation of q between its limits.

On the principle of the mechanical transformability of the stationary states we shall therefore assume, following Ehrenfest, that (10) holds true for:

• a Planck’s vibrator
• any periodic system of one degree of freedom which can be formed in a continuous manner from a linear harmonic vibrator by a gradual variation of the field of force in which the particle moves.

This condition is immediately seen to be fulfilled by all such systems in which the motion is of oscillating type i. e. where the moving particle during a period passes twice through any point of its orbit once in each direction.

If, however, we confine ourselves to systems of one degree of freedom, it will be seen that systems in which the motion is of rotating type, i. e. where the particle during a period passes only once through every point of its orbit, cannot be formed in a continuous manner from a linear harmonic vibrator without passing through singular states in which the period becomes infinite long and the result becomes ambiguous.

This difficulty which has been pointed out by Ehrenfest. It disappears when we consider systems of several degrees of freedom, where a simple generalisation of (10) holds for any system for which every motion is periodic.

As regards the application of (9) to statistical problems, it was assumed in Planck’s theory that the different states of the vibrator corresponding to different values of n are a-priori equally probable.

This assumption was strongly supported by the agreement obtained on this basis with the measurements of the specific heat of solids at low temperatures.

It follows from the considerations of Ehrenfest, mentioned in the former section, that the a-priori probability of a given stationary state is not changed by a continuous transformation.

We shall therefore expect that for any system of one degree of freedom the different states corresponding to different entire values of n in (10) are a-priori equally probable.

As pointed out by Planck in connection with the application of (9), it is simply seen that statistical considerations, based on the assumption of equal probability for the different states given by (10), will show the necessary relation to considerations of ordinary statistical mechanics in the limit where the latter theory has been found to give results in agreement with experiments.

Let the configuration and motion of a mechanical system be characterised by s independent variables q1, . . . , qs and corresponding momenta p1, . . . , ps.

Let the state of the system be represented in a 2s-dimensional phase-space by a point with coordinates q1, . . . , qs, p1, . . . , ps.

Then, according to ordinary statistical mechanics, the probability for this point to lie within a small element in the phase-space is independent of the position and shape of this element and simply proportional to its volume, defined in the usual way by δW = Z dq1 . . . dqs dp1 . . . dps. (11)

In the quantum theory, however, these considerations cannot be directly applied, since the point representing the state of a system cannot be displaced continuously in the 2sdimensional phase-space, but can lie only on certain surfaces of lower dimensions in this space.

For systems of one degree of freedom the phase-space is a two-dimensional surface, and the points representing the states of some system given by (10) will be situated on closed curves on this surface.

In general, the motion will differ considerably for any two states corresponding to successive entire values of n in (10), and a simple general connection between the quantum theory and ordinary statistical mechanics is therefore out of question.

In the limit, however, where n is large, the motions in successive states will only differ very little from each other, and it would therefore make little difference whether the points representing the systems are distributed continuously on the phase-surface or situated only on the curves corresponding to (10), provided the number of systems which in the first case are situated between two such curves is equal to the number which in the second case lies on one of these curves.

But it will be seen that this condition is just fulfilled in consequence of the above hypothesis of equal a-priori probability of the different stationary states, because the element of phase-surface limited by two successive curves corresponding to (10) is equal to

…

so that on ordinary statistical mechanics the probabilities for the point to lie within any two such elements is the same.

We see consequently that the hypothesis of equal probability of the different states given by (10) gives the same result as ordinary statistical mechanics in all such applications in which the states of the great majority of the systems correspond to large values of n.

Considerations of this kind have led Debye to point out that condition (10) might have a general validity for systems of one degree of freedom, already before Ehrenfest, on the basis of his theory of the mechanical transformability of the stationary states, had shown that this condition forms the only rational generalisation of Planck’s condition (9).

We shall now discuss the relation between the theory of spectra of atomic systems of one degree of freedom, based on (1) and (10), and the ordinary theory of radiation, and we shall see that this relation in several respects shows a close analogy to the relation, just considered, between the statistical applications of (10) and considerations based on ordinary statistical mechanics.

Since the values for the frequency ω in two states corresponding to different values of n in (10) in general are different, we see at once that we cannot expect a simple connection between the frequency calculated by (1) of the radiation corresponding to a transition between two stationary states and the motions of the system in these states, except in the limit where n is very large, and where the ratio between the frequencies of the motion in successive stationary states differs very little from unity.

Consider now a transition between the state corresponding to n = n 0 and the state corresponding to n = n 00, and let us assume that n 0 and n 00 are large numbers and that n 0 − n 00 is small compared with n 0 and n 00.

In that case we may in (8) for δE put

…

We get therefore from (1) and (10) for the frequency of the radiation emitted or absorbed during the transition between the two states

In a stationary state of a periodic system the displacement of the particles in any given direction may always be expressed by means of a Fourier-series as a sum of harmonic vibrations:

where the C’s and c’s are constants and the summation is to be extended over all positive entire values of τ .