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Oxford University Press eBooks

Fundamental principles

N. J. B. Green

Published Sep 11, 1997• 7 citations
Publisher

Abstract

This chapter discusses some of the fundamental principles in quantum mechanics. Towards the end of the nineteenth century, it had been thought that all the fundamental laws of nature were well-understood. However, the next 20 years saw the development of both quantum mechanics and relativity. The chapter then looks at the representation of a particle by a wavefunction. It also considers operators, observables, eigenfunctions, and eigenvalues, as well as commutation and uncertainty. Since it is not possible for a wavefunction to have characteristic values of two observables with non-commuting operators, it is not possible to specify exact simultaneous values for these observables. It should be noted that the uncertainty principle is rather deeper than the usual physical explanation might imply, it is actually inherent in the description of matter in terms of wavefunctions. Finally, the chapter explains expansion in eigenfunctions; the variational principle; and transition probabilities.

DOI
10.1093/hesc/9780198557616.003.0002

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This chapter discusses some of the fundamental principles in quantum mechanics. Towards the end of the nineteenth century, it had been thought that all the fundamental laws of nature were well-understood. However, the next 20 years saw the development of both quantum mechanics and relativity. The chapter then looks at the representation of a particle by a wavefunction. It also considers operators, observables, eigenfunctions, and eigenvalues, as well as commutation and uncertainty. Since it is not possible for a wavefunction to have characteristic values of two observables with non-commuting operators, it is not possible to specify exact simultaneous values for these observables. It should be noted that the uncertainty principle is rather deeper than the usual physical explanation might imply, it is actually inherent in the description of matter in terms of wavefunctions. Finally, the chapter explains expansion in eigenfunctions; the variational principle; and transition probabilities.
Abstract In this chapter, the main features of quantum theory are presented. The chapter begins with a historical account of the invention of quantum mechanics. The meaning of position and momentum in quantum mechanics is discussed and non-commuting operators are introduced. The Schrödinger equation is presented and solved for a free particle and for a harmonic oscillator potential in one dimension. The meaning of the wavefunction is considered and the probabilistic interpretation is presented. The mathematical machinery and language of quantum mechanics are developed, including Hermitian operators, observables and expectation values. The uncertainty principle is discussed and the uncertainty relations are presented. Scattering and tunnelling by potential wells and barriers is considered. The use of variational principles to estimate ground state energies is explained and illustrated with a simple example.
This chapter explains that a central feature of quantum theory is its representation of observables by ‘operators’, which act on the wavefunction and extract the information it contains. It shows how operators are constructed and used. Observables are represented by hermitian operators, which have real eigenvalues and orthogonal eigenfunctions. When the system is not described by a single eigenfunction of an operator, it may be expressed as a superposition of such eigenfunctions. One consequence of the use of operators is the ‘uncertainty principle’, one of the most profound departures of quantum mechanics from classical mechanics. The uncertainty principle restricts the precision with which complementary observables may be specified and measured simultaneously.
Part 1: the need for the quantum mechanics quantization of radiation the special theory of relativity wave-particle duality the nature of matter waves the uncertainty principle the Bohr atom. Part 2 Schrodinger's equation: the postulates of quantum mechanics the wave equation the wave equation in three dimensions the time-independent Schrodinger equation stationary states normalisation probability flux physical constraints on the wave-function wave packets Gaussian wave-packets and the uncertainty principle. Part 3 One-dimensional potential wells: potential wells the one-dimensional infinite square well discussion of infinite square-well results eigenvalue equations the finite square well finite square well eigenfunctions the control of the wave-function by the wave equation numerical integration of Schrodinger's equation continuum solutions of the square-well potential interpretation of unbound-particle wave functions the double potential well the quantum bouncer electric fields and the tilted well the harmonic oscillator - introduction the quantum oscillator