《量子力学》Exclusion principle and quantum mechanics

WOLFGANG PAULI Exclusion principle and quantum mechanics Nobel Lecture december 13. 1946 The history of the discovery of the exclusion principle m, for which I have received the honor of the Nobel Prize award in the year 1945, goes back to my students days in Munich. While, in school in Vienna, I had already ob- tained some knowledge of classical physics and the then new Einstein rel- ativity theory, it was at the University of Munich that I was introduced by Sommerfeld to the structure of the atom- somewhat strange from the point of view of classical physics. I was not spared the which every physicist, accustomed to the classical way of thinking, experienced when he came to know of Bohrs u basic postulate of quantum theory m for the first time. At that time there were two approaches to the difficult problems con- nected with the quantum of action. One was an effort to bring abstract order to the new ideas by looking for a key to translate classical mechanics and electrodynamics into quantum language which would form a logical gen- eralization of these. This was the direction which was taken by Bohrs u correspondence principle > Sommerfeld, however, preferred, in view of the dificulties which blocked the use of the concepts of kinematical models, a direct interpretation, as independent of models as possible, of the laws of spectra in terms of integral numbers, following, as Kepler once did in his nvestigation of the planetary system, an inner feeling for harmony. Both methods, which did not appear to me irreconcilable, influenced me. The series of whole numbers 2, 8, 18, 32.. giving the lengths of the periods he natural system of chemical elements, was zealously discussed in Munich, ncluding the remark of the Swedish physicist, Rydberg, that these numbers are of the simple form 2 n, if n takes on all integer values. Sommerfeld tried especially to connect the number 8 and the number of corners of a cu A new phase of my scientific life began when I met Niels Bohr personally for the first time. This was in 1922, when he gave a series of guest lectures at Gottingen, in which he reported on his theoretical investigations on the Peri- odic System of Elements. I shall recall only briefly that the essential progress made by Bohrs considerations at that time was in explaining, by means of he spherically symmetric atomic model, the formation of the intermediate

W OLFGANG P AUL I Exclusion principle and quantum mechanics Nobel Lecture, December 13, 1946 The history of the discovery of the « exclusion principle », for which I have received the honor of the Nobel Prize award in the year 1945, goes back to my students days in Munich. While, in school in Vienna, I had already ob￾tained some knowledge of classical physics and the then new Einstein rel￾ativity theory, it was at the University of Munich that I was introduced by Sommerfeld to the structure of the atom - somewhat strange from the point of view of classical physics. I was not spared the shock which every physicist, accustomed to the classical way of thinking, experienced when he came to know of Bohr’s « basic postulate of quantum theory » for the first time. At that time there were two approaches to the difficult problems con￾nected with the quantum of action. One was an effort to bring abstract order to the new ideas by looking for a key to translate classical mechanics and electrodynamics into quantum language which would form a logical gen￾eralization of these. This was the direction which was taken by Bohr’s « correspondence principle ». Sommerfeld, however, preferred, in view of the dificulties which blocked the use of the concepts of kinematical models, a direct interpretation, as independent of models as possible, of the laws of spectra in terms of integral numbers, following, as Kepler once did in his investigation of the planetary system, an inner feeling for harmony. Both methods, which did not appear to me irreconcilable, influenced me. The series of whole numbers 2, 8, 18, 32... giving the lengths of the periods in the natural system of chemical elements, was zealously discussed in Munich, including the remark of the Swedish physicist, Rydberg, that these numbers are of the simple form 2 n 2 , if n takes on all integer values. Sommerfeld tried especially to connect the number 8 and the number of corners of a cube. A new phase of my scientific life began when I met Niels Bohr personally for the first time. This was in 1922, when he gave a series of guest lectures at Göttingen, in which he reported on his theoretical investigations on the Peri￾odic System of Elements. I shall recall only briefly that the essential progress made by Bohr’s considerations at that time was in explaining, by means of the spherically symmetric atomic model, the formation of the intermediate

