In classical physics it was assumed that a system of two bodies, both having the same mechanical and electrical properties, would show the same behavior regardless of their complete identity. This assumption, that gave rise to the paradoxes of Gibbs and of Einstein which are here discussed, must be abandoned in atomic physics. We must distinguish sharply between identical particles and those that are only quasiidentical, in the sense that they are the same in mass, charge and the potential fields to which they give rise. All of the results which have hitherto been obtained in quantum mechanics are valid even if we assume that two particles such as two electrons or two helium nuclei are only quasi-identical. The postulate of complete identity is a new mathematical condition which, operating simultaneously with the wave equation and its boundary conditions, restricts the number of possible solutions of the equation and thus gives rise mathematically to a number of exclusion laws such as those which have been empirically obtained.The assumption of identity is equivalent to the mathematical assumption that the ordinary configuration space is degenerate, in the sense that one portion is a mere reflection of another. By this method it is shown that if the wave function is a scalar, the laws of exclusion are obtained without ambiguity. Thus a perfectly symmetrical diatomic gas in its simplest electronic state can have only even rotational levels. It is pointed out that the experimental results with ${\mathrm{O}}_{2}$(${\mathrm{O}}_{16}$${\mathrm{O}}_{16}$) and ${\mathrm{He}}_{2}$ show that these atoms exist in a single nuclear state. It is predicted that with heavier atoms both even and odd states will be found owing to the existence of isobars. The monatomic gas is then discussed and it is shown that, if the Schr\"odinger function is a scalar, only the Bose-Einstein statistics are possible.The phenomenon of spin introduces a vector character into the wave function. If we assume that the spin factor changes sign upon reflection when the number of units of spin is odd but not when the number of units of spin is even, we have all the cases of exclusion now known, including the case of the nitrogen molecule. As an example of the method, the problem of the helium atom is discussed.
In classical physics it was assumed that a system of two bodies, both having the same mechanical and electrical properties, would show the same behavior regardless of their complete identity. This assumption, that gave rise to the paradoxes of Gibbs and of Einstein which are here discussed, must be abandoned in atomic physics. We must distinguish sharply between identical particles and those that are only quasiidentical, in the sense that they are the same in mass, charge and the potential fields to which they give rise. All of the results which have hitherto been obtained in quantum mechanics are valid even if we assume that two particles such as two electrons or two helium nuclei are only quasi-identical. The postulate of complete identity is a new mathematical condition which, operating simultaneously with the wave equation and its boundary conditions, restricts the number of possible solutions of the equation and thus gives rise mathematically to a number of exclusion laws such as those which have been empirically obtained.The assumption of identity is equivalent to the mathematical assumption that the ordinary configuration space is degenerate, in the sense that one portion is a mere reflection of another. By this method it is shown that if the wave function is a scalar, the laws of exclusion are obtained without ambiguity. Thus a perfectly symmetrical diatomic gas in its simplest electronic state can have only even rotational levels. It is pointed out that the experimental results with O2({\mathrm{O}}_{16}$${\mathrm{O}}_{16}) and He2 show that these atoms exist in a single nuclear state. It is predicted that with heavier atoms both even and odd states will be found owing to the existence of isobars. The monatomic gas is then discussed and it is shown that, if the Schr"odinger function is a scalar, only the Bose-Einstein statistics are possible.The phenomenon of spin introduces a vector character into the wave function. If we assume that the spin factor changes sign upon reflection when the number of units of spin is odd but not when the number of units of spin is even, we have all the cases of exclusion now known, including the case of the nitrogen molecule. As an example of the method, the problem of the helium atom is discussed.
The postulates 1 to 6 of quantum mechanics as stated in Sections 3.7 and 7.2 apply to multi-particle systems provided that each of the particles is distinguishable from the others. For example, the nucleus and the electron in a hydrogen-like atom are readily distinguishable by their differing masses and charges. When a system contains two or more identical particles, however, postulates 1 to 6 are not sufficient to predict the properties of the system. These postulates must be augmented by an additional postulate. This chapter introduces this new postulate and discusses its consequences. Permutations of identical particles Particles are identical if they cannot be distinguished one from another by any intrinsic property, such as mass, charge, or spin. There does not exist, in fact and in principle, any experimental procedure which can identify any one of the particles. In classical mechanics, even though all particles in the system may have the same intrinsic properties, each may be identified, at least in principle, by its precise trajectory as governed by Newton's laws of motion. This identification is not possible in quantum theory because each particle does not possess a trajectory; instead, the wave function gives the probability density for finding the particle at each point in space. When a particle is found to be in some small region, there is no way of determining either theoretically or experimentally which particle it is.
For a quantum mechanical system of N particles, the Hamiltonian is a sum of the total kinetic energy and the potential energy. In classical physics, it is possible to keep track of individual particles. A system of N identical particles is said to satisfy Bose-Einstein statistics if its wave function is symmetric under the exchange of particles, and these particles are known as bosons. In the nonrelativistic quantum mechanics, this principle must be accepted as an empirical postulate. In the relativistic quantum theory, it can be proven that half-integer-spin particles cannot be bosons and integer-spin particles cannot be fermions. The Pauli exclusion principle plays a critical role in programming physical properties of electrons or more generally fermions.
