Fundamental phenomena of quantum mechanics are now studied by using very precise techniques. Modern views on an interpretation of quantum mechanics are reduced to a set of statements that allow one to explain the result of a measurement by avoiding the intrusion of the observer into the process of that measurement. Submicroscopic mechanics will help us to cope with difficulties associated with the necessity for uniting quantum theory with gravity. Quantum mechanics is based on postulates, above all its formalism prescribes that the state of a quantum system either pure or mixed is given by a vector in a complex vector space, namely, a Hilbert space. Particle physicists study really fundamental things, but somehow forget that their studies are precisely rooted in quantum mechanical formalism. When the velocity v0 of a moving particle is close to the speed of light c, then the system under consideration is characterized by the relativistic Lagrangian.
Fundamental phenomena of quantum mechanics are now studied by using very precise techniques. Modern views on an interpretation of quantum mechanics are reduced to a set of statements that allow one to explain the result of a measurement by avoiding the intrusion of the observer into the process of that measurement. Submicroscopic mechanics will help us to cope with difficulties associated with the necessity for uniting quantum theory with gravity. Quantum mechanics is based on postulates, above all its formalism prescribes that the state of a quantum system either pure or mixed is given by a vector in a complex vector space, namely, a Hilbert space. Particle physicists study really fundamental things, but somehow forget that their studies are precisely rooted in quantum mechanical formalism. When the velocity v0 of a moving particle is close to the speed of light c, then the system under consideration is characterized by the relativistic Lagrangian.
Quantum mechanics (QM) is not just a theory of micro-objects: it is our current fundamental theory of motion. It expresses a deeper understanding of Nature than classical mechanics. Precisely as classical mechanics, the conventional formulation of QM describes evolution of states and observables in time. Precisely as classical mechanics, this is not sufficient to deal with general relativistic systems, because these systems do not describe evolution in time; they describe correlations between observables. Therefore, a formulation of QM slightly more general than the conventional one – or a quantum version of the relativistic classical mechanics discussed in the previous chapter – is needed. In this chapter I discuss the possibility of such a formulation. In the last section I discuss the general physical interpretation of QM. QM can be formulated in a number of more or less equivalent formalisms: canonical (Hilbert spaces and self-adjoint operators), covariant (Feynman's sum-over-histories), algebraic (states as linear functionals over an abstract algebra of observables) and others. Generally, but not always, we are able to translate these formalisms into one another, but often what is easy in one formulation is difficult in another. A generalrelativistic sum-over-histories formalism has been developed by Jim Hartle. Here I focus on the canonical formalism, because the canonical formalism has provided the mathematical completeness and precision needed to explicitly construct the mathematical apparatus of quantum gravity. Later I will consider alternative formalisms.
The behavior of classical macroscopic particles – from baseballs to planets – is governed by Newtonian mechanics as we have described in the first chapters of this book. The behavior of very small particles, on the other hand, is governed by an entirely different collection of equations and rules that go under the name of “quantum mechanics” and these particles are referred to as quantum particles. These rules have analogies to classical mechanics, but are fundamentally different and somewhat mysterious as we discussed in Chapter 8. The fundamental principles of quantum mechanics cannot be proven to be correct. They were developed based on the results of experimental observation, by analogy with classical mechanics, and through the consideration of logical implications of the formalisms used. There is no way to derive the fundamental elements of quantum mechanics, any more than one could predict from first principles that Newtonian forces should add as vectors. On the other hand, quantum mechanics is among the most rigorously tested theories in science, and the predictions of quantum mechanics have, so far, always been in agreement with experiment. In this chapter, we will present quantum mechanics as a series of postulates and examine their implications in application to simple one-dimensional systems. In the next chapter we will examine quantum effects in real systems including atoms, molecules, crystals, and artificially engineered nano-structures. Postulates of quantum mechanics QM Postulate 1: A quantum particle is described by a wave function Ψ( x , y , z , t ) which is a function of spatial coordinates and time. All physically determinable quantities can be derived from the wave function Ψ( x , y , z , t ) ; quantities which cannot be found from the wave function are not physically meaningful. Note that some things that one could ask about a quantum particle, for example the exact position and simultaneous exact momentum of a particle, are not derivable from the wave function and are therefore considered indeterminate (or uncertain ). These are things which cannot ever be known about a particle, and, as such, they are meaningless to contemplate.
