Elements of quantum mechanics Quantum mechanics was discovered in 1925 through groping efforts to compromise two apparently contradictory pictures on the fundamental entities in nature. One was the wave picture for light which was later extended to matter by de Broglie, another was the corpuscular picture of matter which was later extended to light by Einstein. Schrödinger's wave equation came as a natural development of the first stream, while Heisenberg's matrix mechanics was presented as a unique proposal from the second stream. In spite of completely different appearances, the two theories proved, within a couple of years after their discoveries, to be equivalent. This is a most beautiful example that the physical reality exists independent of the mathematical framework formulated for its description. In this chapter, we will give a very brief review of the principles of quantum mechanics, 1–3 mainly with the harmonic oscillator as a model system for the following reasons. The first is historical: the electromagnetic wave, whose interaction with matter is the subject of this book, is a harmonic oscillator, a system which was for the first time subject to “quantization”, thus opening a way to the discovery of quantum mechanics. The second is technical: the harmonic oscillator is one of very few examples of analytically soluble problems in quantum mechanics. The third is pedagogical: the harmonic oscillator is a system best suited for realization of the equivalence of the two different pictures mentioned above and hence for a deeper understanding of the principles of quantum mechanics.
Elements of quantum mechanics Quantum mechanics was discovered in 1925 through groping efforts to compromise two apparently contradictory pictures on the fundamental entities in nature. One was the wave picture for light which was later extended to matter by de Broglie, another was the corpuscular picture of matter which was later extended to light by Einstein. Schrödinger's wave equation came as a natural development of the first stream, while Heisenberg's matrix mechanics was presented as a unique proposal from the second stream. In spite of completely different appearances, the two theories proved, within a couple of years after their discoveries, to be equivalent. This is a most beautiful example that the physical reality exists independent of the mathematical framework formulated for its description. In this chapter, we will give a very brief review of the principles of quantum mechanics, 1–3 mainly with the harmonic oscillator as a model system for the following reasons. The first is historical: the electromagnetic wave, whose interaction with matter is the subject of this book, is a harmonic oscillator, a system which was for the first time subject to “quantization”, thus opening a way to the discovery of quantum mechanics. The second is technical: the harmonic oscillator is one of very few examples of analytically soluble problems in quantum mechanics. The third is pedagogical: the harmonic oscillator is a system best suited for realization of the equivalence of the two different pictures mentioned above and hence for a deeper understanding of the principles of quantum mechanics.
Schrödinger's work on wave mechanics in 1926 appears to have been driven by the idea that one could give a purely wave-theoretical description of matter. Key elements in this picture were the idea of particles as wave packets (Section 4.3) and the possible implications for the problem of radiation (Section 4.4). This pure wave theory, in contrast to de Broglie's work, did away with the idea of particle trajectories altogether (Section 4.5). The main conflict, however, was between Schrödinger and the proponents of quantum mechanics (in particular Heisenberg, Section 4.6), both in its form at the time of Schrödinger's papers and in its further developments as sketched in the previous chapter. For reference, we provide a brief chronology of Schrödinger's writings relating to wave mechanics up to the Solvay conference: Paper on Einstein's gas theory, submitted 15 December 1925, published 1 March 1926 (Schrödinger 1926a). First paper on quantisation, submitted 27 January 1926, addendum in proof 28 February 1926, published 13 March 1926 (Schrödinger 1926b). Second paper on quantisation, submitted 23 February 1926, published 6 April 1926 (Schrödinger 1926c). Paper on the relation between wave and matrix mechanics (‘equivalence paper’), submitted 18 March 1926, published 4 May 1926 (Schrödinger 1926d). Paper on micro- and macromechanics (coherent states for the harmonic oscillator), published 9 July 1926 (Schrödinger 1926e). […]
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 presents the origins of quantum mechanics. The story of how people hit on the highly non-intuitive world picture of quantum mechanics, in which the physical state of a system is represented by an element in an abstract linear space and its observable properties by operators in the space, is fascinating and exceedingly complicated. The much greater change from the classical world picture of Newtonian mechanics and general relativity to the quantum world picture came in many steps taken by many people, often against the better judgment of participants. There are three major elements in the story. The first is the experimental evidence that the energy of an isolated system can only assume special discrete or quantized values. The second is the idea that the energy is proportional to the frequency of a wave function associated with the system. The third is the connection between the de Broglie relation and energy quantization through the mathematical result that a wave equation with fixed boundary conditions allows only discrete quantized values of the frequency of oscillation of the wave function (as in the fundamental and harmonics of the vibration of a violin string).
This chapter begins by understanding the classical view of an electromagnetic wave and proceed step-by-step in exploring the historical development of quantum mechanics. The frequency of the electromagnetic wave is the same as the frequency of mechanical oscillation of the charge, and the intensity of the electromagnetic wave is the square of the amplitude of the mechanical oscillation of the charge. The origin of quantum mechanics is linked to the discovery of then new laws of radiation by Max Planck in 1900. The explanation of the empirical facts of radiation by Planck using an extraordinarily powerful yet simple model lead to the discovery of the "quantum" nature of radiation. The work to develop a mathematical theory of quantum mechanics continued for a very long time. Two mathematical approaches for the formalism of quantum mechanics emerged. One was developed in 1925 by Werner Heisenberg, called matrix mechanics, and the other was developed by Erwin Schrodinger in 1926 called wave mechanics.
This chapter starts with the historical development of quantum mechanics and explanations of key concepts behind fundamental discoveries such as Planck's hypothesis, the photoelectric effect, Bohr's atomic model, Compton scattering, the de Broglie hypothesis, the Pauli exclusion principle, the Schrödinger equation, Heisenberg quantum mechanics and the uncertainty principle, the Born interpretation of wave function, the Davisson and Germer experiment, and Dirac relativistic quantum mechanics, which made important contributions to the foundation of modern quantum mechanics. Concepts like the particle properties of radiation, the wave properties of matter, quantized energy and momentum, and the idea that one can no longer know exactly where a single particle like an electron is at any instant of time are brought out clearly through this chapter. A list of important quantum mechanical parameters along with solved examples and unsolved exercises on the key concepts are included towards the end.
In completing the story of spin, we have run far ahead of the continued development of the understanding of the matrix, operator and wave mechanical approaches to quantum mechanics. The reconciliation of these approaches was described in Chap. 15, but there remained the issue of the interpretation of the wavefunction and the deeper implications of the theory. The understanding came gradually with Born's interpretation of the wavefunction, Ehrenfest's demonstration of the equivalence of the classical and quantum pictures and Heisenberg's enunciation of the uncertainty principle. These led to what became known as the Copenhagen interpretation of quantum mechanics. At the same time, the formal mathematical foundations of the different approaches to quantum phenomena were set on a secure foundation thanks to the efforts of Hilbert and many others. These developments resulted in what may be referred to as the completion of quantum mechanics, in the sense that it laid the foundations for all the future development of physics at the atomic and subatomic level – some of these achievements are summarised in Chap. 18. Schrödinger's interpretation (1926) Schrödinger regarded wave mechanics as superior to the matrix mechanical approach to quantum physics, not only because it was based upon the well-known eigenfunction techniques of classical physics, but also because it was much more visualisable. His first attempt at interpreting the wavefunction appeared in the final Sect. 7 of the fourth part of his great series of papers (Schrödinger, 1926f) and was entitled On the physical significance of the field scalar.