The Quantum Cookbook shows that whilst quantum mechanics is mathematically challenging, some basic knowledge and a bit of effort will carry you a long way. It also explains how quantum mechanics was derived from the physics. The abstract formalism based on state vectors in Hilbert space was introduced only when it was deemed desirable to lend the theory greater mathematical consistency, and to reject some of its historical baggage. The best way to come to terms with this formalism is to understand how and why it came about. Debates about interpretation continue to this day and, by providing some historical context, you should get the impression that any lack of comprehension of its meaning on your part is absolutely not your fault. Quantum mechanics challenges our comprehension of what any (and all) scientific theories are meant to be telling us about the nature of reality. It’s okay to have doubts.
The Quantum Cookbook shows that whilst quantum mechanics is mathematically challenging, some basic knowledge and a bit of effort will carry you a long way. It also explains how quantum mechanics was derived from the physics. The abstract formalism based on state vectors in Hilbert space was introduced only when it was deemed desirable to lend the theory greater mathematical consistency, and to reject some of its historical baggage. The best way to come to terms with this formalism is to understand how and why it came about. Debates about interpretation continue to this day and, by providing some historical context, you should get the impression that any lack of comprehension of its meaning on your part is absolutely not your fault. Quantum mechanics challenges our comprehension of what any (and all) scientific theories are meant to be telling us about the nature of reality. It’s okay to have doubts.
Quantum Mechanics is an essential topic in today's physics curriculum at both the undergraduate and graduate levels. Quantum mechanics can explain the microscopic world with fantastic accuracy; the fruits from its insights have created technologies that have revolutionized the world. Computers, lasers, mobile telephones, optical communications are but a few examples. The language of quantum mechanics is now an accepted part of the language of physics and day-to-day usage of this language provides physicists with the intuition that is essential for achieving meaningful results. Nevertheless, most physicists acknowledge that, at least once in their scientific career, they have had difficulties understanding the foundations of quantum theory, perhaps even the impression that a really satisfactory and convincing formulation of the theory is still lacking. Numerous quantum mechanics textbooks are available for explaining quantum formalism and applying it to understand problems such as the properties of atoms, molecules, liquids, and solids; the interactions between matter and radiation; and more generally to understand the physical world that surrounds us. Other texts are available for elucidating the historical development of this discipline and describing the steps through which it went before quantum mechanics reached its modern formulation. In contrast, books are rare that review the conceptual difficulties of the theory and then provide a comprehensive overview of the various attempts to reformulate quantum mechanics in order to solve these difficulties. The present text by Franck Laloë does precisely this.
The difficulties of the old quantum theory were largely dispelled by the development of quantum mechanics in the 1920s. First Heisenberg (1925) developed the rather abstract matrix mechanics and soon after Schro¨dinger (1926; see also Eckart, 1926) developed the physically more appealing wave mechanics. Subsequently he showed them to be mathematically equivalent. The physicists were greatly relieved; now they had once again a mathematical formalism that enabled the results of experiments to be calculated. It was possible to move forward. It is certainly a very elegant and indeed beautiful formalism, and does enable the results of many physical measurements to be calculated with great accuracy. It is used daily by atomic and nuclear physicists; there is no other way to make calculations. From the earliest days there has been intense debate about the interpretation of quantum mechanics, and it shows no signs of being settled; if anything it is becoming more intense over the years. As Penrose has remarked, no one understands quantum mechanics. Feynman (1967; see also Sykes, 1994) also admitted that ‘nobody understands quantum mechanics’, and Einstein once remarked: ‘I have thought a hundred times as much about the quantum problems as I have about general relativity theory’ (Pais, 1994, p. 57). It should therefore be evident that the most that can be done here is to try to show where the real difficulties lie, and how the failure to recognize that quantum mechanics is an essentially statistical theory has led to a series of confusing quantum paradoxes. It is just these paradoxes that have often been used to support theological speculations. It is useful to ask what ‘understanding quantum mechanics’ actually means. Certainly it means being able to calculate the results of many measurements. The same applies to Newton’s theory of gravitation, which we understand in the sense that we can use it as a basis of calculation, although we do not really understand the gravitational force itself. Very often, especially in modern physics, the mathematical formalism does not give us the sort of intuitive physical feeling for what is going on that we often obtain from classical mechanics. Perhaps we can never achieve more than this for quantum mechanics. In classical mechanics we can assume that a property that is measured objectively exists prior to the interaction of the measuring apparatus with the observed system. Quantum mechanics, however, ‘is incompatible with the proposition that measurement discovers some unknown but pre-existing property’ (Peres, 1997, p. 14). There is an important distinction between mysteries and absurdities. We may be forced to live with mysteries, but there is no reason why we should put up with absurdities such as the wave-particle duality, tunnelling through potential barriers, acausal events and the collapse of the wave function. These are the quantum paradoxes that will be discussed later on. They arise within the Copenhagen interpretation of quantum mechanics and are solved by other interpretations. These alternative interpretations enable us to make conjectures about what is going on, even though we cannot understand it in detail. This is acceptable and a stimulus to further research. It is certainly difficult enough to understand quantum mechanics, but there is no need to try to visualize logical absurdities in the effort to do so. Quantum mechanics is a mathematical formalism that makes it possible to calculate the results of some experiments. Anything more than this, such as questions about the interpretation of what is measured and whether it tells us what is actually going on in the system being studied, belongs to philosophy. Thus we are concerned here with what is essentially a philosophical debate; if it were just a matter of physics the debate would have been settled long ago. One of the difficulties is that it invokes the results of experiments and uses the language of quantum mechanics. Inevitably this language uses philosophical terms and then it is easy to give the impression that the philosophical propositions are in some way entailed by the physical results. These in turn can influence theological thinking. In order to be quite clear which arguments are justified and which are not, it is essential to separate the three levels of discourse: physical, philosophical and theological. Our task is to relate these together, and this immediately raises the traditional problems of the philosophy of science, including the relation of experiment to theory, and of theory to explanation. We begin by discussing experiments, distinguishing between those that are real and those that are imaginary.
