Bohr's first model of the periodic table

We now need to retrace our steps and follow Bohr's activities from his great Trilogy of 1913 to his model for the periodic table of 1922. In 1914, Bohr petitioned the Danish government to create a professorship for him in theoretical physics and this was granted two years later. In the meantime, he returned to Manchester as Schuster Reader in physics before taking up his appointment as Professor of Theoretical Physics in Copenhagen in 1916. In 1917, Bohr successfully petitioned the physics faculty of Copenhagen University to found an Institute for Theoretical Physics with Bohr as its founding director. The Institute was officially opened in 1921, but the strain of setting up the new institute combined with his continuing, almost obsessive, research programme into the fundamentals of quantum theory, took a heavy toll and in 1921 he suffered a serious bout of ill-health. Despite this, he remained the driving force behind the attack on the problems of quantum theory on a very broad front. Thanks to his tireless efforts and inspiration, Copenhagen became one of the two major centres for the development of quantum theory, the other being in Göttingen, through the 1920s when the foundations of the old theory were to be cut away and replaced by completely new concepts. The Institute for Theoretical Physics, commonly referred to as the ‘Bohr Institute’, became formally the Niels Bohr Institute in 1965, three years after his death.

In completing the story of spin, we have run far ahead of the continued development of the understanding of the matrix, operator and wave mechanical approaches to quantum mechanics. The reconciliation of these approaches was described in Chap. 15, but there remained the issue of the interpretation of the wavefunction and the deeper implications of the theory. The understanding came gradually with Born's interpretation of the wavefunction, Ehrenfest's demonstration of the equivalence of the classical and quantum pictures and Heisenberg's enunciation of the uncertainty principle. These led to what became known as the Copenhagen interpretation of quantum mechanics. At the same time, the formal mathematical foundations of the different approaches to quantum phenomena were set on a secure foundation thanks to the efforts of Hilbert and many others. These developments resulted in what may be referred to as the completion of quantum mechanics, in the sense that it laid the foundations for all the future development of physics at the atomic and subatomic level – some of these achievements are summarised in Chap. 18.

Schrödinger's interpretation (1926)

Schrödinger regarded wave mechanics as superior to the matrix mechanical approach to quantum physics, not only because it was based upon the well-known eigenfunction techniques of classical physics, but also because it was much more visualisable. His first attempt at interpreting the wavefunction appeared in the final Sect. 7 of the fourth part of his great series of papers (Schrödinger, 1926f) and was entitled On the physical significance of the field scalar.

On 13 March 1926, the first of six papers on wave mechanics by Erwin Schrödinger was published in the Annalen der Physik with the title Quantisation as an eigenvalue problem (Part 1) (Schrödinger, 1926b). The startling first paragraph reads:

These papers were the fruits of an extraordinary burst of creativity on Schrödinger's part which resulted from his interactions with Einstein in the latter part of 1925. Central to these exchanges were de Broglie's remarkable researches which culminated in his famous PhD dissertation and published papers of 1924. These events have already be recounted in Chap. 9. The subsequent developments which led to Schrödinger's discovery of the equation which bears his name will be taken up in Sect. 14.2, but let us first understand more about Schrödinger's background.

Schrödinger's background in physics and mathematics

Education and career up to 1925

Unlike Heisenberg, Jordan, Pauli and Dirac, Erwin Schrödinger was not one of the young Turks who developed Knabenphysik, young man's physics.

Following the success of the 1911 Solvay Conference and the rapid dissemination of the proceedings, the emphasis of research shifted towards the understanding of the spectra of atoms and molecules. With the availability of precision spectroscopic techniques, the bewildering variety of spectral features of atoms and molecules became apparent. The efforts described in Sect. 1.6 indicate how regularities were found in the patterns of spectral lines, the culmination of these investigations being the discovery of the formula for the Balmer series and the various formulae to account for the principal, diffuse and sharp series of the lines in the spectra of sodium, potassium, magnesium, calcium and zinc. As Planck remarked in 1902,

The Zeeman effect: Lorentz and Larmor's interpretations

In 1862, Faraday attempted to measure the change in wavelength of spectral lines when the source of the lines was placed in a strong magnetic field, but failed to observe any positive effect (Jones, 1870). Inspired by this negative result, Pieter Zeeman repeated the experiment and discovered the broadening of the D lines of sodium when a sodium flame was placed between the poles of a strong electromagnet (Zeeman, 1896a).

