Following Bohr’s invitation, I went to Copenhagen in the autumn of 1922, where I made a serious effort to explain the so-called “anomalous Zeeman effect”, as the spectroscopists called a type of splitting of the spectral lines in a magnetic field which is different from the normal triplet. On the one hand, the anomalous type of splitting exhibited beautiful and simple laws and Landé had already succeeded to find the simpler splitting of the spectroscopic terms from the observed splitting of the lines. The most fundamental of his results thereby was the use of half-integers as magnetic quantum numbers for the doublet-spectra of the alkali metals. On the other hand, the anomalous splitting was hardly understandable from the standpoint of the mechanical model of the atom, since very general assumptions concerning the electron, using classical theory as well as quantum theory, always led to the same triplet. A closer investigation of this problem left me with the feeling that it was even more unapproachable. We know now that at that time one was confronted with two logically different difficulties simultaneously. One was the absence of a general key to translate a given mechanical model into quantum theory which one tried in vain by using classical mechanics to describe the stationary quantum states themselves. The second difficulty was our ignorance concerning the proper classical model itself which could be suited to derive at all an anomalous splitting of spectral lines emitted by an atom in an external magnetic field. It is therefore not surprising that I could not find a satisfactory solution of the problem at that time. I succeeded, however, in generalizing Landé’s term analysis for very strong magnetic fields12, a case which, as a result of the magneto-optic transformation (Paschen-Back effect), is in many respects simpler. This early work was of decisive importance for the finding of the exclusion principle.
Very soon after my return to the University of Hamburg, in 1923, I gave there my inaugural lecture as Privatdozent on the Periodic System of Elements. The contents of this lecture appeared very unsatisfactory to me, since the problem of the closing of the electronic shells had been clarified no further. The only thing that was clear was that a closer relation of this problem to the theory of multiplet structure must exist. I therefore tried to examine again critically the simplest case, the doublet structure of the alkali spectra. According to the point of view then orthodox, which was also taken over by Bohr in his already mentioned lectures in Göttingen, a non-vanishing angular momentum of the atomic core was supposed to be the cause of this doublet structure.
In the autumn of 1924 I published some arguments against this point of view, which I definitely rejected as incorrect and proposed instead of it the assumption of a new quantum theoretic property of the electron, which I called a “two-valuedness not describable classically”3. At this time a paper of the English physicist, Stoner, appeared4 which contained, besides improvements in the classification of electrons in subgroups, the following essential remark: For a given value of the principal quantum number is the number of energy levels of a single electron in the alkali metal spectra in an external magnetic field the same as the number of electrons in the closed shell of the rare gases which corresponds to this principal quantum number.
On the basis of my earlier results on the classification of spectral terms in a strong magnetic field the general formulation of the exclusion principle became clear to me. The fundamental idea can be stated in the following way: The complicated numbers of electrons in closed subgroups are reduced to the simple number one if the division of the groups by giving the values of the four quantum numbers of an electron is carried so far that every degeneracy is removed. An entirely non-degenerate energy level is already “closed”, if it is occupied by a single electron; states in contradiction with this postulate have to be excluded. The exposition of this general formulation of the exclusion principle was made in Hamburg in the spring of 19255, after I was able to verify some additional conclusions concerning the anomalous Zeeman effect of more complicated atoms during a visit to Tübingen with the help of the spectroscopic material assembled there.
With the exception of experts on the classification of spectral terms, the physicists found it difficult to understand the exclusion principle, since no meaning in terms of a model was given to the fourth degree of freedom of the electron. The gap was filled by Uhlenbeck and Goudsmit’s idea of electron spin6 , which made it possible to understand the anomalous Zeeman effect simply by assuming that the spin quantum number of one electron is equal to and that the quotient of the magnetic moment to the mechanical angular moment has for the spin a value twice as large as for the ordinary orbit of the electron. Since that time, the exclusion principle has been closely connected with the idea of spin. Although at first I strongly doubted the correctness of this idea because of its classical-mechanical character, I was finally converted to it by Thomas’ calculations7 on the magnitude of doublet splitting. On the other hand, my earlier doubts as well as the cautious expression “classically non-describable two-valuedness” experienced a certain verification during later developments, since Bohr was able to show on the basis of wave mechanics that the electron spin cannot be measured by classically describable experiments (as, for instance, deflection of molecular beams in external electromagnetic fields) and must therefore be considered as an essentially quantum-mechanical property of the electron8,9.