1945 W.PAULI shells of the atom and the general properties of the rare earths. The question, as to why all electrons for an atom in its ground state were not bound in the innermost shell, had already been emphasized by Bohr as a fundamental problem in his earlier works. In his Gottingen lectures he treated particularly the closing of this innermost K-shell in the helium atom and its essential connection with the two non-combining spectra of helium, the ortho-and para-helium spectra However, no convincing explanation for this phenom enon could be given on the basis of classical mechanics. It made a strong impression on me that Bohr at that time and in later discussions was looking for a general explanation which should hold for the closing of every electron shell and in which the number 2 was considered to be as essential as 8 in contrast to Sommerfeld's approach Following Bohrs 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 Lan- de 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 e anom- alous splitting was hardly understandable from the standpoint of the me- chanical 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 in- to 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 emit- ted 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 suc ceeded, however, in generalizing Lande's term analysis for very strong magnetic fields, a case which, as a result of the magneto-optic transforma- tion(Paschen-Back effect), is in many respects simpler. This early work

2 8 1945 W.PAUL I shells of the atom and the general properties of the rare earths. The question, as to why all electrons for an atom in its ground state were not bound in the innermost shell, had already been emphasized by Bohr as a fundamental problem in his earlier works. In his Göttingen lectures he treated particularly the closing of this innermost K-shell in the helium atom and its essential connection with the two non-combining spectra of helium, the ortho- and para-helium spectra. However, no convincing explanation for this phenom￾enon could be given on the basis of classical mechanics. It made a strong impression on me that Bohr at that time and in later discussions was looking for a general explanation which should hold for the closing of every electron shell and in which the number 2 was considered to be as essential as 8 in contrast to Sommerfeld’s approach. 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 Lan￾déI 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 anom￾alous splitting was hardly understandable from the standpoint of the me￾chanical 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 in￾to 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 emit￾ted 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 suc￾ceeded, however, in generalizing Landé’s term analysis for very strong magnetic fields2 , a case which, as a result of the magneto-optic transforma￾tion (Paschen-Back effect), is in many respects simpler. This early work

EXCLUSION PRINCIPLE AND QUANTUM MECHANICS 29 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 El ements. 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 prob- lem to the theory of multiplet structure must exist. I therefore tried to exam- ine again critically the simplest case, the doublet structure of the alkali spec- tra. According to the point of view then orthodox, which was also taken over by Bohr in his already mentioned lectures in Gottingen, a non-vanish ing 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 u two-valuedness not describable classically > At this time a paper of the English physicist, Stoner, appeared which contained, besides improve- ments 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 al 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 be- came 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 degen- eracy is removed. An entirely non-degenerate energy level is already u closed D, if it is occupied by a single electron; states in contradiction with this postula have to be excluded. The exposition of this general formulation of the ex clusion principle was made in Hamburg in the spring of 1925after I was able to verify some additional conclusions concerning the anomalous Zee- man effect of more complicated atoms during a visit to Tubingen 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

EXCLUSION PRINCIPLE AND QUANTUM MECHANIC S 29 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 El￾ements. 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 prob￾lem to the theory of multiplet structure must exist. I therefore tried to exam￾ine again critically the simplest case, the doublet structure of the alkali spec￾tra. According to the point of view then orthodox, which was also taken over by Bohr in his already mentioned lectures in Göttingen, a non-vanish￾ing 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 improve￾ments 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 be￾came 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 degen￾eracy 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 ex￾clusion principle was made in Hamburg in the spring of 19255, after I was able to verify some additional conclusions concerning the anomalous Zee￾man 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

