The Quantum Rules

The most direct route to an understanding of the rules governing the energy content of substances is by the study of specific heats. 

From the formula for the pressure of a perfect gas, p = and the relation pV — NkT, it follows that the kinetic energy in the three translational degrees of freedom is pT. The allocation for each is thus kT. The equipartition law provides that where the energy is shared between s square terms, the average amount in a molecule is skT. The molecular heat, C , is therefore 

The two square terms, other than those for the translational energy, which occur with the diatomic gases are evidently connected with rotation. They are two rather than three, since one of the three axes of reference is that joining the two atoms, and about this particular axis the molecule will possess just the same kind of inertia as if it were monatomic. Given that monatomic substances do not in fact show rotation, there is no reason why diatomic substances should show it about the axis in question. 

The three vibrational degrees of freedom reasonably attributable to a monatomic solid should account for a constant specific heat of 6, in fair accord with the law of Dulong and Petit. 

In one sense these interpretations of the specific heats of simple substances are very successful. But deeper reflection shows that something fundamental is missing. There is no reason in Newtonian 

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Proceeding in the spirit above it seems reasonable to inquire why s is equal to the number of equivalent rotations, rather than to the total number of symmetry operations for the molecule of interest. Rotational partition functions of the diatomic molecule were discussed immediately above. It was pointed out that symmetry requirements mandate that homonuclear diatomics occupy rotational states with either even or odd values of the rotational quantum number J depending on the nuclear spin quantum number I. Heteronuclear diatomics populate both even and odd J states. Similar behaviors are expected for polyatomic molecules but the analysis of polyatomic rotational wave functions is far more complex than it is for diatomics. Moreover the spacing between polyatomic rotational energy levels is small compared to kT and classical analysis is appropriate. These factors appreciated there is little motivation to study the quantum rules applying to individual rotational states of polyatomic molecules. 

Notice the similarities between the two parts of Table 1.1 and the form of the Periodic Table given in Figure 1.3. The quantum rules, the Pauli exclusion principle and the aufbau principle combine to explain the general structure of the Periodic Table. 

Suppose now that two gaseous atoms are made to approach each other. As long as the electron clouds of the two atoms do not overlap, the electron-energy states continue to follow the quantum rules for gaseous atoms. When, however, the electron clouds begin to overlap and the electrons interact with both atoms, the rules for electron-energy states are upset and they start changing. 

Nucleons have a quantum ordering, analogous to the quantum energy levels that electrons occupy. However, the quantum rules for protons and neutrons are much more complicated and heyond our purposes here. 

It has been shown in the last section, and will further appear in succeeding sections, that the hypothesis of the spinning electron has made it possible to understand the splitting of terms (multiplets), a phenomenon which the orbital picture by itself was quite incapable of explaining. The phenomenon in fact depends upon the possession of angular momentum by the electron itself this internal angular momentum, by the quantum rules, can be directed in different ways with respect to the direction of the orbital moment, or with respect to a direction marked out by external means. 

The rigid rotator in space can be described by polar coordinates of the figure axis,

quantum rules it is found that the total angular momentum is given by Equation 6-8, and the component of angular momentum along the z axis by... 

For this crystal it is seen that a cycle for the coordinate z is the identity distance d, so that (p, being constant in the absence of forces acting on the crystal) the quantum rule becomes... 

In this communication I wish to show, first for the simplest case of the non-relativistic and unperturbed hydrogen atom, that the usual rules of quantization can be replaced by another postulate, in which there occurs no mention of whole numbers. Instead, the introduction of integers arises in the same natural way as, for example, in a vibrating string, for which the number of nodes is integral. The new conception can be generalized, and I belieVe that it penetrates deeply into the true nature of the quantum rules. 

A much more rigorous test of the quantum rules is made possible by applying Bohr s frequency condition to the frequencies of spectral lines. 

The quantitative side of the matter is less definite as far as solid crystals are concerned, because the vibrations of the solid are in reality very complex and can only be described in rough approximation by a single frequency. In fact a complicated spectrum of frequencies must be invoked to do justice to the finer details of behaviour. Nevertheless, the operation of the first of the quantum rules is clearly shown by what has been described. 

The quantization of translational energy has already been considered. For vibrational systems the equation is found to yield physically admissible solutions only for values of E defined by the relation E = n+ )hv. The successive energy levels differ by hv as required. The lowest value occurs when the integer n is zero, so that E = Jiv. Schrodinger s equation, unlike the quantum rule which it has superseded, predicts the existence of a so-called zero-point energy. The assumption that there is such a thing is in fact required for the explanation of certain phenomena, so that in this respect the new equation possesses an important advantage. 

The duality referred to, while cutting us off completely from the possibility of describing the invisible in terms of the visible, has the great simplicity that translational motion becomes subject in a not wholly unexpected way to the quantum rules. All kinds of molecular states fall, as a result, into discrete series, and the calculation of absolute probabilities acquires a meaning. There is thus a prospect of answering the fundamental question as to what determines the forms, physical and chemical, into which atoms and molecules eventually settle down. 

The greater precision thus achieved is, of course, at the cost of several particular assumptions. The quantum rules are admirably summarized in the wave equation, but the formulation of this expression and the rules for its application do remain postulates of a specialized kind. 

Before this matter is dealt with, however, the verification of the rules themselves by application to some characteristic spectroscopic phenomena may be profitably considered. These phenomena are, of course, important in themselves, but even more so in so far as they confirm the quantum rules, which will presently provide the key to the whole structure of the periodic system of the elements. 

Spectra and atomic structure in the light of the quantum rules... 

In condensed phases a minimum potential energy is not the sole consideration. The molecules aU execute motions of various kinds, consistently with their structiu e and the quantum rules, and room must be found in the space lattice for movements of the required amplitude. Rise in temperature increases the motions and leads to increased entropy. Sometimes this is compatible with the maintenance of the original lattice, and then the entropy chcmges are manifested in the specific heat. Sometimes, however, a new lattice must be formed to accommodate the liveher and more diverse movements, and then there is a change of phase, the entropy increase being now revealed in the absorption of latent heat. 

The quantum rules are statements of the permitted values of the quantum numbers, n, I and m. 

The quantum rules were described in terms of the permitted values of the three quantum numbers / , / and m. ... 

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