The subsequent developments were determined by the occurrence of the new quantum mechanics. In 1925, the same year in which I published my paper on the exclusion principle, De Broglie formulated his idea of matter waves and Heisenberg the new matrix-mechanics, after which in the next year Schrödinger’s wave mechanics quickly followed. It is at present unnecessary to stress the importance and the fundamental character of these discoveries, all the more as these physicists have themselves explained, here in Stockholm, the meaning of their leading ideas10. Nor does time permit me to illustrate in detail the general epistemological significance of the new discipline of quantum mechanics, which has been done, among others, in a number of articles by Bohr, using hereby the idea of “complementarity” as a new central concept11. I shall only recall that the statements of quantum mechanics are dealing only with possibilities, not with actualities. They have the form “This is not possible” or “Either this or that is possible”, but they can never say “That will actually happen then and there”. The actual observation appears as an event outside the range of a description by physical laws and brings forth in general a discontinuous selection out of the several possibilities foreseen by the statistical laws of the new theory. Only this renouncement concerning the old claims for an objective description of the physical phenomena, independent of the way in which they are observed, made it possible to reach again the self-consistency of quantum theory, which actually had been lost since Planck’s discovery of the quantum of action. Without discussing further the change of the attitude of modern physics to such concepts as “causality” and “physical reality” in comparison with the older classical physics I shall discuss more particularly in the following the position of the exclusion principle on the new quantum mechanics.
As it was first shown by Heisenberg12, wave mechanics leads to qualitatively different conclusions for particles of the same kind (for instance for electrons) than for particles of different kinds. As a consequence of the impossibility to distinguish one of several like particles from the other, the wave functions describing an ensemble of a given number of like particles in the configuration space are sharply separated into different classes of symmetry which can never be transformed into each other by external perturbations. In the term “configuration space” we are including here the spin degree of freedom, which is described in the wave function of a single particle by an index with only a finite number of possible values. For electrons this number is equal to two; the configuration space of N electrons has therefore 3 N space dimensions and N indices of “two-valuedness”. Among the different classes of symmetry, the most important ones (which moreover for two particles are the only ones) are the symmetrical
class, in which the wave function does not change its value when the space and spin coordinates of two particles are permuted, and the antisymmetrical class, in which for such a permutation the wave function changes its sign. At this stage of the theory three different hypotheses turned out to be logically possible concerning the actual ensemble of several like particles in Nature.
I. This ensemble is a mixture of all symmetry classes.
II. Only the symmetrical class occurs.
III. Only the antisymmetrical class occurs.
As we shall see, the first assumption is never realized in Nature. Moreover, it is only the third assumption that is in accordance with the exclusion principle, since an antisymmetrical function containing two particles in the same state is identically zero. The assumption III can therefore be considered as the correct and general wave mechanical formulation of the exclusion principle. It is this possibility which actually holds for electrons.
This situation appeared to me as disappointing in an important respect. Already in my original paper I stressed the circumstance that I was unable to give a logical reason for the exclusion principle or to deduce it from more general assumptions. I had always the feeling and I still have it today, that this is a deficiency. Of course in the beginning I hoped that the new quantum mechanics, with the help of which it was possible to deduce so many half-empirical formal rules in use at that time, will also rigorously deduce the exclusion principle. Instead of it there was for electrons still an exclusion: not of particular states any longer, but of whole classes of states, namely the exclusion of all classes different from the antisymmetrical one. The impression that the shadow of some incompleteness fell here on the bright light of success of the new quantum mechanics seems to me unavoidable. We shall resume this problem when we discuss relativistic quantum mechanics but wish to give first an account of further results of the application of wave mechanics to systems of several like particles.
In the paper of Heisenberg, which we are discussing, he was also able to give a simple explanation of the existence of the two non-combining spectra of helium which I mentioned in the beginning of this lecture. Indeed, besides the rigorous separation of the wave functions into symmetry classes with respect to space-coordinates and spin indices together, there exists an approximate separation into symmetry classes with respect to space coordinates alone. The latter holds only so long as an interaction between the spin and the orbital motion of the electron can be neglected. In this way the para- and ortho-helium spectra could be interpreted as belonging to the class of symmetrical and antisymmetrical wave functions respectively in the space coordinates alone. It became clear that the energy difference between corresponding levels of the two classes has nothing to do with magnetic interactions but is of a new type of much larger order of magnitude, which one called exchange energy.
Of more fundamental significance is the connection of the symmetry classes with general problems of the statistical theory of heat. As is well known, this theory leads to the result that the entropy of a system is (apart from a constant factor) given by the logarithm of the number of quantum states of the whole system on a so-called energy shell. One might first expect that this number should be equal to the corresponding volume of the multidimensional phase space divided by hf, where h is Planck’s constant and f the number of degrees of freedom of the whole system. However, it turned out that for a system of N like particles, one had still to divide this quotient by N! in order to get a value for the entropy in accordance with the usual postulate of homogeneity that the entropy has to be proportional to the mass for a given inner state of the substance. In this way a qualitative distinction between like and unlike particles was already preconceived in the general statistical mechanics, a distinction which Gibbs tried to express with his concepts of a generic and a specific phase. In the light of the result of wave mechanics concerning the symmetry classes, this division by N!, which had caused already much discussion, can easily be interpreted by accepting one of our assumptions II and III, according to both of which only one class of symmetry occurs in Nature. The density of quantum states of the whole system then really becomes smaller by a factor N! in comparison with the density which had to be expected according to an assumption of the type I admitting all symmetry classes.
