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Classical mechanics

M. Kachelrieß

Published Oct 26, 2017• 0 citations
Publisher

Abstract

Abstract This chapter reviews those concepts of classical mechanics which are essential for progressing towards quantum theory. The Lagrangian and Hamiltonian formulation of classical mechanics are derived from action principles. The Green function method is illustrated and the action of a relativistic point particle recalled.
DOI
10.1093/oso/9780198802877.003.0001

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The Lagrangian could be written in terms of kinetic and potential energy, functions of coordinates and velocity and the amplitude of the fields at that point. The essential condition is that the Lagrangian assumed for the field, when studied with Lagrange’s equation, gives Maxwell’s equations as solution and leads to the same wave equation. There are many other fields of physics and many mechanical systems to which the Lagrangian approach can be applied. The goal of the present text is to introduce the beginner to the least action principle and the Lagrangian approach in an elementary way. The chapter argues that the Lagrangian chosen for the fields involved must be an expression that leads to those fields and equations. In wave mechanics, particle properties are described by a so-called wave representing the classical matter field ψ used in Schrodinger’s approach of wave mechanics.
# Least Action in Classical Mechanics: # The Concept of Functional # The Principle of Least Action # Conservation of Energy. Constants of the Motion # Particles Interacting Through an Intermediate Oscillator # Least Action in Quantum Mechanics: # The Lagrangian in Quantum Mechanics # The Calculation of Matrix Elements in the Language of a Lagrangian # The Equations of Motion in Lagrangian Form # Translation to the Ordinary Notation of Quantum Mechanics # The Generalization to Any Action Function # Conservation of Energy. Constants of the Motion # The Role of the Wave Function # Transition Probabilities # Expectation Values for Observables # Application to the Forced Harmonic Oscillator # Particles Interacting Through an Intermediate Oscillator # Space-Time Approach to Non-Relativistic Quantum Mechanics # The Lagrangian in Quantum Mechanics
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Abstract This book provides an innovative and mathematically sound treatment of the foundations of analytical mechanics and the relation of classical mechanics to relativity and quantum theory. A distinguishing feature of the book is its integration of special relativity into teaching of classical mechanics. After a thorough review of the traditional theory, the book introduces extended Lagrangian and Hamiltonian methods that treat time as a transformable coordinate rather than the fixed parameter of Newtonian physics. Advanced topics such as covariant Langrangians and Hamiltonians, canonical transformations, and Hamilton-Jacobi methods are simplified by the use of this extended theory. And the definition of canonical transformation no longer excludes the Lorenz transformation of special relativity. This is also a book for those who study analytical mechanics to prepare for a critical exploration of quantum mechanics. Comparisons to quantum mechanics appear throughout the text. The extended Hamiltonian theory with time as a coordinate is compared to Dirac’s formalism of primary phase space constraints. The chapter on relativistic mechanics shows how to use covariant Hamiltonian theory to write the Klein-Gordon and Dirac equations. The chapter on Hamilton-Jacobi theory includes a discussion of the closely related Bohm hidden variable model of quantum mechanics. Classical mechanics itself is presented with an emphasis on methods, such as linear vector operators and dyadics, that will familiarise the student with similar techniques in quantum theory. Several of the current fundamental problems in theoretical physics, such as the development of quantum information technology and the problem of quantising the gravitational field, require a rethinking of the quantum-classical connection.
In this chapter we discuss theories which are rigorous in their formulation but which in order to be useful need to be modified by introducing approximations of some kind. The approximations we are interested in are those which involve introduction of classical mechanical concepts, that is, the classical picture and/or classical mechanical equations of motion in part of the system. At this point, we wish to distinguish between “the classical picture,” which is obtained by taking the classical limit ħ → 0 and the appearance of “classical equations of motion.” The latter may be extracted from the quantum mechanical formulation without taking the classical limit—but, as we shall see later by introducing a certain parametrization of quantum mechanics. Thus there are two ways of introducing classical mechanical concepts in quantum mechanics. In the first method, the classical limit is defined by taking the limit ħ → 0 either in all degrees of freedom (complete classical limit) or in some degrees of freedom (semi-classical theories). We note in passing that the word semi-classical has been used to cover a wide variety of approaches which have also been referred to as classical S-matrix theories, quantum-classical theories, classical path theory, hemi-quantal theory, Wentzel Kramer-Brillouin (WKB) theories, and so on. It is not the purpose of this book to define precisely what is behind these various acronyms. We shall rather focus on methods which we think have been successful as far as practical applications are concerned and discuss the approximations and philosophy behind these. In the other approach, the ħ-limit is not taken—at least not explicitly— but here one introduces “classical” quantities, such as, trajectories and momenta as parameters, and derives equations of motion for these parameters. The latter method is therefore one particular way of parameterizing quantum mechanics. We discuss both of these approaches in this chapter. The Feynman path-integral formulation is one way of formulating quantum mechanics such that the classical limit is immediately visible [3]. Formally, the approach involves the introduction of a quantity S, which has a definition resembling that of an action integral [101].
This chapter attempts to expound on basic and essential ideas (for further use in the book) from both classical and quantum mechanics. The chapter is somewhat technical in nature but only requires an elemental knowledge of calculus. The first three sections take a review of some of the elements of classical mechanics and classical statistical mechanics – the Euler–Lagrange and the Hamilton–Jacobi equations, the idea of an ensemble in the classical context, and the continuity equation for particle density. The remaining part is devoted to the elements of quantum mechanics – the connection between the Hamilton–Jacobi equation and the Schrödinger equation, the idea of an ensemble in the quantum context, the free particle wave function and operators, the uncertainty principle and the idea of the expectation value of an operator, and the concept of a wave packet.