The Dreams That Stuff is Made of

by Stephen Hawking

  If we consider the X-ray terms as functions of the atomic numbers Z we obtain in general smooth curves, first discovered by Moseley and Darwin. Only at places where any irregularity in the introduction of electrons occurs are there slight kinks. In this way we can verify the arrangement of the electrons derived from the study of optical spectra. The chief result of this discussion of observed spectra is the following: It is not at all true that all electrons are first introduced in the orbits n = 1, then in those for which n = 2, n = 3, and so on, but on the contrary it is possible that, with electrons already in the orbits n = 4, new electrons of higher azimuthal quantum number k fill up an inner shell, for instance n = 3. This can be deduced partly from spectroscopic, partly from chemical evidence. If two neighboring elements differ only in that the number of inner electrons, for instance those for which n = 3, differ by one, while the number of outer electrons, i.e. n = 4, remains constant (for instance, two), then we should expect that these elements are chemically very similar. We have such groups of similar elements in the fourth period, in the group Sc, Ti, . . .Ni, which have the common property of paramagnetism or ferromagnetism. Even more remarkable are the rare earths which are similar in every respect. This is seen in the following presentation of the periodic system of the elements:

  FIG. 11

  As a result of all these considerations we give Bohr’s table for the arrangement of the electrons:

  FIG. 12

  We see here how the electronic shell n = 3, k = 1, 2, which was completed with 8 electrons with Sc (Z = 21) begins to increase again for n = 3, k = 3. The same occurs with Y (Z = 39) for n = 4, and with La (Z = 57) for n = 5.

  The X-ray spectra confirm the assumption that internal changes begin with these elements. There are obvious kinks in the curves expressing the relation between X-ray terms and atomic number, which otherwise are quite smooth, for the elements of atomic number Z = 21, 39, 57 (Fig. 13).

  FIG. 13

  We may therefore assume that Bohr’s arrangement of electronic shells is correct, at any rate as far as the numbers n, k are concerned.

  We know nothing about the dynamic mechanism which results in these simple laws. Above all it cannot be explained mechanically why a certain group with a certain main quantum number n is “filled” by a certain limited number of electrons, first 2, then 8, then 18, or why the sub-groups defined by k can also take only a definite number of electrons.

  LECTURE 9

  Sommerfeld’s inner quantum numbers—Attempts toward their interpretation by means of the atomic angular momentum—Breakdown of the classical theory—Formal interpretation of spectral regularities—Stoner’s definition of subgroups in the periodic system—Pauli’s introduction of four quantum numbers for the electron—Pauli’s principle of unequal quantum numbers—Report on the development of the formal theory.

  The view of the atom just described has recently led much further in the investigation of the so-called multiplets. Many spectral lines which we have considered here as though they were simple are in fact multiple. For instance, the D-line of sodium is double. Sommerfeld first resolved these lines into terms by introducing a new inner quantum number j and giving a selection rule for this number. The possibility of a third quantum number of the optical electron is indicated by the fact that it has three degrees of freedom: It need only be supposed that the core is not spherically symmetrical but only symmetrical about an axis. Then the optical electron no longer moves in a central field and therefore the orbit is no longer plane, but, to a first approximation, the motion can be described thus: Assume that the orbit is plane for a single revolution and has the angular momentum k. Then this orbit together with the axis of the atomic core, regarded as a rigid system, is endowed with a precession of angular momentum R around the total momentum J considered fixed in space. K , R , J , as easily shown, are action variables conjugate to the corresponding angles of rotation. We place, therefore,K = kh , R = r h , J = j h ,

  where k is the azimuthal quantum number, already introduced above, of the optical electron in its orbit. The quantum number r characterizes the constitution of the core, for, given r and k, j cannot have any value, but only those between |k − r | and |k + r |. Also, j , as precessional momentum, can only make jumps where the jumps j → j ± 1 correspond to oscillations of the electric moment perpendicular to the J-axis and j → j corresponds to oscillations parallel to the J-axis.

  FIG. 14

  We have thus the possibility of explaining multiplets, and the selection rule found empirically by Sommerfeld for the inner quantum number agrees with that found theoretically; but the number of components for given k and r is not verified by experiments. For instance, we are inclined to ascribe to the inert gases, which are certainly highly symmetrical, the angular momentum zero and hence the same to the core of the alkali atoms. But then they should show no separation. If we assume for the inert gases r = 1, the values of j lie between k − 1 and k + 1, therefore j = k − 1, j = k and j = k + 1. But the alkali atoms have no triplets. In the s-states (k = 1) they have single lines andin allotherstates(k = 2,3...)doublets.

  This constitutes a violation of Bohr’s principle of selection. The number of possible states of a system consisting of an ion to which an electron has been added is not equal to the product of the number of the states of the ion by the number of possible electronic orbits, but one less. Bohr calls this a “non-mechanical constraint” and has repeatedly emphasized that this is a most important deviation from mechanical laws. The diffculty has been overcome formally by in troducing half-quantum numbers . . . , . . . and a cabalistic rule.