We have reached the goal I set myself when this journey began. From about 1927 onwards, the quantum theory in its modern guise for non-relativistic quantum mechanics was essentially complete, although there remained problems of interpretation which took a number of years to unravel – some of them are still hotly debated. But much of the apparatus was already in place and the subsequent developments changed completely the face of physics. Jammer (1989) summarises the achievement as follows:

As noted by Mehra and Rechenberg (2001), the 1930s also saw the beginning of the compartmentalisation of physics into separate quantum disciplines. Thus, during the 1930s,with the general acceptance and success of quantum mechanics, the quantum physicists began to specialise in disciplines such as atomic physics, molecular physics, solid state physics, including metal and semiconductor physics, condensed matter physics and low temperature physics, while at high energies, nuclear, particle and cosmic ray physics developed as disciplines in their own right. Whereas the pioneers of quantum mechanics regarded the whole province of quantum physics as their domain, the various branches of quantum physics became fragmented into these specialisms, not so different from those encountered in any physics department today.

The key role of experimental technique

1895 was not an arbitrary choice as the cut-off year for the history recounted in Chap. 1. Over the following years a number of experimental discoveries were made which were to change dramatically the face of physics. The origins of these discoveries can be traced to the need for increased precision in experimental physics. The industrial revolution and the widespread availability of electricity and electrical communication required more exact understanding of the physical properties of materials and also the establishment of international standards. These necessitated a much more professional approach to the teaching of experimental physics and its associated theory. As part of that movement the Clarendon Laboratory was founded in Oxford in 1868 and the Cavendish Laboratory in Cambridge in 1874. Of particular significance for this chapter was the foundation of the Physikalisch-Technische Reichsanstaldt in Berlin in 1887 with the task of providing precise measurements of the physical properties of materials which would be of importance to industry. There was an expectation that these laboratories would develop new techniques for undertaking precision measurements.

New experimental techniques were developed thanks to the development of new technologies. To mention only a few of the more important of these, better vacuums were available to physicists through the invention of the Geissler pump, invented by Johann Heinrich Wilhelm Geissler, a brilliant inventor and glass-blower, in about 1855. The vacuum was produced by trapping air in a mercury column and forcing it down the column by the force of gravity.

Optical spectroscopy, multiplets and the splitting of spectral lines

The achievements described in Chap. 6 represented a remarkable advance in the understanding of quantum phenomena, but there remained major challenges which were ultimately to undermine the successes of the old quantum theory. Continuing advances in spectroscopy enabled high resolution spectra to be obtained and, with the ability to place the sources of emission in strong electric and magnetic fields, the full complexity of atomic and molecular spectra became apparent. Atomic spectra display regularities, for example, the series spectra of elements such as sodium and calcium which could be described by the Rydberg formula (Sect. 1.6). Some of the most prominent spectral features consisted, however, of multiplets, meaning the splitting of a line into a number of separate lines with similar wavelengths. Examples of multiplets are illustrated in Fig. 7.1, derived from observations of the photosphere of the Sun from the Pic du Midi observatory.

The simplest lines are singlets, the example of the Hα line of the Balmer series of hydrogen being shown in Fig. 7.1a. In fact, the line is a very narrow doublet, which Sommerfeld attributed to the effects of special relativity upon the circular and elliptical orbits of electrons of the same principal quantum number (Sect. 5.3). The splittings we are interested in here are very much larger effects. The classic example of a doublet is the splitting of the sodium D line into two bright components labelled D1 (589.592 nm) and D2 (588.995 nm) (Fig. 7.1b).

Einstein in 1905

The next great steps were taken by Albert Einstein and it is no exaggeration to state that he was the first person to appreciate the full significance of quantisation and the reality of quanta. He showed that these are fundamental aspects of all physical phenomena, rather than just a ‘formal device’ for accounting for the Planck distribution. From 1905 onwards, he never deviated from his belief in the reality of quanta – it was some considerable time before the great figures of the day conceded that Einstein was indeed correct.