1945 W. PAULI the electron. The gap was filled by Uhlenbeck and Goudsmit's idea of elec tron spin, which made it possible to understand the anomalous Zeeman effect simply by assuming that the spin quantum number of one electron is ual to y2 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 con- nected with the idea of spin. Although at first I strongly doubted the correct- ness of this idea because of its classical-mechanical character, I was finally converted to it by Thomas calculations on the magnitude of doublet slitting. On the other hand, my earlier doubts as well as the cautious ex pression u classically non-describable two-valuedness m 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 clas- sically 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 electron 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 Schrodingers wave mechanics quickly followed. It is at present un- necessary 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 ideas. 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 concept. I shall only recall that the statements of quantum me- chanics are dealing only with possibilities, not with actualities. They have the form This is not possible p or< Either this or that is possible m, but they can never say That will actually happen then and there p. The actual observa- tion 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 pos- sibilities foreseen by the statistical laws of the new theory. Only this renounce- ment 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 ac

30 1945 W.PAUL I the electron. The gap was filled by Uhlenbeck and Goudsmit’s idea of elec￾tron 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 ½ 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 con￾nected with the idea of spin. Although at first I strongly doubted the correct￾ness 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 ex￾pression « 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 clas￾sically 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 un￾necessary 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 concept II. I shall only recall that the statements of quantum me￾chanics 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 observa￾tion 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 pos￾sibilities foreseen by the statistical laws of the new theory. Only this renounce￾ment 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 ac-

EXCLUSION PRINCIPLE AND QUANTUM MECHANICS 31 tually had been lost since Plancks discovery of the quantum of action. with- out discussing further the change of the attitude of modern physics to such concepts as causality m and physical reality in comparison with the older classical physics I shall discuss more particularly in the following osition of the exclusion principle on the new quantum mechanics As it was first shown by Heisenberg, wave mechanics leads to quali tatively different conclusions for particles of the same kind(for instance for electrons)than for particles of different kinds. As a consequence of the im- possibility to distinguish one ofseveral 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 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 Pmm么、emt, and the antisymmetrical class, in which for such wave function changes its sign. At this stage of the theory ee 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 Il. Only the symmetrical class occurs Ill. 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 prin ciple, since an antisymmetrical function containing two particles in the same state is identically zero. The assumption Ill 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

EXCLUSION PRINCIPLE AND QUANTUM MECHANIC S 31 tually had been lost since Planck’s discovery of the quantum of action.With￾out 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 quali￾tatively different conclusions for particles of the same kind (for instance for electrons) than for particles of different kinds. As a consequence of the im￾possibility to distinguish one ofseveral 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 prin￾ciple, 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

1945 W.PAULI 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 quan- tum 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 ex- clusion 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 re- spect to space-coordinates and spin indices together, there exists an approx- imate separation into symmetry classes with respect to space coordinates alone. The latter holds only so long as an interaction between the spin and he 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 cor- responding levels of the two classes has nothing to do with magnetic inter actions but is of a new type of much larger order of magnitude, which 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 multi dimensional phase space divided by It, where h is Plancks constant and fthe 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

32 1945 W.PAUL I 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 quan￾tum 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 ex￾clusion 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 re￾spect to space-coordinates and spin indices together, there exists an approx￾imate 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 cor￾responding levels of the two classes has nothing to do with magnetic inter￾actions 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 multi￾dimensional phase space divided by h f , 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

EXCLUSION PRINCIPLE AND QUANTUM MECHANICS 33 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 al- eady much discussion, can easily be interpreted by accepting one of our assumptions II and Ill, 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 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 Dirac the symmetrical class the same had been done already before the discovery of the new quantum mechanics by Einstein and Bose The former case could be applied to the electrons in a metal and could be used for the inter- pretation 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 n) was already known long ago, namely the photons. This is not only an immediate con- equence of Plancks 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 rge and not accurately fixed number of photons is present in a single quantum state. We note that the symmetrical class for photons occurs to- gether with the integer value i for their spin, while the antisymmetrical class for the electron occurs together with the half-integer value 12 for the spin. he 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 21+ 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

EXCLUSION PRINCIPLE AND QUANTUM MECHANIC S 33 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 al￾ready 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 Dirac 13, 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 inter￾pretation 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 con￾sequence 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 to￾gether with the integer value I for their spin, while the antisymmetrical class for the electron occurs together with the half-integer value ½ 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 + I 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

