The Pauli Exclusion Principle

Origin, Verifications, and Applications

Ilya G. Kaplan

• Publisher: Wiley
• Year: 2016
• ISBN:
  9781118795248
• Edition: 1
• Topic: Physical Sciences
• Subtopic: Physical & Theoretical Chemistry

This is the first scientic book devoted to the Pauli exclusion principle, which is a fundamental principle of quantum mechanics and is permanently applied in chemistry, physics, and molecular biology. However, while the principle has been studied for more than 90 years, rigorous theoretical foundations still have not been established and many unsolved problems remain.

Following a historical survey in Chapter 1, the book discusses the still unresolved questions around this fundamental principle. For instance, why, according to the Pauli exclusion principle, are only symmetric and antisymmetric permutation symmetries for identical particles realized, while the Schrödinger equation is satisfied by functions with any permutation symmetry? Chapter 3 covers possible answers to this question. The construction of function with a given permutation symmetry is described in the previous Chapter 2, while Chapter 4 presents effective and elegant methods for finding the Pauli-allowed states in atomic, molecular, and nuclear spectroscopy. Chapter 5 discusses parastatistics and fractional statistics, demonstrating that the quasiparticles in a periodical lattice, including excitons and magnons, are obeying modified parafermi statistics.

With detailed appendices, The Pauli Exclusion Principle: Origin, Verifications, and Applications is intended as a self-sufficient guide for graduate students and academic researchers in the fields of chemistry, physics, molecular biology and applied mathematics. It will be a valuable resource for any reader interested in the foundations of quantum mechanics and its applications, including areas such as atomic and molecular spectroscopy, spintronics, theoretical chemistry, and applied fields of quantum information.

1
Historical Survey

1.1 Discovery of the Pauli Exclusion Principle and Early Developments

Wolfgang Pauli formulated his principle before the creation of the contemporary quantum mechanics (1925–1927). He arrived at the formulation of this principle trying to explain regularities in the anomalous Zeeman effect in strong magnetic fields. Although in his Princeton address [1], Pauli recalled that the history of the discovery goes back to his student days in Munich. At that time the periodic system of chemical elements was well known and the series of whole numbers 2, 8, 18, 32… giving the lengths of the periods in this table was zealously discussed in Munich. A great influence on Pauli had his participation in the Niels Bohr guest lectures at Göttingen in 1922, when he met Bohr for the first time. In these lectures Bohr reported on his theoretical investigations of the Periodic System of Elements. Bohr emphasized that the question of why all electrons in an atom are not bound in the innermost shell is the fundamental problem in these studies. However, no convincing explanation for this phenomenon could be given on the basis of classical mechanics.

In his first studies Pauli was interested in the explanation of the anomalous type of splitting in the Zeeman effect in strong magnetic fields. As he recalled [1]:

The anomalous type of splitting was especially fruitful because it exhibited beautiful and simple laws, but on the other hand it was hardly understandable, 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, it was even more unapproachable. A colleague who met me strolling rather aimlessly in the beautiful streets of Copenhagen said to me in a friendly manner, ‘You look very unhappy’; whereupon I answered fiercely, ‘How can one look happy when he is thinking about the anomalous Zeeman effect?’

Pauli decided to analyze the simplest case, the doublet structure of the alkali spectra. In December 1924 Pauli submitted a paper on the Zeeman effect [2], in which he showed that Bohr’s theory of doublet structure based on the nonvanishing angular moment of a closed shell, such as K‐shell of the alkali atoms, is incorrect and closed shell has no angular and magnetic moments. Pauli came to the conclusion that instead of the angular momentum of the closed shells of the atomic core, a new quantum property of the electron had to be introduced. In that paper he wrote, remarkable for that time, prophetic words. Namely:

According to this point of view, the doublet structure of alkali spectra … is due to a particular two‐valuedness of the quantum theoretic properties of the electron, which cannot be described from the classical point of view.

This nonclassical two‐valued nature of electron is now called spin. In anticipating the quantum nature of the magnetic moment of electron before the creation of quantum mechanics, Pauli exhibited a striking intuition.

After that, practically all was ready for the formulation of the exclusion principle. Pauli also stressed the importance of the paper by Stoner [3], which appeared right at the time of his thinking on the problem. Stoner noted that the number of energy levels of a single electron in the alkali metal spectra for the given value of the principal quantum number in an external magnetic field is the same as the number of electrons in the closed shell of the rare gas atoms corresponding to this quantum number. On the basis of his previous results on the classification of spectral terms in a strong magnetic field, Pauli came to the conclusion that a single electron must occupy an entirely nondegenerate energy level [1].

In the paper submitted for publication on January 16, 1925 Pauli formulated his principle as follows [4]:

In an atom there cannot be two or more equivalent electrons, for which in strong fields the values of all four quantum numbers coincide. If an electron exists in an atom for which all of these numbers have definite values, then this state is ‘occupied.’

In this paper Pauli explained the meaning of four quantum numbers of a single electron in an atom, n, l,

, and m_j (in the modern notations); by n and l he denoted the well known at that time the principal and angular momentum quantum numbers, by j and m_j —the total angular momentum and its projection, respectively. Thus, Pauli characterized the electron by some additional quantum number j, which in the case of

was equal to

. For the fourth quantum number of the electron Pauli did not give any physical interpretations, since he was sure, as we discussed above, that it cannot be described in terms of classical physics.

Introducing two additional possibilities for electron states, Pauli obtained

possibilities for the set (n, l, j, m_j). That led to the correct numbers 2, 8, 18, and 32 for the lengths of the periods in the Periodic Table of the Elements.

As Pauli noted in his Nobel Prize lecture [5]: “…physicists found it dif...

Table of contents

• Cover
• Title Page
• Table of Contents
• Preface
• 1 Historical Survey
• 2 Construction of Functions with a Definite Permutation Symmetry
• 3 Can the Pauli Exclusion Principle Be Proved?
• 4 Classification of the Pauli‐Allowed States in Atoms and Molecules
• 5 Parastatistics, Fractional Statistics, and Statistics of Quasiparticles of Different Kind
• Appendix A: Necessary Basic Concepts and Theorems of Group Theory
• Appendix B: The Permutation Group
• Appendix C: The Interconnection between Linear Groups and Permutation Groups
• Appendix D: Irreducible Tensor Operators
• Appendix E: Second Quantization
• Index
• End User License Agreement