Einstein completed what we would now call his undergraduate studies in August 1900. Between 1902 and 1904, he wrote three papers on the foundations of Boltzmann's statistical mechanics. In 1905, Einstein was 26 and employed as ‘technical expert, third class’ at the Swiss patent office in Bern. In that year, he completed his doctoral dissertation on A new determination of molecular dimensions, which he presented to the University of Zurich on 20 July 1905. In the same year, he published three papers which are among the greatest classics in the literature of physics. Any one of them would have ensured that his name remained a permanent fixture in the scientific literature. These papers are:

1. (1) On a heuristic point of view concerning the production and transformation of light (Einstein, 1905a);

2. (2) On the motion of small particles suspended in stationary liquids required by the molecular-kinetic theory of heat (Einstein, 1905b);

3. (3) On the electrodynamics of moving bodies (Einstein, 1905c).

The pilot wave function and its uses outside of quantum mechanics

Let us recall our discussion on Bohmian mechanics in Chapter 6. The idea of using Bohmian mechanics outside of quantum mechanics is not new anymore. Khrennikov [1] provides for an overview on how this model can contribute in precise terms to areas such as economics and finance.1 Historically, the work by Bohm and Hiley [2] and also Hiley and Pylkkanen [3] brought forward the idea that the pilot wave function could be seen as a wave function containing information. In Khrennikov [4], Choustova [5], and Haven [6], the idea of using pilot wave theory to finance was investigated. Khrennikov [1] (p. 160) remarks that “the force induced by the pilot wave field does not depend on the amplitude of the wave.” He cites the argument made by Bohm and Hiley [2] that because of this property the pilot wave is an information wave.

The various interpretations (with applications) of the wave function will be covered in detail in Chapter 13.

Here are some of the key features which can be of use in an financial/economics setting. When the wave function is not factorized, then a change in the price of a stock i will affect the prices of stocks, j, with j ≠ i. See Khrennikov [1] (p. 161). The Bohmian theory, as we remarked already in Chapters 1 and 6, is non-local.

This chapter attempts to delve deeper into the question on how quantum mechanical techniques can be brought closer into the realm of economics and finance.

The (non-)Hermiticity of finance-based operators?

Hermiticity of operators was discussed in Chapter 4 of the book. We again take up this very important concept in the context of financial asset pricing. It is a classical result from quantum mechanics that the existence of Hermiticity of the Hamiltonian operator is intimately linked with the notion of conservation of probability. The existence of Hermiticity is also known to be closely linked to the concept of spatial localization. Please see below.

Baaquie [1] makes the important argument that the Black-Scholes Hamiltonian is non-Hermitian and this condition provides for the need to satisfy the martingale condition. Please recall that the martingale property was covered in Chapter 2, Section 2.9. It is also important to mention that Luigi Accardi has indicated that it is white noise which may be the cause of non-Hermiticity in a finance context.

One can argue that within an economics/finance context, the equivalent of the state function, using Baaquie [1] [2], can be the option price function. A similar interpretation can also be found in the paper by Li and Zhang [3] (see also Haven [4]). Please note that the work of Li and Zhang is covered in Section 13.12 of this chapter.

In this chapter and Chapters 11, 12, and 13, we will discuss quantum-like models with financial applications in mind. The very last chapter of this book will analyze the sources of quantum-like processing in the brain.

The concept of non-arbitrage, as we have indicated before (see Section 4.18.3) is of key importance in financial models. We reiterate its importance in this chapter. We also briefly mention a popular finance model. In the coming chapters, the wave function will sometimes have an information interpretation. Therefore we also discuss the informational aspects of the classical financial theory (see Section 10.1). This chapter is meant to introduce the reader to some of the key achievements in financial theory. By doing so, we hope to convince the reader, in later chapters, that some of those very achievements can possibly be reformulated with the help of the wave function approach. Section 10.3 in this chapter will give a flavor of how Fourier integration can possibly be used to model George Soros' concept of reflexivity.

Relevance of the concepts of efficiency and non-arbitrage: a brief discussion

With the 2008 financial crisis, the concept of efficiency may have been seriously questioned. Indeed the financial market in September 2008 did take a tremendous hit and the ensuing price movements would not be consistent with a so-called efficient market.