1945 W.PAULI was discovered, I proposed to use the assumption of a nuclear spin to inter- pret the hyperfine-structure of spectral lines.T is 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 Zee. man 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 atoms. 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 nu umber are symmetric and antisymmetric respectively for a permutation of the space coordinates of the two nuclei. Further there exist among the(2 1+ 1)spin states of the pair of nuclei, (21+ 1)(+ 1) states symmetrical and (21+ 1)I states antisymmetrical in the spins, since the(2 1+ 1)states with two spins in the same direction are necessarily sym- metrical. 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 (1+ 1) 1. In the reverse case of an antisymmetrical total wave function of the iclei. the ratio is 1: (1+ 1). Transitions between one state with even and another state with an odd rotational quantum number will be extremely rare as they can only be caused by 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 al- ternating The first application of this method was the result that the protons have the spin 12 and fulfill the exclusion principle just as the electrons. The initial difficulties to understand quantitatively the specific heat of hydrogen mole cules at low temperatures were removed by Dennison's hypothesis, that at this low temperature the thermal equilibrium between the two modifications gen molecule (ortho-H= odd rotational quantum numbers

34 1945 W.PAUL I was discovered, I proposed to use the assumption of a nuclear spin to inter￾pret 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 Zee￾man 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 sym￾metrical. 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 by 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 al￾ternating. The first application of this method was the result that the protons have the spin ½ and fulfill the exclusion principle just as the electrons. The initial difficulties to understand quantitatively the specific heat of hydrogen mole￾cules 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

EXCLUSION PRINCIPLE AND QUANTUM MECHANICS 35 parallel proton spins; para-H2; even rotational quantum numbers, antipar allel 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 mod ification 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 in- terest. If we consider a compound system consisting of numbers A, 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 A, A,+... is even odd. This holds regardless of the parity of S. Earlier one tried the assumption of protons and electrons, so that M is the number of pro- tons,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 ave the spin iand symmetrical states. After the discovery of the neutron, the nuclei have been considered, however, as composed of protons and neu trons 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, how ever, 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 most important and most simple crucial example for a nucleus with a different parity of M and Z is the heavy hydrogen or deuteron with M=2 and Z= l which has symmetrical states and the spin 1=1, as could be proved by the investigation of the band spectra of a molecule with two deu- tenons. From the spin value iof the deuteron can be concluded that the neutron must have a half-integer spin. The simplest possible assumption that nis spin of the neutron is equal to 12, just as the spin of the proton and of th electron turned out to be correct. There is hope, that further experiments with light nuclei, especially with protons, neutrons, and deuterons will give us further information about the nature of the forces between the constituents of the nuclei, which, at present

EXCLUSION PRINCIPLE AND QUANTUM MECHANIC S 35 parallel proton spins; para-H2: even rotational quantum numbers, antipar￾allel 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 mod￾ification 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 in￾terest. 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 AI + 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 pro￾tons, 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 states 18. After the discovery of the neutron, the nuclei have been considered, however, as composed of protons and neu￾trons 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, how￾ever, 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 most important and most simple crucial example for a nucleus with a different parity of M and Z is the heavy hydrogen or deuteron with M = 2 and Z = 1 which has symmetrical states and the spin I = 1, as could be proved by the investigation of the band spectra of a molecule with two deu￾terons 19. From the spin value I of the deuteron can be concluded that the neutron must have a half-integer spin. The simplest possible assumption that this spin of the neutron is equal to ½, just as the spin of the proton and of the electron, turned out to be correct. There is hope, that further experiments with light nuclei, especially with protons, neutrons, and deuterons will give us further information about the nature of the forces between the constituents of the nuclei, which, at present

