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Fundamental theories of physics

More on Cogwheels

Gerard ’t Hooft

Published Jan 1, 2016• 0 citations
Publisher

Abstract

The prototype of the simplest periodic deterministic systems is a model whose states move on a circle. The prototype of the quantum mechanical periodic system is the harmonic oscillator. In line with the philosophy on which this book it based, these two models can be mapped one onto the other. However, we need an intermediate case: the harmonic rotator . We illustrate the use of the group \(\mathit{SU}(2)\) for this model, and we explain how to take the ‘continuum limit’. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

DOI
10.1007/978-3-319-41285-6_12

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The prototype of the simplest periodic deterministic systems is a model whose states move on a circle. The prototype of the quantum mechanical periodic system is the harmonic oscillator. In line with the philosophy on which this book it based, these two models can be mapped one onto the other. However, we need an intermediate case: the harmonic rotator . We illustrate the use of the group (\mathit{SU}(2)) for this model, and we explain how to take the ‘continuum limit’. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
To begin to describe quantum algorithms, we need an understanding of the physical phenomena that motivate quantum mechanics, a simple mathematical model of a potential quantum storage device and the notation to describe and manipulate such a model. In this chapter we present the context in which we will work, motivate the need for quantum mechanics and provide a quick introduction to the essential mathematical tools and notation to be used throughout the book. As part of the price of brevity, we will work with a static model of quantum mechanics and will defer additional mathematical definitions until they are needed. The reader who is already familiar with this material can easily skip to Chapter 2, giving only a cursory glance at Chapter 1 to identify any idiosyncratic notation. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
An outline is given of the interpretation of quantum mechanical expressions if one assumes that what is called Quantum Mechanics today, refers to an underlying reality as in Chap. 2 , even if many authors claim to have evidence to the contrary, as is usually brought forward. If there exists a sub-microscopic classical world of real things, is that the same classical world as the world of large things such as planets, rocks and people? How does the notion of probability enter into quantum mechanics? These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
One of the most commonly used quantum mechanical systems is that of the simple harmonic oscillator . The prototype system is the simple pendulum. Classically, the motion of the system exhibits harmonic, or sinusoidal oscillations. The sinusoidal motion of classical mechanics is the simplest system that produces oscillatory behavior, and therefore is found in almost an infinity of physical systems. Thus, the properties of the harmonic oscillator become important for their use in describing quite diverse physical systems. In a solid, the motion of the atoms is much like the motion of free particles, with the important exception that the atoms are forced on average to retain positions within the solid that specify a particular crystal structure. The motion remains that of a harmonic oscillator, but now the shift of the wave function and the energy levels are hybrids of the confinement harmonic oscillator and the magnetic harmonic oscillator.
In classical, deterministic systems, we may have classical symmetries, transforming ontological states into other ontological states. In line with our general philosophy, we now extend the group of symmetries using quantum symmetries, where ontological states are compared with superimposed states. The groups of symmetry transformations obtained this way may become much larger than that of the classical symmetries. Some of the mathematical procedures, borrowed from ‘ordinary’ quantum mechanics, are explained. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.