discussion of harmonic oscillator eigenvalues and eigenfunctions the classical probability density more potential wells. Part 4 Schrodinger's equation in three dimensions: particle in a three dimensional box degeneracy Schrodingers equation in spherical polar coordinates - the hydrogen atom normalisation. Part 5 Formal development of quantum mechanics: eigenvalues equations, operators and measurements the momentum operator, eigenvalues and eigenfunctions the position operator and its eigenfunctions Hermitian operators orthogonality of eigenfunctions the expansion postulate example - the b-decay of tritium expectation values compatibility of observables - commutators non-commuting operators time-dependence of expectation values - Ehrenfest's theorem abstract operators - the parity operator, matrix mechanics. Part 6 Angular momentum: angular momentum operators angular momentum commutators angular momentum eigenvalues and eigenfunctions interpretation of angular momentum eigenvalues interpretation of angular momentum eigenfunctions the Stern-Gerlach apparatus and electron spin spin operators - the Pauli Spin matrices the Zeeman effect measurements in quantum mechanics. Part 7 Spherically symmetric potential wells: the rigid rotator the radial equation the hydrogen atom and hydrogen like ions the isotropic harmonic oscillator molecular vibrations - the Morse potential the spherical square well general properties of spherically symmetric potential wells. (Part contents)
Quantum Mechanics is one of the most important scientific developments of the twentieth century. It serves as the basis for understanding how matter and energy behave on very small scales. Unlike classical mechanics, quantum mechanics explains physical phenomena through probabilistic models, wave functions, operators, and discrete quantized states. This paper offers a thorough study of the fundamentals of quantum mechanics, covering its historical background, postulates, mathematical framework, wave-particle duality, Schrödinger equation, uncertainty principle, and operator theory. The paper also looks at how quantum mechanics is used in modern technologies such as quantum computing, lasers, semiconductors, and medical imaging. The goal of this paper is to provide a clear and detailed foundation for students and researchers who want to grasp the main principles of quantum theory. General Terms Quantum Mechanics, Physics, Wave Function, Uncertainty, Quantum Theory, Measurement, Operators.
In this chapter we shall mainly present the basic formalism that was initially developed by Heisenberg, also known as matrix mechanics (see Subsec. 1.5.7). We will first introduce in Sec. 2.1 the concept of quantum observables. Then, the problem of discrete and continuous spectra will be discussed and the basic non-commutability of quantum-mechanical observables will be deduced. While in Sec. 2.1 we discuss observables on a general formal level, in Sec. 2.2 some basic quantum-mechanical observables will be defined, and then different representations discussed and commutation relations derived. In Sec. 2.3 a basic uncertainty relation is derived. In the same section the relationship between uncertainty, superposition, and complementarity will be discussed. Finally, in Sec. 2.4 complete subsets of commuting observables will be shown to be Boolean subalgebras pertaining to a quantum algebra which is not Boolean. BASIC FEATURES OF QUANTUM OBSERVABLES This section is devoted to a general and formal exposition of quantum observables. In Subsec. 2.1.1 we shall learn how one can mathematically represent quantum observables as Hermitian operators. In Subsec. 2.1.2 we shall see how to change a basis, while in Subsec. 2.1.3 we shall find the relationship between eigenvalues of the observables and probabilities and learn how to calculate mean values. In Subsec. 2.1.4 we shall deal with an operator diagonalization. Finally, in Subsec. 2.1.5, the basic non-commutability of quantum observables will be presented by means of an example.
The laws of quantum mechanics were formulated in the year 1925 through the work of Werner Heisenberg, followed by Max Born, Pascual Jordan, Paul Dirac, and Wolfgang Pauli. A separate but equivalent approach was independently developed by Erwin Schrödinger in early 1926. The laws governing quantum mechanics were highly mathematical and their aim was to explain many unresolved problems within the framework of a formal theory. The conceptual foundation emerged in the subsequent 2–3 years that indicated how radically different the new laws were from classical physics. In this chapter some of these salient features of quantum mechanics are discussed. The topics include the quantization of energy, wave–particle duality, the probabilistic nature of quantum mechanics, Heisenberg uncertainty relations, Bohr’s principle of complementarity, and quantum superposition and entanglement. This discussion should indicate how different and counterintuitive its fundamentals are from those of classical physics.