Abstract The discussion in previous chapters has centred on the use of trajectorybased information to add quantum corrections to the classical picture, in a mainly time-independent picture. This chapter takes an explicitly timedependent standpoint by examining the relevance of various types of wavepacket to classical mechanics and vice versa; Littlejohn (1986) gives a much fuller mathematical account. The intuitive connection is obvious, in the sense that a unit normalized wavepacket corresponds most nearly to a single quantum mechanical particle, but the associated uncertainty relations imply a closer connection with a classical ensemble. Analysis of this connection is one of the main concerns of the first half of the chapter.
Abstract The discussion in previous chapters has centred on the use of trajectorybased information to add quantum corrections to the classical picture, in a mainly time-independent picture. This chapter takes an explicitly timedependent standpoint by examining the relevance of various types of wavepacket to classical mechanics and vice versa; Littlejohn (1986) gives a much fuller mathematical account. The intuitive connection is obvious, in the sense that a unit normalized wavepacket corresponds most nearly to a single quantum mechanical particle, but the associated uncertainty relations imply a closer connection with a classical ensemble. Analysis of this connection is one of the main concerns of the first half of the chapter.
In the first two chapters we have examined the basic principles – superposition (p. 18), complementarity (p. 19), quantization (p. 44), statistical algorithm (p. 57), and correspondence (p. 72) (see also Subsec. 2.3.4) – and the basic entities, observables and states, of quantum mechanics, as well as the main differences with respect to classical mechanics. While what we have discussed so far is rather a static picture of observables and states, in this chapter we shall deal with quantum dynamics, i.e. with the time evolution of quantum-mechanical systems. Historically, after Bohr had provided a quantized description of the atom (see Subsec. 1.5.4), Einstein showed the quantized nature of photons (see Subsec. 1.2.1), and de Broglie hypothized the wave-like nature of matter (see Subsec. 1.5.5), the first building block of quantum mechanics was provided by the commutation relations, proposed by Heisenberg in 1925, whose consequence is represented by the uncertainty relation (see Subsec. 2.2.7 and Sec. 2.3). This was the subject of the previous chapters. The dynamical part of the theory was proposed by Schrödinger in 1926, and is known as the Schrödinger equation. It is also known as wave mechanics (see Subsec. 1.5.7). In this chapter we shall show that Heisenberg's and Schrödinger's formulations are only two different aspects of the same theory. We shall also come back to this point in Sec. 8.1.1. Here, first we shall derive the fundamental equation which rules quantum dynamics (Sec. 3.1), and, in Sec. 3.2, we shall summarize the main properties of the Schrödinger equation.
This chapter demonstrates how one can find the time evolution of the operators from the knowledge of the time evolution of the state vector, and vice versa. It introduces a unitary operator and discusses the concept of a unitary transformation to show how one could transfer the time dependence from the state vectors to the operators. The possibility that either the state vectors or the operators can depend explicitly on time will lead to introduce the Schrodinger, Heisenberg, and interaction pictures. The chapter discusses the fundamental differences between these pictures. It also discusses the Ehrenfest theorem, which shows under which conditions quantum mechanics predicts the same results for measured physical quantities as classical physics. It illustrates unitary transformations of an arbitrary operator Â(t) which can depend explicitly on time. Quantum mechanics produces the same result as classical mechanics for a system in which particles are represented by well-localized wave functions.
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.
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.
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 forms a natural divide half-way into the book. In the first half, we discussed the principles of discrete time (DT) classical mechanics (CM). In the second half we focus mainly on quantum principles. This chapter is a good place in which to take stock of what we have done, what we plan to do, and how the two halves of the book are related. The common theme is Lee's approach to DT mechanics, discussed in Chapter 10. As we implied previously, theorists whose work is relevant to us can be classified into two types: the applied mathematicians and the mathematical physicists, who may also be called fundamentalists. This division is based not on any value judgements but on the motivations and ambitions driving a theorist's work, which are generally easy to identify. The applied mathematicians fall into two categories. The first consists of those interested in finding better ways of understanding CM and, if necessary, finding ever better approximations to it. They explore DT CM with that in mind and are generally not remotely interested in quantum mechanics (QM). The second category consists of quantum theorists, such as Bender, who develop DT numerical simulation to approximate standard QM. Lattice gauge theorists also fall into the applied group, since their discretization of spacetime is regarded at all times as an approximation to the continuum.