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Oxford University Press eBooks

Wavefunctions

Peter Joshua Atkins, Julio de Paula, James Keeler

Published Dec 5, 2022• 0 citations
Publisher

Abstract

This chapter examines the interpretation of the wavefunction, and specifically what it reveals about the location of a particle. In quantum mechanics, all the properties of a system are expressed in terms of a wavefunction, which is obtained by solving the equation proposed by Erwin Schrödinger. Indeed, wavefunctions provide the essential foundation for understanding the properties of electrons in atoms and molecules, and are central to explanations in chemistry. The chapter then considers how, according to the Born interpretation, the probability density at a point is proportional to the square of the wavefunction at that point. A wavefunction is normalized if the integral over all space of its square modulus is equal to 1. Ultimately, a wavefunction must be single-valued, continuous, not infinite over a finite region of space, and have a continuous slope. The quantization of energy stems from the constraints that an acceptable wavefunction must satisfy.

DOI
10.1093/hesc/9780198847816.003.0034

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This chapter examines the interpretation of the wavefunction, and specifically what it reveals about the location of a particle. In quantum mechanics, all the properties of a system are expressed in terms of a wavefunction, which is obtained by solving the equation proposed by Erwin Schrödinger. Indeed, wavefunctions provide the essential foundation for understanding the properties of electrons in atoms and molecules, and are central to explanations in chemistry. The chapter then considers how, according to the Born interpretation, the probability density at a point is proportional to the square of the wavefunction at that point. A wavefunction is normalized if the integral over all space of its square modulus is equal to 1. Ultimately, a wavefunction must be single-valued, continuous, not infinite over a finite region of space, and have a continuous slope. The quantization of energy stems from the constraints that an acceptable wavefunction must satisfy.
This chapter addresses quantum mechanics and spectroscopy. Two key ideas emerge from the theory of quantum mechanics: the first is that the electron is described by a wavefunction, and the second is that the energy of the electron is quantized. The wavefunction is important since its square gives the probability density of the electron. Quantization means that the energy can only have certain discrete values, rather than being allowed to vary continuously. The chapter looks at how these wavefunctions and energy levels arise in quantum mechanics, and how they can actually be computed in some simple cases. This will involve setting up and then solving the famous Schrödinger equation. Quantum mechanics is not restricted to the description of electrons in atoms and molecules, but can also be used to describe the motion of molecules in space (translation), as well as the rotation and vibration of molecules.
Schrödinger hoped that his wave mechanics would help to re-establish some sense of ‘visualizability’ of the physics going on inside the atom. In searching for a suitable interpretation of the wavefunction, he focused on the density of electrical charge, which he associated with the wavefunction ψ‎ multiplied by its complex conjugate. Hidden in his words is the interpretation that would eventually come to dominate our understanding of the wavefunction. Max Born had no hesitation in concluding that the only way to reconcile wave mechanics with the particle description is to interpret the modulus-square of the wavefunction as a probability density. It was Wolfgang Pauli who proposed to interpret this not only as a transition probability or as the probability for the system to be in a specific state, as Born had done, but as the probability of ‘finding’ the electron at a specific position in its orbit inside an atom.
Abstract Introduces the Schrödinger wave equation as the key equation governing the behaviour of electrons, and presents simple solutions (known as wave functions) as electron waves, and packets of electron waves, which can represent localised, particle-like electrons. Considers the behaviour of electron waves on meeting a potential barrier and trapped in potential wells with finite and infinite walls, and introduces the idea of energy levels. Discusses the accepted interpretation of the electron wave function, namely that its modulus square leads to the probability of finding an electron in a given region of space. Considers further the uncertainty relationship and associated philosophical implications.
Introduction In this chapter we discuss the basic physical concepts and equations related to the behavior of particles in the nanoworld. We introduce the Schrödinger wave equation for particles and determine the ways to calculate observable physical quantities. We find that, in wave mechanics, the motion of a particle confined to a finite volume is always characterized by discrete values of the energy and standing-wave-like wavefunctions, i.e., such a motion is quantized. While motion in an infinite space (i.e., free motion) is not quantized and is described by propagating waves, the energy of the particle is characterized by a continuous range of values. Keeping in mind the diverse variants of nanostructures, by using wave mechanics we analyze some particular examples, which highlight important quantum properties of particles. Many of the examples analyzed can serve as the simplest models of nanostructures and will be exploited in following chapters to understand the fundamentals of processes in nanoelectronics. The Schrödinger wave equation From the previous chapter, we conclude that nanosize physical systems are quantum-mechanical systems, inasmuch as their sizes are comparable to typical de Broglie wavelengths of the particles composing these systems. In dealing with quantum-mechanical systems, one aims at determining the wavefunction of a single particle or of the whole system. As we will demonstrate in the subsequent discussion, knowledge of the wavefunction in quantum mechanics is sufficient to describe completely a particle or even a system of particles.
In quantum mechanics, the physical state of an electron is described by a wave function. According to the standard probability interpretation, the wave function of an electron is probability amplitude, and its modulus square gives the probability density of finding the electron in a certain position in space. In this article, we show that this central assumption of quantum mechanics may have an ontological extension. It is argued that microscopic particles such as electrons are indeed particles, but their motion is not continuous but discontinuous and random. On this view, the modulus square of the wave function not only gives the probability density of the particles being found in certain locations, but also gives the probability density of the particles being there. In other words, the wave function in quantum mechanics can be regarded as a representation of the state of random discontinuous motion of particles, and at a deeper level, it may represent the dispositional property of the particles that determines their random discontinuous motion.
This chapter discusses the central concepts and equations of quantum theory which demonstrates how particles spread through space like waves and are described mathematically by a wavefunction. It examines how the wavefunction is interpreted and introduces the uncertainty principle, which is one of the most profound departures of quantum mechanics from classical mechanics. It also highlights the dynamical properties of a system that are contained in the wavefunction and are obtained by solving the Schrödinger equation. The chapter reconciles the facts that atoms and molecules can possess only certain energies, that waves exhibit the properties of particles, and that particles exhibit the properties of waves. It reviews the classical concept that have been accommodated by the development of quantum mechanics, in which equations are set up to treat a particle as spread through space like a wave.