1945 WPAULI is not yet sufficiently clear. Already now we can say, however, that these in- teractions are fundamentally different from electromagnetic interactions The comparison between neutron-proton scattering and proton-proton scattering even showed that the forces between these particles are in good approximation the same, that means independent of their electric charge. If one had only to take into account the magnitude of the interaction energy, one should therefore expect a stable di-proton or 2He(M=2,Z= 2)with forbidden by the exclusion principle in accordance with experience, because his state would acquire a wave function symmetric with respect to the two protons. This is only the simplest example of the application of the exclusion principle to the structure of compound nuclei, for the understanding of which this principle is indispensable, because the constituents of these heavier nuclei, the protons and the neutrons, fullfil In order to prepare for the discussion of more fundamental questions, we want to stress here a law of Nature which is generally valid, namely, the onnection between spin and symmetry class. A half-integer value of the spin quantum number is always connected with antisymmetrical states(exclusion prin ciple), an integer spin with symmetrical states. This law holds not only for pro- tons and neutrons but also for protons and electrons. Moreover, it can easily be seen that it holds for compound systems, if it holds for all of its constit uents. If we search for a theoretical explanation of this law, we must pass to the discussion of relativistic wave mechanics, since we saw that it can cer tainly not be explained by non-relativistic wave mechanics We first consider classical fields, which, like scalars, vectors, and tensors transform with respect to rotations in the ordinary space according to a one- valued representation of the rotation group. We may, in the following, call such fields briefly u one-valued fields So long as interactions of different kinds of field are not taken into account, we can assume that all field com- ponents will satisfy a second-order wave equation, permitting a superpo tion of plane waves as a general solution. Frequency and wave number of these plane waves are connected by a law which, in accordance with De Broglie's fundamental assumption, can be obtained from the relation be tween energy and momentum of a particle claimed in relativistic mechanics by division with the constant factor equal to Planck's constant divided by 2T. Therefore, there will appear in the classical field equations, in general,a new constant H with the dimension of a reciprocal length, with which the

36 1945 W.PAUL I is not yet sufficiently clear. Already now we can say, however, that these in￾teractions are fundamentally different from electromagnetic interactions. The comparison between neutron-proton scattering and proton-proton scattering even showed that the forces between these particles are in good approximation the same, that means independent of their electric charge. If one had only to take into account the magnitude of the interaction energy, one should therefore expect a stable di-proton or :He (M = 2, Z = 2) with nearly the same binding energy as the deuteron. Such a state is, however, forbidden by the exclusion principle in accordance with experience, because this state would acquire a wave function symmetric with respect to the two protons. This is only the simplest example of the application of the exclusion principle to the structure of compound nuclei, for the understanding of which this principle is indispensable, because the constituents of these heavier nuclei, the protons and the neutrons, fullfil it. In order to prepare for the discussion of more fundamental questions, we want to stress here a law of Nature which is generally valid, namely, the connection between spin and symmetry class. A half-integer value of the spin quantum number is always connected with antisymmetrical states (exclusion prin￾ciple), an integer spin with symmetrical states. This law holds not only for pro￾tons and neutrons but also for protons and electrons. Moreover, it can easily be seen that it holds for compound systems, if it holds for all of its constit￾uents. If we search for a theoretical explanation of this law, we must pass to the discussion of relativistic wave mechanics, since we saw that it can cer￾tainly not be explained by non-relativistic wave mechanics. We first consider classical fields20 , which, like scalars, vectors, and tensors transform with respect to rotations in the ordinary space according to a one￾valued representation of the rotation group. We may, in the following, call such fields briefly « one-valued » fields. So long as interactions of different kinds of field are not taken into account, we can assume that all field com￾ponents will satisfy a second-order wave equation, permitting a superposi￾tion of plane waves as a general solution. Frequency and wave number of these plane waves are connected by a law which, in accordance with De Broglie’s fundamental assumption, can be obtained from the relation be￾tween energy and momentum of a particle claimed in relativistic mechanics by division with the constant factor equal to Planck’s constant divided by 2p. Therefore, there will appear in the classical field equations, in general, a new constant m with the dimension of a reciprocal length, with which the