Our purpose in this chapter is to present the key concepts of quantum mechanics in the language of Hilbert spaces. The reader who has not previously met the physical ideas motivating quantum mechanics, and some of the more elementary applications of Schrödinger's equation, is encouraged to read any of a number of excellent texts on the subject such as [1–4]. Otherwise, the statements given here must to a large extent be taken on trust – not an altogether easy thing to do, since the basic assertions of quantum theory are frequently counterintuitive to anyone steeped in the classical view of physics. Quantum mechanics is frequently presented in the form of several postulates, as though it were an axiomatic system such as Euclidean geometry. As often presented, these postulates may not meet the standards of mathematical rigour required for a strictly logical set of axioms, so that little is gained by such an approach. We will do things a little more informally here. For those only interested in the mathematical aspects of quantum mechanics and the role of Hilbert space see [5–8]. Many of the standard applications, such as the hydrogen atom, will be omitted here as they can be found in all standard textbooks, and we leave aside the enormous topic of measurement theory and interpretations of quantum mechanics. This is not to say that we need be totally comfortable with quantum theory as it stands. Undoubtedly, there are some philosophically disquieting features in the theory, often expressed in the form of socalled paradoxes.
The Quantum theory deals with the microscopic structure of the Universe that describes the finer details of nature at the subatomic domain.To a physicist, quantum mechanics is one of the three great pillars supporting our intuition of evocative, bewildering and fascinating natural World as we continuously struggle with the inescapable strangeness of the subatomic domain as if something strange is always afoot.It is well accepted fact that quantum mechanics is a physical theory that describes the way things behave at subatomic level which is quite dazzling in its precision and explanatory power.A test of quantum electrodynamics, the oldest and best understood of the modern quantum theory that involves measuring the way an electron behaves in the vicinity of a magnetic force. Many of the technological and social changes that have revolutionized our lives have actually risen out of the fundamental theoretical research carried out with only motivation to understand the World in a better way than ever. For example, the invention of transistor actually came with the fundamental theoretical research on the Uncertainty Principle which states that the uncertainty of the position of a particle times the uncertainty of its velocity is always a constant. If we double the certainty of the position of the particle, we have to make half the certainty of its velocity and vice versa. Quantum theory is perhaps the prime example of the infinitely esoteric in the sense that it describes a World in which a particle really can be in several places at once and it moves from one place to another by exploring the entire Universe simultaneously-the theory of simultaneity. The diverse and complex Universe is actually made of a few tiny particles according to the rules of quantum theory which is so simple that can be summarized on the back of an envelope. The more we understand about the elemental nature of the Universe, the simpler it looks. We have already discovered that everything in the diverse and complex Universe is nothing but the different assemblies of atoms, and the wide variety of atoms are constructed using the three building blocks at the sub-atomic domain i.e. subatomic particles: protons, neutrons and electrons. We have also discovered that protons and neutrons ate made up of combination of three entities called quarks. And that is where things stop as far as recent discovery is concerned. But the recent progress in particle physics inspiring the quest for the ultimate unified theory of physics that can describe everything in the Universe in one go. The discovery of Boson particle certainty will revolutionaries the Newton's mass concept and gravity into whole new looks. To an optimistic point of view, we must hope the best out of the progress in physics.
A type of mechanics will be presented that possesses some distinctive properties. On the one hand, its physical description & rules of operation are readily comprehensible & intuitively clear. On the other, it fully satisfies all observable predictions of non-relativistic quantum mechanics. Within it, particles exist at points in space, follow continuous, piecewise differentiable paths, and their linear momentum is equal to their mass times their velocity along their path. Yet the probabilities for position and momentum, conditioned on the state of the particle's environment, follow the rules of quantum theory. Indeed, all observable consequences of quantum theory are satisfied; particles can be entangled, have intrinsic spin, this spin is not local to the particle, particle identity can effect probabilities, and so forth. All the rules of quantum mechanics are obeyed, and all arise in a straightforward fashion. After this is established, connections will be drawn out between this type of mechanics and other types of quantum worlds; those that obey Bohmian mechanics, stochastic mechanics, the many worlds interpretation, and physical collapse. In the final section, a relativistic version of the mechanics will be presented.
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.