This chapter begins by exploiting the Einstein–de Broglie relation and the notion of symbol. The following steps are the local and global conservation laws associated with the Schrödinger equation, probabilistic interpretation of the wave function and probability distribution, the spreading of wave packets and transformation properties of wave functions. In the last part, the Heisenberg picture is outlined at an introductory level. From classical to wave mechanics In the previous chapter we have seen how to go from wave equations on space–time to Hamilton equations on phase space by means of an appropriate approximation requirement on wave solutions. In more formal language, we could say we have considered a procedure to associate a Hamiltonian ordinary differential equation on phase space starting with a partial differential equation on configuration space. This association produces a correspondence between differential operators and polynomials on phase space, which relies on the notion of symbol (see below). Now we are aiming to pass from a function on phase space, i.e. the classical Hamiltonian, to an operator which, by analogy, is called the Hamiltonian operator. Such a transition is physically non-trivial because it amounts to building a wave equation from the equation on rays. From the formal point of view it appears to contain some ambiguities. Such a transition is always possible if polynomials in p have constant coefficients.
This chapter begins by exploiting the Einstein–de Broglie relation and the notion of symbol. The following steps are the local and global conservation laws associated with the Schrödinger equation, probabilistic interpretation of the wave function and probability distribution, the spreading of wave packets and transformation properties of wave functions. In the last part, the Heisenberg picture is outlined at an introductory level. From classical to wave mechanics In the previous chapter we have seen how to go from wave equations on space–time to Hamilton equations on phase space by means of an appropriate approximation requirement on wave solutions. In more formal language, we could say we have considered a procedure to associate a Hamiltonian ordinary differential equation on phase space starting with a partial differential equation on configuration space. This association produces a correspondence between differential operators and polynomials on phase space, which relies on the notion of symbol (see below). Now we are aiming to pass from a function on phase space, i.e. the classical Hamiltonian, to an operator which, by analogy, is called the Hamiltonian operator. Such a transition is physically non-trivial because it amounts to building a wave equation from the equation on rays. From the formal point of view it appears to contain some ambiguities. Such a transition is always possible if polynomials in p have constant coefficients.
Till 1925 the idea of what the quantum theory would be was unknown. During 1925 three different but equivalent versions of quantum theory were proposed – Schrödinger-proposed wave mechanics; Heisenberg-developed matrix mechanics and Dirac-introduced operator theory. Considering the de Broglie’s matter waves Erwin Schrödinger, an Austrian physicist, argued that if a particle like an electron behaves as a wave then the equation of wave motion could be successfully applied to it. He postulated a function varying in both space and time in a wave-like manner (hence called wave function and denoted it as ψ). This function is generally complex and assumed to contain information about a system. Schrödinger set up a linear and time-dependent wave-like equation, called Schrödinger wave equation, to describe the wave aspect of a particle taking account of de Broglie’s relation for wavelength. Physically, |ψ(X, t)|2, where ψ(X, t) is the solution of the Schrödinger equation, is interpreted as position probability density. That is, |ψ(X, t)|2 is the probability density of observing a particle at position X at time t. ψ does not give exact outcomes of observations but helps us to know all possible events and their probabilities . Further, the probability interpretation allows us to find the average or expected result of a set of measurements on a quantum system.
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.
In the first part of this chapter, the Hamiltonian function and Hamilton-Jacobi equation are presented. The latter is the starting point of Schrödinger’s work to obtain his wave equation. Next, the derivation of the wave equation is given along with the foundations of Schrödinger’s wave mechanics, i.e., the Hamilton’s analogy between optics and mechanics. It is also introduced some applications of the Schrödinger’s new quantum theory to several problems (radial function of the hydrogen atom, Compton effect and quantum harmonic oscillator). Other applications are also indicated in this chapter. The Dirac notation is introduced followed by the derivation of the expectation value in quantum mechanics and its derivation with respect to time. Next, Born’s probability theory and the superposition principle are introduced followed by the derivation of the quantum linear momentum operator. Afterwards, the normalization of a wave function and the orthogonality of a wave function are given followed by the properties of the Hermitian operator. In the end of the chapter, several examples and exercises are presented (168 words).
Abstract The second chapter introduces the notion of wave functions and their equation of motion proposed by Erwin Schrödinger. The interpretation and normalization of the wave function are discussed, leading to the continuity equation. The chapter also introduces basic mathematical operations that can be carried out for wave functions and operators, describing e.g., how to calculate scalar products and commutators, as well as expectation values and variances of observables. With these tools at hand, Heisenberg’s uncertainty principle is re-examined, focusing on the example of a Gaussian wave packet, which is studied both in position space and momentum space. We also discuss the time evolution of general states and of the exemplary wave packet. Finally, we will see how classical physics is recovered as a limiting case of the quantum-mechanical description
We found in the previous chapter that the molecular partition function is required to determine the thermodynamic properties of an ideal gas. To evaluate the partition function, specification of pertinent energy levels and degeneracies is necessary. Such knowledge demands that we investigate at least the rudiments of quantum mechanics, and especially those quantum concepts required for subsequent applications to statistical thermodynamics. For this reason, we concentrate in the next few chapters on the Schrödinger wave equation, whose various solutions provide the ε j 's and g j 's needed for the eventual calculation of thermodynamic properties. Depending on your academic background, you might thus consider reviewing classical mechanics (Appendix G) and operator theory (Appendix H) in preparation for your upcoming study of quantum mechanics. We begin this chapter with a historical review of the developments leading to the formulation of quantum mechanics, subsequently focusing on the Bohr model for atomic hydrogen and the de Broglie hypothesis for matter waves. We then introduce the Schrödinger wave equation, the basic postulates of quantum mechanics, and salient insights from these postulates germane to the development of statistical thermodynamics. We next apply the Schrödinger wave equation to the translation energy mode of an atom or molecule. This application conveniently explains both quantum states and quantum numbers, including their relation to our previous notions of microstate and macrostate. We end this chapter by discussing the Heisenberg uncertainty principle, including its utility in defining indistinguishability and symmetry conditions for multiparticle systems.
This chapter presents some of the features of the phase space picture. For quantum mechanical systems Eugene Paul Wigner introduced a quantum analogue of a phase space probability distribution called Wigner distribution function or phase space picture. Using the Wigner distribution function it is possible to perform a canonical transformation in quantum mechanics. The phase space picture is a useful candidate to illustrate the transition from classical to quantum mechanics. The Poisson brackets and canonical transformations are part of the effective formulation of classical mechanics. In the infinite height potential video sequences of the evolution of Wigner functions for a wave function of a wave packet and its mirror wave packet is constructed. The chapter also presents the application of the canonical transformations and phase space distribution function in the study of wave packet spread and coherent and squeezed states.