The Quantum Cookbook provides a unique bridge between popular exposition and formal textbook presentation, based on derivations of the iconic equations of quantum mechanics. This is a book for curious readers with some background in physics and sufficient mathematical capability. It aims not to teach readers how to do quantum mechanics but rather helps them to understand how to think about quantum mechanics. Each derivation is presented as a ‘recipe’ with listed ingredients, including standard results from the mathematician’s toolkit, set out in a series of easy-to-follow steps. The recipes have been written sympathetically, for readers who—like the author—will often struggle to follow the logic of a derivation which misses out steps that are ‘obvious’, or which use techniques that readers are assumed to know. The simple truth is that quantum mechanics did not suddenly materialize overnight in the minds of its creators, fully formed, complete with all its axioms and principles. It was instead tortured from much more familiar classical physical descriptions, over a period of decades, as physicists struggled to interpret a series of ever more baffling experimental results. Only later was a much higher level of abstraction introduced into quantum mechanics, in an attempt to establish a secure mathematical foundation that would eradicate all the confusing classical misconceptions inherited from its birth and early childhood. Whilst there are some obvious exceptions, for the most part these derivations are triumphs of physical intuition over mathematical rigour and consistency.
The odd concepts of quantum physics are nearly a century old. They may seem difficult and forbidding at first sight, but the barrier to understanding is not their difficulty but their differentness . Unlike classical models, quantum theory does not attempt to simulate Nature. It gives us information about observations of Nature. But it does contain a “theoretical entity” that claims to contain all information, observation notwithstanding, namely the quantum state vector in Hilbert space. Scientists routinely use the language of Hilbert space, and nonscientists do not use it at all, which is understandable but unfortunate. A sort of baby-talk has become the lingua franca for much of the popular journalism of quantum physics, an awkward patois that combines some of the early groping language of Bohr, de Broglie, Schrödinger, and Heisenberg with more modern words about symmetries and states. Earnest amateurs still ask me about the mystery of Bohr orbits. There is no mystery because there are no orbits, except as historical curiosities. Same for “wave–particle duality.” Same for “quantum jumps.” A friend once asked me to address a philosophy seminar about the “paradoxes of quantum mechanics.” He said he had approached C. N. Yang first, who replied “What paradoxes?” There are some mathematical and logical rough spots in our current best theory of matter, and perhaps some of them can be framed as paradoxes. But the theory is actually rather straightforward, and it has a perfectly clear interpretation linked to experiments that, in their simplest form, anyone can perform (and students do routinely in college laboratories). No empirical observations are known that are inconsistent with the quantum framework.
The mathematical formalism of quantum theory in terms of vectors and operators in infinite-dimensional complex vector spaces is very abstract. The definitions of many mathematical quantities used do not seem to have an intuitive meaning, which makes it difficult to appreciate the mathematical formalism and understand quantum mechanics. This book provides intuition and motivation to the mathematics of quantum theory, introducing the mathematics in its simplest and familiar form, for instance, with three-dimensional vectors and operators, which can be readily understood. Feeling confident about and comfortable with the mathematics used helps readers appreciate and understand the concepts and formalism of quantum mechanics. This book is divided into four parts. Part I is a brief review of the general properties of classical and quantum systems. A general discussion of probability theory is also included which aims to help in understanding the probability theories relevant to quantum mechanics. Part II is a detailed study of the mathematics for quantum mechanics. Part III presents quantum mechanics in a series of postulates. Six groups of postulates are presented to describe orthodox quantum systems. Each statement of a postulate is supplemented with a detailed discussion. To make them easier to understand, the postulates for discrete observables are presented before those for continuous observables. Part IV presents several illustrative applications, which include harmonic and isotropic oscillators, charged particle in external magnetic fields and the Aharonov–Bohm effect. For easy reference, definitions, theorems, examples, comments, properties and results are labelled with section numbers. Various symbols and notations are adopted to distinguish different quantities explicitly and to avoid misrepresentation. Self-contained both mathematically and physically, the book is accessible to a wide readership, including astrophysicists, mathematicians and philosophers of science who are interested in the foundations of quantum mechanics.
In this paper we derive a theorem which proves that the physical interpretation implied by the first postulate of quantum mechanics (QM) is inconsistent with the orthodox formalism. In order to expose this inconsistency we will analyze how the concept of ‘physical system’ is built within classical theories through the notion of invariance and explain in what sense a vector in Hilbert space is not capable of fulfilling these same mathematical conditions. Through an analysis of the mathematical formalism we derive a No Dynamical Invariance (NDI) theorem which proves that, contrary to what is claimed in the first postulate, a vector in Hilbert space cannot be interpreted coherently as the state of a physical (quantum) system. We conclude the paper by analyzing the consequences of the NDI theorem with respect to several ongoing debates in QM.