A simple explanation of Pauli's principle was first given with the wave mechanics. Its interpretation in the new theory was that the wave functions of Schrödinger were antisymmetrical in all the electrons concerned. Thus when the interactions of the electrons may be neglected, the wave function (for a system of n electrons) can never be of the form in nature, but only of the form Hence σ, τ,…ω must be all distinct.
A simple explanation of Pauli's principle was first given with the wave mechanics. Its interpretation in the new theory was that the wave functions of Schrödinger were antisymmetrical in all the electrons concerned. Thus when the interactions of the electrons may be neglected, the wave function (for a system of n electrons) can never be of the form in nature, but only of the form Hence σ, τ,…ω must be all distinct.
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.
The Pauli exclusion principle is a key postulate of the quantum theory and informs much of what we know about matter. In terms of electronic structure, the lone, deceptively simple mathematical requirement is that the total wave function be antisymmetric with respect to the exchange of any two electrons. However, visualizing the effect on the electron distribution presents a formidable challenge to students because of the complexity and number of variables involved. Approaches to gaining insight developed here include a quantitative analysis of two-dimensional configuration spaces as well as a qualitative assessment of interference patterns. For 2p x 1 2p y 1 states as representative examples, the analysis reveals that the Pauli principle directs electrons with opposing spins to codistribute along axes passing through the nucleus in the plane of the occupied orbitals. On the other hand, electrons with the same spin are never simultaneously in the same place or on the same axis; instead, they preferentially distribute in perpendicular directions.
The modern state of the Pauli Exclusion Principle (PEP) is discussed. PEP can be considered from two viewpoints. On the one hand, it asserts that particles with half-integer spin (fermions) are described by antisymmetric wave functions, and particles with integer spin (bosons) are described by symmetric wave functions. This is the so-called spin-statistics connection (SSC). As we will discuss, the physical reasons why SSC exists are still unknown. On the other hand, according to PEP, the permutation symmetry of the total wave functions can be only of two types: symmetric or antisymmetric, both belong to one-dimensional representations of the permutation group, all other types of permutation symmetry are forbidden; whereas the solution of the Schrödinger equation may have any permutation symmetry. It is demonstrated that the proof in some widespread textbooks on quantum mechanics that only symmetric and antisymmetric states (one-dimensional representations of the permutation group) can exist is wrong. However, the scenarios, in which an arbitrary permutation symmetry (degenerate permutation states) is permitted lead to contradictions with the concepts of particle identity and their independence. Thus, the existence in our nature particles only in nondegenerate permutation states (symmetric and antisymmetric) is not accidental and so-called symmetrization postulate may not be considered as a postulate, since all other symmetry options for the total wave function may not be realized. From this an important conclusion follows: we may not expect that in future some unknown elementary particles can be discovered that are not fermions or bosons.
The Pauli exclusion principle (PEP) can be considered from two aspects. First, it asserts that particles that have half-integer spin (fermions) are described by antisymmetric wave functions, and particles that have integer spin (bosons) are described by symmetric wave functions. It is called spin-statistics connection (SSC). The physical reasons why SSC exists are still unknown. On the other hand, PEP is not reduced to SSC and can be consider from another aspect, according to it, the permutation symmetry of the total wave function can be only of two types: symmetric or antisymmetric. They both belong to one-dimensional representations of the permutation group, while other types of permutation symmetry are forbidden. However, the solution of the Schrödinger equation may have any permutation symmetry. We analyze this second aspect of PEP and demonstrate that proofs of PEP in some wide-spread textbooks on quantum mechanics, basing on the indistinguishability principle, are incorrect. The indistinguishability principle is insensitive to the permutation symmetry of wave function. So, it cannot be used as a criterion for the PEP verification. However, as follows from our analysis of possible scenarios, the permission of states with permutation symmetry more general than symmetric and antisymmetric leads to contradictions with the concepts of particle identity and their independence. Thus, the existence in our Nature particles only in symmetric and antisymmetric permutation states is not accidental, since all symmetry options for the total wave function, except the antisymmetric and symmetric, cannot be realized. From this an important conclusion follows, we may not expect that in future some unknown elementary particles that are not fermions or bosons can be discovered.
The approach is to start with some of the elementary ideas of classical mechanics and to translate these concepts to their equivalents in quantum mechanics. This will be done by changing the classical components of the total energy of particle into their quantum mechanical equivalents and arriving at the Schrodinger equation. The conjugate variables, position and momentum, in the Hamiltonian functions can be determined precisely for classical mechanical applications. This chapter deals with the Schrodinger model of the behavior of single electrons as affected by potential wells. The wave mechanic approach is extended to provide an outline of the interactions of many electrons with nuclei. In terms of the Pauli exclusion principle, the four quantum numbers of the electron states must be different. As additional electrons are added, the states available for occupation can be filled only according to the rules governing the quantum numbers.
Abstract It is well-known that the problem of the collision between two particles interacting according to the inverse square law is exactly soluble on the wave mechanics, and that the solution yields the same scattering laws as the classical theory. If, however, the two particles are identical, e.g., two electrons or two α-particles, this is not necessarily the case; for the wave functions used must be antisymmetrical or symmetrical in the co-ordinates of the two particles; and this may affect the scattering laws. In this paper we shall discuss the collision between two particles possessing spin, such as electrons, and also between two particles without spin, such as α-particles. Assuming an inverse square law force between the particles, and neglecting the actual spin forces, we shall deduce from the symmetry properties of the wave functions a scattering law differing considerably from the classical. We shall also mention the various methods by which the effect could be observed, and give some experimental evidence in its favour. The application of the exclusion principle to collision problems has been discussed by the author in a previous paper. Suppose we wish to describe the motion of two particles interacting in any field of force. We obtain a solution w(r1 r2) of the wave equation, where r1 refers to the position of the first particle, and r2 to that of the second. If we did not use antisymmetrical wave functions, we should argue that the probability that the first particle should be at r1 and the second at r 2 would be |w (r1 r2)|2, and therefore the probability that one particle should be at r1 and the other at r2 would be |w (r1 r2)|2 + |w(r2 r1)|