  If a mechanically “logical” system leads to the quantum numbers−3, −2, −1, 0, +1, +2, +3

  then, by this rule, we replace this series by the one given below:

  in which the number of terms is one less. In this way the doublets of the alkalies are “explained.” The three positions of the core of inert gas type (r = 1) with respect to the electronic orbits give only 2 j-values,

  The absolute value of j is of course still arbitrary. Instead of we can write j = k − 1, k , or choose any other normalization which is convenient for the purpose on hand. Only the number of possible values of j is important.

  The anomalous Zeeman effect is quite similar. Here the classical theory gives for an atom with the total angular momentum , 2j + 1 orientations in a magnetic field, namely all the values of the magnetic quantum number m between −j and j. In fact, however, only 2j terms exist, corresponding to the scheme,

  In this way important advances have been made in the systematization of spectra. We shall now review these briefly.

  Stoner has given an important generalization of Bohr’s theory of the periodic system. He recognized that the electrons in the closed inert gas configurations can be arranged in the following groups:

  By dividing the completed number of electrons in each group by 2, the numbers labeled “quantum number j ” are obtained. For every j > 1, two neighboring values of j are combined to form a larger group with a “quantum number k ” equal to the larger j. In this way a one-to-one correspondence is established between quantum numbers and electrons. Stoner was able to show that this development of Bohr’s scheme leads to the explanation of many properties of atoms, especially of their spectra in the optical and X-ray regions.

  To give only one example: The ionized carbon atom C+ has a doublet spectrum, which has been analyzed by Fowler. The lines of this spectrum which correspond to jumps to the fundamental orbit give information about this orbit, that is about the normal state of the singly ionized carbon atom C+. Bohr had originally attributed to the carbon atom four equivalent electrons with n = 2, k = 1 because chemical facts seemed to demand this equivalence. That would leave to the C+-ion three equivalent electrons n = 2, k = 1. Stoner’s scheme evidently gives for the carbon atom two electrons n = 2, k = 1 and two electrons n = 2, k = 2, of which the former are more strongly attached because they belong to ellipses. The ion C+ therefore has two electrons n =
2, k = 1 and one electron n = 2, k − 2. The jumps of the latter give rise to the spectrum of C+. The fundamental orbit must hence be shown by its combinations to be a p-term (k = 2). In fact observations verify this conclusion and not Bohr’s original hypothesis. In the table on p. 57 this result has already been taken account of. We cannot enter into further details here.

  Pauli showed Stoner’s arrangement to be a consequence of a very general principle. He started from the assumption that spectra behave as though four quantum numbers belonged to each electron, that is, that beside the three numbers n, k, j used so far there exists still a fourth m which determines the magnetic separation. Until now j has been interpreted as the resultant of the angular momentum of the electron and the core. Pauli abandons this idea and ascribes all four quantum numbers to one electron. The difficulty is that the electron, according to our ordinary ideas, has only three degrees of freedom. It will be seen later that the newest development of the quantum theory seems to lead to a fourth degree of freedom, i. e. an axial rotation, for the electron. Let us forego for the moment a physical explanation and describe briefly Pauli’s method. His quantum numbers are normalized somewhat differently from those introduced above. He employs n and k in the usual way, denoting however the latter by k1, and using instead of j a number k2 which can always have exactly two values k2 = k1 − 1 and k2 = k1. Each single electron behaves therefore as the optical electron of the alkalies and gives a doublet. The same must hold also for the magnetic quantum number m. For the alkalies m takes, according to observation, 2 k2 different values—this can be shown from our cabalistic rule, if k2 is interpreted as the total momentum. Therefore, all terms belonging to a given k1 take in all 2(k1 − 1) + 2k1 = 2(2k1 − 1) values. Pauli next observed that Bohr’s method of building up atoms by successive steps can be kept in this way. The number of possible states is simply the sum of those of the core and of those of the newly-entering electron (permanence of quantum numbers). If we pass, for instance, from an alkali atom to the neighboring alkaline earth, then the doublet system of the first becomes a system of single and triplet terms. In the singlet system a state with given n, k1 is decomposed into 1 × (2k1 − 1) terms, in the triplet system into 3(2k1 − 1) terms. This has been interpreted up to the present as meaning that, in strong fields, there corresponds to the optical electron in spite of mechanics, 2k1 − 1 orientations in every case, while the core is oriented in the single terms along one direction, in the triplet terms along three directions. The latter contradicts the principle of permanence because the free alkali atom can have, in the unexcited state (s-state k1 = 1), only two such orientations. The totality of the 4(2k1 − 1) states of the atom can be interpreted as meaning that the core, as in the free alkali atom, can have two states and the optical electron, as in the alkalies, 2(2k1 − 1) states. A corresponding explanation can be given in general, but we shall pass over all these details and consider now the connection between Pauli’s ideas and Stoner’s classification of the periodic system.