Abstract One of the outstanding differences between classical theory and quantum mechanics is the importance which the latter attaches to the identity of particles. Although perfectly well realised before the introduction of modern quantum theory, the fact that two electrons are experimentally indistinguishable did not introduce any novel features into the dynamics of many electron systems on Bohr’s theory. However, Heisenberg showed that these identity relations are of fundamental importance in many body problems, and was able to explain the formerly very puzzling characteristic of the spectrum of neutral helium, its division into two non-combining term systems. Owing to the identity of the particles, the wave function representing such a system must have certain symmetry properties; for the particles known in Nature this symmetry characteristic is that the function be either symmetric or anti-symmetric in the co-ordinates of the particles. For particles with a spin such as the electron and proton, only antisymmetry is allowed (Pauli’s principle), while for α-particles the wave function must be symmetrical. Since Heisenberg’s explanation of the helium spectrum the theory of anti-symmetric electron wave functions has been extended to describe the general features of the spectra of all atoms, while it appears from the work of Heitler and London that the theory is of fundamental importance in the explanation of chemical combination between atoms. Again, in furnishing a new statistics it is an essential feature of the theory of metals. As the effect of the identity relations depends on the fact that exchanging the electrons does not alter the system in any way, these effects are known as exchange phenomena.
Abstract Section I.—The exclusion principle of Pauli was introduced into the old quantum theory as an empirical fact that had been brought to light in the ordering of spectra. The new mechanics has provided some sort of explanation of the principle, for in a closed system with many electrons the mathematically possible stationary states can be separated into a number of groups, with the property that transitions between stationary states in different groups cannot occur. One of these group is made up of the stationary states corresponding occur. One of the these groups is made up of the stationary states corresponding to wave functions that are antisymmetrical in the co-ordinates of the electrons. Apart from accidental degeneracies, the stationary states in different groups have different energy values. The exclusion principle then states that only energy values belonging to the antisymmertical group are found in nature. The exclusion principle has had great success not only in explaining the spectra of helium and of more complicated atoms, but also, under the form of the Fermi-Dirac statistics, in accounting for metallic conduction and ferromagnetism. In all these phenomena we are dealing with systems is stationary states, possessing energy values which are discrete, although they may lie very close together. Now, as was first emphasised by Oppenheimer, we must also use antisymmetrical wave-functions to describe aperiodic phenomena, such as the collision between an electron and an atom. If we do not, we obtain probabilities for the formation of atoms whose wave-funtions are not antisymmetrical, as we shall show in section 4, where we consider the collision between an electron and a helium atom. A helium atom described by a symmetrical wave-function would show a singlet series in palce of the observed triplets and triplet series in place of the observed singlets. The wave-functions of open systems are essentially degenerate; the symmetical and antisymmetrical solution are not separated from one another by a finite energy difference; but for any arbitrary value of the energy (and of the other integrals of the motion) we can form a symmetrical and an antisymmetrical solution. This is somewhat fundamental difference between open and closed systems. For closed systems containing two electrons there exist only the symmetrical and the antisymmetical solution; but for open systems we might take any combination of the two. In fact, to describe an observable phenomenon such as a collision, the wave-function that it would first occur to us to use is a combination of the two. To fix our ideas we shall discuss the collision between two electrons. Our arguments could equally well be applied to the collision between an electron and a hydrogen atom, the problem originally discussed by Born; but the former is the simpler case, and perhaps illustrates our theory better.
Extending the one-particle Schrodinger equation from one to three spatial dimensions enabled us to describe electron states in the hydrogen atom and in other hydrogen-like systems. This chapter generalizes the Schrodinger equation from one to two and then to many particles bound as a single system. For all particles of half-integral spin, the total wave function, as for electrons, is always antisymmetric with respect to interchange, whereas with all particles of integral spin the total wave function is always symmetric. The chemical and spectroscopic characteristics of an element depend on the details of the quantum state of its outermost electrons. The chemical valence depends upon the extent to which the total number of electrons in an atom represents an excess or a deficit with respect to the more stable configuration of a completed shell. The Schrödinger equation for helium contains no explicit mention of the spin states of the electrons.
Bose–Einstein and Fermi–Dirac statistics versus Boltzmann statistics It was remarked in Chapter 2, §2.1, that there are circumstances in which the “Boltzmann statistics” for an ideal gas – the approximation in Eq. (2.6), in which the total partition function is decomposed into a product of single-molecule partition functions, with division by N 1 ! N 2 ! … to correct for particle identity – is inaccurate even if the gas is still ideal (non-interacting particles). That happens when the number of single-particle states within the energy kT of the ground state is no longer very much greater than the number of particles. (Recall Fig. 2.1 and the accompanying discussion.) We noted that when the Boltzmann statistics is no longer an adequate approximation one must distinguish the particles as fermions, no two of which can be in the same single-particle state (counting states of different spin as different), or as bosons, for which there is no restriction on multiple occupancy of the singleparticle states. Fermions are particles of half-odd-integer spin quantum number; they could be composite particles consisting of an odd number of spin-½ particles. Protons, neutrons, and electrons, all of which are of spin ½, are fermions. Bosons are particles of integer spin; they could be spinless, or of spin 1, etc., or could be composites consisting of an even number of fermions. The nuclei of helium atoms of mass number 4 consist of two protons and two neutrons, and the neutral atom has two electrons outside the nucleus; 4 He atoms are bosons.