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1989 | Buch

Experiments, Measure and Integration Kapitel lesen Erstes Kapitel lesen

An Introduction to Hilbert Space and Quantum Logic

verfasst von: David W. Cohen

Verlag: Springer New York

Buchreihe : Problem Books in Mathematics

Enthalten in: Professional Book Archive

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Über dieses Buch

Historically, nonclassical physics developed in three stages. First came a collection of ad hoc assumptions and then a cookbook of equations known as "quantum mechanics". The equations and their philosophical underpinnings were then collected into a model based on the mathematics of Hilbert space. From the Hilbert space model came the abstaction of "quantum logics". This book explores all three stages, but not in historical order. Instead, in an effort to illustrate how physics and abstract mathematics influence each other we hop back and forth between a purely mathematical development of Hilbert space, and a physically motivated definition of a logic, partially linking the two throughout, and then bringing them together at the deepest level in the last two chapters. This book should be accessible to undergraduate and beginning graduate students in both mathematics and physics. The only strict prerequisites are calculus and linear algebra, but the level of mathematical sophistication assumes at least one or two intermediate courses, for example in mathematical analysis or advanced calculus. No background in physics is assumed.

Inhaltsverzeichnis

Frontmatter
Chapter 1. Experiments, Measure and Integration
Abstract
We learn about our physical universe by doing experiments. That is, first we do something such as flip a coin, or touch a hot stove, or measure how long it takes a marble to drop from a certain height. Then we record what happens after we do it—the coin comes up heads, we get burned, the marble takes 6 seconds to drop. What we record is called an outcome of the experiment. We identify an experiment by its outcome set, so we can write C = (heads, tails) to denote the coin flip experiment.
David W. Cohen
Chapter 2. Hilbert Space Basics
Abstract
We assume that you know the definition of a vector space V = (V, +, ·) over the field ℭ of complex numbers. Perhaps you recall defining an “inner product” between vectors in a vector space and using the inner product to consider angles between vectors. A ‘Hilbert space’ is a vector space with an inner product. We begin by defining an inner product space.
David W. Cohen
Chapter 3. The Logic of Nonclassical Physics
Abstract
In this chapter we introduce a mathematical formulation for the foundations of quantum physics.
David W. Cohen
Chapter 4. Subspaces in Hilbert Space
Abstract
In Chapter 3 we used two-dimensional Hilbert space to model a manual for measuring electron spin, because every experiment had only two outcomes: spin-up and spin-down. Of course, many physical experiments have outcome sets that are not finite. A measurement of energy, position, or momentum of a moving particle usually means obtaining an outcome from an infinite number of possibilities. In this chapter we shall learn about infinite dimensional Hilbert spaces. In particular, we shall consider the subspace structure of infinite dimensional Hilbert spaces. These provide the logics for orthodox quantum physics.
David W. Cohen
Chapter 5. Maps on Hilbert Spaces
Abstract
As with all mathematical structures, it is important to know what kinds of functions from one Hilbert space to another preserve the important properties of the structure. Besides mathematical interest, the functions on Hilbert space that we study in this chapter provide a crucial link to the notion of an “observable physical quantity” in the Hilbert space formulation of quantum mechanics.
David W. Cohen
Chapter 6. State Space and Gleason’s Theorem
Abstract
We begin this chapter by defining the state space of a general logic. We examine the geometric structure of the state space and use it to define the notions of pure states, mixtures of states, and physical properties. Next we define an observable on a logic, allowing us to consider physical experiments whose outcome sets are more general than the finite subsets of ℜ we saw in Chapter 1. We shall be guided by Lemma 1B.10, however, when we define the expected value of an observable as the integral of the identity function on ℜ with respect to a measure determined by a state on the logic.
David W. Cohen
Chapter 7. Spectrality
Abstract
In the last chapter we saw how Gleason’s theorem provided us with out first connection between physical observables and Hermitian operators. Beginning with a Hilbert space H and its projection logic, we defined an observable as a logic-valued function on the real Borel sets. Then through expected values every observable was shown to be associated with a member of M σ * (H), and then every member of M σ * (H) was associated with a Hermitian operator on H by Gleason’s theorem and the spectral theorem.
David W. Cohen
Chapter 8. The Hilbert Space Model for Quantum Mechanics and the EPR Dilemma
Abstract
Throughout this book we have used the phrase “quantum physics” to mean the collection of all the philosophical viewpoints and all the alternative ways of dealing with a physical universe behaving in contradiction to classical Newtonian physics, whereas “quantum mechanics” means a specific set of working hypotheses about the physical universe. Up to now we have been exploring quantum physics in general. Part A of this chapter is a brief history of quantum mechanics. In Part B we outline a set of working hypotheses for quantum mechanics. In Part C we discuss a philosophical dilemma arising from these hypotheses and articulated in a famous paper written in 1935 by Einstein, Podolsky, and Rosen.
David W. Cohen
Backmatter
Metadaten
Titel
An Introduction to Hilbert Space and Quantum Logic
verfasst von
David W. Cohen
Copyright-Jahr
1989
Verlag
Springer New York
Electronic ISBN
978-1-4613-8841-8
Print ISBN
978-1-4613-8843-2
DOI
https://doi.org/10.1007/978-1-4613-8841-8