Even for an ideal gas, in which the interaction energy between molecules can be neglected, deviations from the ordinary equation of state have to be expected for the reason that only one class of symmetry is possible as soon as the mean De Broglie wavelength of a gas molecule becomes of an order of magnitude comparable with the average distance between two molecules, that is, for small temperatures and large densities. For the antisymmetrical class the statistical consequences have been derived by Fermi and Dirac13, for the symmetrical class the same had been done already before the discovery of the new quantum mechanics by Einstein and Bose14. The former case could be applied to the electrons in a metal and could be used for the interpretation of magnetic and other properties of metals.
As soon as the symmetry classes for electrons were cleared, the question arose which are the symmetry classes for other particles. One example for particles with symmetrical wave functions only (assumption II) was already known long ago, namely the photons. This is not only an immediate consequence of Planck’s derivation of the spectral distribution of the radiation energy in the thermodynamical equilibrium, but it is also necessary for the applicability of the classical field concepts to light waves in the limit where a large and not accurately fixed number of photons is present in a single quantum state. We note that the symmetrical class for photons occurs together with the integer value I for their spin, while the antisymmetrical class for the electron occurs together with the half-integer value for the spin.
The important question of the symmetry classes for nuclei, however, had still to be investigated. Of course the symmetry class refers here also to the permutation of both the space coordinates and the spin indices of two like nuclei. The spin index can assume 2 I + 1 values if I is the spin-quantum number of the nucleus which can be either an integer or a half-integer. I may include the historical remark that already in 1924, before the electron spin was discovered, I proposed to use the assumption of a nuclear spin to interpret the hyperfine-structure of spectral lines15. This proposal met on the one hand strong opposition from many sides but influenced on the other hand Goudsmit and Uhlenbeck in their claim of an electron spin. It was only some years later that my attempt to interpret the hyperfine-structure could be definitely confirmed experimentally by investigations in which also Zeeman himself participated and which showed the existence of a magneto-optic transformation of the hyperfine-structure as I had predicted it. Since that time the hyperfine-structure of spectral lines became a general method of determining the nuclear spin.
In order to determine experimentally also the symmetry class of the nuclei, other methods were necessary. The most convenient, although not the only one, consists in the investigation of band spectra due to a molecule with two like atoms16. It could easily be derived that in the ground state of the electron configuration of such a molecule the states with even and odd values of the rotational quantum number are symmetric and antisymmetric respectively for a permutation of the space coordinates of the two nuclei. Further there exist among the (2 I + 1)2 spin states of the pair of nuclei, (2 I + 1) (I + 1) states symmetrical and (2 I + 1)I states antisymmetrical in the spins, since the (2 I + 1) states with two spins in the same direction are necessarily symmetrical. Therefore the conclusion was reached: If the total wave function of space coordinates and spin indices of the nuclei is symmetrical, the ratio of the weight of states with an even rotational quantum number to the weight of states with an odd rotational quantum number is given by (I + 1): I. In the reverse case of an antisymmetrical total wave function of the nuclei, the same ratio is I: (I + 1). Transitions between one state with an even and another state with an odd rotational quantum number will be extremely rare as they can only be caused b
y an interaction between the orbital motions and the spins of the nuclei. Therefore the ratio of the weights of the rotational states with different parity will give rise to two different systems of band spectra with different intensities, the lines of which are alternating.
The first application of this method was the result that the protons have the spin and fulfill the exclusion principle just as the electrons. The initial difficulties to understand quantitatively the specific heat of hydrogen molecules at low temperatures were removed by Dennison’s hypothesis17, that at this low temperature the thermal equilibrium between the two modifications of the hydrogen molecule (ortho-H2: odd rotational quantum numbers, parallel proton spins; para-H2 : even rotational quantum numbers, antiparallel spins) was not yet reached. As you know, this hypothesis was later, confirmed by the experiments of Bonhoeffer and Harteck and of Eucken, which showed the theoretically predicted slow transformation of one modification into the other.
Among the symmetry classes for other nuclei those with a different parity of their mass number M and their charge number Z are of a particular interest. If we consider a compound system consisting of numbers A1, A2, . . . of different constituents, each of which is fulfilling the exclusion principle, and a number S of constituents with symmetrical states, one has to expect symmetrical or antisymmetrical states if the sum A1 + A2 + . . . is even or odd. This holds regardless of the parity of S. Earlier one tried the assumption that nuclei consist of protons and electrons, so that M is the number of protons, M − Z the number of electrons in the nucleus. It had to be expected then that the parity of Z determines the symmetry class of the whole nucleus. Already for some time the counter-example of nitrogen has been known to have the spin I and symmetrical states18. After the discovery of the neutron, the nuclei have been considered, however, as composed of protons and neutrons in such a way that a nucleus with mass number M and charge number Z should consist of Z protons and M − Z neutrons. In case the neutrons would have symmetrical states, one should again expect that the parity of the charge number Z determines the symmetry class of the nuclei. If, however, the neutrons fulfill the exclusion principle, it has to be expected that the parity of M determines the symmetry class: For an even M, one should always have symmetrical states, for an odd M, antisymmetrical ones. It was the latter rule that was confirmed by experiment without exception, thus proving that the neutrons fulfill the exclusion principle.
The Dreams That Stuff is Made of Page 40