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Cambridge University Press eBooks

Dynamical states

Richard A. Healey

Published Nov 24, 1989• 0 citations
Publisher

Abstract

THE HIERARCHY OF QUANTUM SYSTEMS It is frequently noted that some quantum systems are composed of others – their subsystems: For example, a hydrogen atom is composed of a proton and an electron, and an α-particle is composed of two protons and two neutrons. It is less frequently noted that all such quantum systems may also be thought of as subsystems of more extensive systems: Thus, any ten α-particles constitute a quantum system, no matter how widely dispersed – and indeed this system has the important property that its quantum state is required to be symmetric under exchange of any pair of α-particles. In the context of nonrelativistic quantum mechanics, it is reasonable to assume that there is a fixed set of quantum systems, none of whose members has any nontrivial subsystems (i.e., none except itself and the null subsystem Ø , which is a subsystem of every quantum system); any system in this set will be called atomic (note that atoms are not atomic quantum systems in this sense – even a hydrogen atom is composed of an electron and a proton: An electron may be atomic in this sense, but the assumption that it is forms no part of quantum mechanics). All atomic systems together compose the universal quantum system ω , which may be finite or infinite, depending on the total number of atomic systems.

DOI
10.1017/cbo9780511624902.004

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THE HIERARCHY OF QUANTUM SYSTEMS It is frequently noted that some quantum systems are composed of others – their subsystems: For example, a hydrogen atom is composed of a proton and an electron, and an α-particle is composed of two protons and two neutrons. It is less frequently noted that all such quantum systems may also be thought of as subsystems of more extensive systems: Thus, any ten α-particles constitute a quantum system, no matter how widely dispersed – and indeed this system has the important property that its quantum state is required to be symmetric under exchange of any pair of α-particles. In the context of nonrelativistic quantum mechanics, it is reasonable to assume that there is a fixed set of quantum systems, none of whose members has any nontrivial subsystems (i.e., none except itself and the null subsystem Ø , which is a subsystem of every quantum system); any system in this set will be called atomic (note that atoms are not atomic quantum systems in this sense – even a hydrogen atom is composed of an electron and a proton: An electron may be atomic in this sense, but the assumption that it is forms no part of quantum mechanics). All atomic systems together compose the universal quantum system ω , which may be finite or infinite, depending on the total number of atomic systems.
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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.
The hydrogen spectrum is explored as a sensitive probe of a massive object being in a superposition of two distinct locations. Here I solve the simpler but instructive semiclassical problem of the proton hopping stochastically between two sites but forbidden to be anywhere else, analogous to the superposition over two locations. Although the model is semiclassical, it captures symmetry properties and the structure of the energy spectrum that one would expect in a fully quantum treatment. The hydrogen atom is a well-understood system whose spectroscopy reaches exceptional precision, and its semiclassical description displays symmetry with respect to two fixed locations. I develop a classical model for an electron and a proton, with the proton stochastically hopping between two fixed sites; and describe its classical dynamics mathematically via a one-particle effective Hamiltonian. For this Hamiltonian I identify a set of invariance properties that are characteristic of heteronuclear diatomic molecules and are not as extensive as the SO(4) symmetries present in the traditional models of the hydrogen atom. I connect the classical description of the proton-hopping model with the two-fixed-centre problem with time-dependent charges. I find that there is a lower limit to the hopping distance of the proton below which state-of-the-art measurements cannot resolve the spectral signature predicted; and also an upper limit above which approximation methods are not applicable. I gain meaningful insight into the mechanisms that shape the quantum dynamics of proton tunnelling in hydrogen by analyzing the classical counterpart, leveraging the shared symmetries between quantum and classical analogous systems. My results predict an observable spectral shift in the ground-state energy when the proton is delocalized over two sites separated by a distance large enough to be considered macroscopically distinct. With this research, I support the idea that proton tunnelling inside the hydrogen atom is worth exploring as a probe for a delocalized massive object.
Because of considerable mathematical difficulties the Schrödinger equation can be solved exactly for only a few simple quantum systems. In particular for atoms with at least two electrons no such exact solutions exist. The most important system where an exact solution is indeed possible consists of a single electron with charge q 1 = - e in motion under the influence of a Coulomb potential about a nucleus with a positive charge q 2 = Ζe , for Ζ ≥ 1 an integer. Such a system is called a hydrogenic or hydrogen-like atom , with atomic number Z . One may view the nucleus as consisting of Ζ protons. For Ζ = 1 we obtain the hydrogen atom. For Ζ = 2, for example, the system is singly ionized helium He + ; this is helium with its two protons as usual, but stripped of one of its two electrons. Since it has only one electron but two protons it is positively charged, as indicated by the plus sign +. Similarly the lithium atom Li consists of three electrons and three protons (and four neutrons). If we strip off two electrons we obtain the ion Li ++ —doubly ionized lithium, which is the case Ζ = 3. 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.
From the outset statistical mechanics will be framed in the language of quantum theory. The typical macroscopic system is composed of multiple constituents, and hence described in some many-particle Hilbert space. In general, not much is known about such a system, certainly not the precise preparation of all its microscopic details. Thus, its description requires a more general notion of a quantum state, a so-called mixed state. This chapter begins with a brief review of the basic axioms of quantum theory regarding observables, pure states, measurements, and time evolution. Particular attention is paid to the use of projection operators and to the most elementary quantum system, a two-level system. The chapter then motivates the introduction of mixed states and examines in detail their mathematical representation and properties. It also dwells on the description of composite systems, introducing, in particular, the notions of statistical independence and correlations.