The World According to Quantum Mechanics

Why the Laws of Physics Make Perfect Sense After All

Ulrich Mohrhoff, Manu Jaiswal

• Publisher: WSPC
• Year: 2018
• eBook ISBN: 9789813273719
• Edition: 2
• Topic: Physical Sciences
• Subtopic: Quantum Theory

Apart from providing a lucid introduction to the mathematical formalism and conceptual foundations of quantum mechanics, we explain why the laws of physics have the form that they do. In addition, we present a new and unique look at the quantum world, steering clear of two common errors: the error of the ψ-ontologists, who reify a calculational tool; and the error of the anti-realists, for whom physical theories are simply devices for expressing regularities among observations.

The new edition of this acclaimed text adds around 200 pages on a variety of topics, such as how the founders sought to make sense of quantum mechanics, Kant's theory of science, QBism, Everettian quantum mechanics, de Broglie–Bohm theory, environmental decoherence, contextuality, nonlocality, and the paradox of subjectivity — the curious fact that the world seems to exist twice, once for us, in our minds, and once by itself, independently of us.

Contents:

• Overview:
  • Probability: Basic Concepts and Theorems
  • A Brief History of the 'Old' Theory
  • The 'New' Theory
  • The Feynman Route to Schrödinger (Stage 1)
  • Special Relativity in a Nutshell
  • The Feynman Route to Schrödinger (Stage 2)
• A Closer Look:
  • Quantum Mechanics: A Probability Calculus
  • The Classical Forces: Effects
  • The Classical Forces: Causes
  • Quantum Mechanics Resumed
  • Spin 1/2
  • Angular Momentum and the Hydrogen Atom
  • Composite Systems
  • Contextuality, Locality, Realism, and All That
  • Quantum Statistics
  • Relativistic Particles
• Making Sense:
  • How the Founders Sought to Make Sense of Quantum Mechanics
  • Quantum Mechanics and Consciousness
  • QBism
  • ψ-Ontology
  • Environment-Induced Decoherence
  • Space, Time, and the Macroworld
  • Questions of Substance
  • Manifestation
  • Why the Laws of Physics are Just So
  • Consciousness and the Physical World
  • Quanta and Vedanta
  • Appendix A: Some Mathematical Tools
  • Appendix B: Solutions to Selected Problems

Readership: Students, lecturers in university undergraduate physics courses related to quantum mechanics.
Key Features:

• Presents the formalism of quantum mechanics in a clear and compelling manner
• Useful to both students (undergraduate as well as graduate) and teachers (including higher secondary)
• Serves as the missing link between the popular literature on quantum mechanics and the academic literature

PART 2

A Closer Look

Chapter 7

Quantum mechanics: A probability calculus

7.1The classical probability calculus

A concise and exhaustive description of a classical physical system at any one time consists of the values of a fixed number N of coordinates and an equal number of momenta. Such a description is usually called the state of the system. The state of a classical system with N degrees of freedom can therefore be represented by a point

in a 2N-dimensional space

known as phase space.

As an example consider the classical harmonic oscillator (Fig. 7.1). It has one degree of freedom, a 2-dimensional phase space, and the following equation of motion:

Problem 7.1. In Fig. 7.1, which way does the oscillator’s state

move? Clockwise or counterclockwise?

The mathematical apparatus of quantum mechanics serves to assign probabilities to the possible outcomes of measurements. With the exception of measurements that have discrete sets of possible outcomes, no real-world experiment has an exact outcome. (Otherwise one could experimentally establish whether the value of a physical quantity with a continuous range of possible values is rational rather than irrational.) If measurement outcomes are digitally displayed, as may reasonably be assumed, then every measurement has a finite number of possible outcomes. A (successful) measurement is therefore equivalent to a finite number of simultaneous elementary tests (one for each outcome), each having two possible outcomes. An elementary test associated with a continuous physical quantity X typically answers the question: does the value of X lie in the interval I? If the possible outcomes of a measurement are the disjoint intervals I_k, k = 1, . . . , n, then the actual outcome of one elementary test will be positive, while the actual outcomes of the remaining n − 1 tests will be negative.

Fig. 7.1Phase space diagram of a harmonic oscillator. Observe that when |x| is at a maximum, p changes its sign (the oscillator reverses its motion), and when x = 0, |p| is at a maximum (the oscillator moves fastest). See the main text for the significance of the gray strips.

In our oscillator example, some of the possible outcomes of elementary tests are intervals of the x axis, and some are intervals of the p axis. In the system’s phase space, the former intervals correspond to vertical strips, the latter to horizontal strips, as indicated in Fig. 7.1. The intersection of a horizontal and a vertical strip also corresponds to the outcome of an elementary test, one that determines the simultaneous truth of the propositions “the value of x lies in the vertical strip” and “the value of p lies in the horizontal strip.” Generalizing this possibility by allowing every subset of phase space to be the possible outcome of an elementary test, we arrive at the following characterization of the classical probability calculus.

•An elementary te...

Table of contents

• Cover Page
• Title Page
• Copyright Page
• Contents
• From the preface to the first edition
• Preface to the second edition
• Overview
• A Closer Look
• Making Sense
• Appendix A Some mathematical tools
• Appendix B Solutions to selected problems
• Bibliography
• Index