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CRC Press eBooks

The Schrödinger Equation

David K. Ferry

Published Nov 6, 2020• 0 citations
Publisher

Abstract

The introductory basics of quantum mechanics were said to arise from the changes from classical mechanics in the first chapter. In some sense, these changes are brought to an observable level by the smallness of some parameter, such as the size scale. In another sense, however, quantum mechanics introduces some fundamental changes, that are not at all present in classical mechanics. Perhaps the most important effect is the appearance of operators in place of dynamical variables and the non-commuting nature of these operators. While these properties are present in normal mathematics, they are not usually associated with discussions of classical mechanics.

DOI
10.1201/9781003717195-2

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The introductory basics of quantum mechanics were said to arise from the changes from classical mechanics in the first chapter. In some sense, these changes are brought to an observable level by the smallness of some parameter, such as the size scale. In another sense, however, quantum mechanics introduces some fundamental changes, that are not at all present in classical mechanics. Perhaps the most important effect is the appearance of operators in place of dynamical variables and the non-commuting nature of these operators. While these properties are present in normal mathematics, they are not usually associated with discussions of classical mechanics.
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.
Quantum mechanics is the fundamental theory of nature at small scales and low energies of atoms and subatomic particles. Quantum mechanics is inescapable. Nearly all fields of modern physics, from elementary particles to the big bang, from semiconductors to solar energy cells, are related to quantum physics. Particles have characteristic trait of both particles and waves. The Schrodinger equation is the basic equation of quantum mechanics. The Schrodinger equation was established based on the quantization of energy conservation using quantum operators and the de Broglie relation. Wave function plays an important role in quantum mechanics. The probability interpretation assumes that the norm of the wave function measures the probability of the system in a particular coordinate or representation, in which the total probability is normalized.
This chapter is a refresher on the fundamental concepts of quantum mechanics. One essential difference between classical and quantum mechanics is that in classical mechanics, we are used to deterministic and precise laws stemming from the fact that the act of measurement can be ignored. Indeed, for all practical purposes, the interaction with the macroscopic system to be measured can be rendered negligibly small. However, at very small scales, the situation changes radically. The assumption that the measurement is ignorable is not true for micro-objects. In order to perform a measurement, we must interact with the system we are measuring. We discovered experimentally that nature has an objective scale, encoded by Planck's constant, that limits our ability to perform perturbation-free measurements. All we can do is perform many measurements on identically prepared systems, which would allow us to determine the probability of a given outcome. Micro-objects behave non-intuitively, described by new laws, those of quantum mechanics. In this chapter, we will briefly cover the postulates of quantum mechanics and introduce the Schrodinger and Heisenberg pictures of quantum mechanics.
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 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).