  Pauli found, that the latter is equivalent to the following general principle: It never happens that any two electrons in the atom have the same four quantum numbers n, k1, k2 , m. If n, k1, k2 are given, the number of possible values of m, as was seen above, is 2k2. Therefore, the greatest number of “equivalent” electrons, that is having the same n, k1, k2, is also 2k2, otherwise m would be equal for two of these electrons. If the quantum number k2 is identified with Stoner’s j, which also takes for every k (or k1) the values k, k − 1, then Stoner’s classification is shown to be a consequence of Pauli’s principle. The object of the theory is therefore to understand Pauli’s principle, that is either to derive it from the laws of quantum mechanics or to show that it belongs to indemonstrable basic postulates.

  Further developments can be briefly described in the following way: According to Pauli electrons aggregate into systems while keeping their own quantum numbers. The energy of such a system, whether a core, a group of outer electrons or a complete atom, depends, however, only on a certain resultant of the quantum numbers of the individual electrons. A distinction must therefore be made between the quantum numbers of the individual electrons and the resultant quantum numbers of the electron groups. If this group is identical with the whole atom, the resulting quantum numbers fix the terms which determine the spectrum. The rules governing the formation of this resultant are mainly of empirical origin. The following is alone mentioned as a theoretical guiding principle. Paschen and Back have discovered that in strong magnetic fields the Zeeman components of a multiplet are displaced with respect to each other so that they correspond to the normal separation (Larmor frequency). This can be interpreted theoretically by assuming that the individual electrons move practically independently of one another in fields where the magnetic energy is much larger than that due to the interaction between the electrons. The electrons therefore precess with normal Larmor frequency. From this follows that in strong fields the magnetic quantum numbers behave additively and the construction of the resultant is referred to this idea.

  The development of this conception was very much aided by the investigations of Russell and Saunders. It was already known, according to Götze, that lines appear which correspond to combinations of p-terms with other p’-terms, therefore having the same azimuthal quantum number. Bohr accounted for this by assuming a simultaneous jump of two electrons, whereby the simple harmonic character of the motion, from which we derive the selection rule, k → k ± 1, is lost. Russell and Saunders found that there exist negative p’-terms, which hence correspond to a state of the atom of higher energy than required for ionization. They were able to explain their observations by introducing a resultant quantum number for the electrons jumping simultaneously and then treating this system with respect to the core in the same way as the optical electron with respect to the core of the alkali atoms. This method was developed systematically by Heisenberg and applied to the practical interpretation of numerous spectra by Hund. The latter succeeded in analyzing completely, among others, the series of the magnetic atoms beginning with scandium and ending with the group iron, cobalt, nickel, deducing not only the completed numbers of the electron groups in the normal states, but also interpreting in a rough way the character of the spectra.

  With this we have come to the limit which can be attained by the development of Bohr’s fundamental ideas. There is material a plenty. It is now time for the theorist to take the initiative again and lay the foundations of a real dynamics of atoms. Heisenberg found a short time ago the key to the gate, closed for such a long time, which kept us from the realm of atomic laws. In his brief paper, the leading physical ideas are clearly stated, but only exemplified on account of the lack of appropriate mathematical equipment. The required machinery Jordan and I have discovered in the matrix calculus. Shortly afterwards, as I learned later, Dirac also found an algorithm which is equivalent to ours, but without noticing its identity with the usual mathematical theory of matrices.

  LECTURE 10

  Introduction to the new quantum theory—Representation of a coördinate by a matrix—The elementary rules of matrix calculus.

  In seeking a line of attack for the remodelment of the theory, it must be borne in mind that weak palliatives cannot overcome the staggering difficulties so far encountered, but that the change must reach its very foundations. It is necessary to search for a general principle, a philosophical idea, which has proved successful in other similar cases. We look back to the time before the advent of the theory of relativity, when the electrodynamics of moving bodies was in difficulties similar to those of the atomic theory of today. Then Einstein found a way out of the difficulty by noting that the existing theory operated with a conception which did not correspond to any observable phenomenon in the physical world, the conception of simultaneity. He showed that it is fundamentally impossible to establish the simultaneity of two events occurring in different localities, but rather that a new definition, prescribing a definite method of measurement is r
equired. Einstein gave a method of measurement adapting itself to the structure of the laws of propagation of light and of electromagnetic phenomena in general. Its success justified the method and with it the initial principle involved: The true laws of nature are relations between magnitudes which must be fundamentally observable. If magnitudes lacking this property occur in our theories, it is a symptom of something defective. The development of the theory of relativity has shown the fertility of this idea, for the attempt to state the laws of nature in invariant form, independently of the system of coördinates, is nothing but the expression of the desire of avoiding magnitudes which are not observable. A similar situation exists in other branches of physics.