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Oxford University Press eBooks

The Schrödinger Equation

Muhammad Suhail Zubairy

Published May 7, 2020• 0 citations
Publisher

Abstract

In this chapter, the Schrödinger equation is “derived” for particles that can be described by de Broglie waves. The Schrödinger equation is very different from the corresponding equation of motion in classical mechanics. In order to illustrate the fundamental differences between the two theories, one of the simplest problems of particle dynamics is solved in both Newtonian and quantum mechanics. This simple example also helps to show that quantum mechanics is the fundamental theory and classical mechanics is an approximation, a remarkably good approximation, when considering macroscopic objects. The solution of the Schrödinger equation is presented for a particle inside a box and the quantization condition is derived. The amazing possibility of quantum tunneling when a particle is incident on a barrier of height larger than the energy of the incident particle is also discussed. Finally the three-dimensional Schrödinger equation is solved for the hydrogen atom.

DOI
10.1093/oso/9780198854227.003.0017

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In this chapter, the Schrödinger equation is “derived” for particles that can be described by de Broglie waves. The Schrödinger equation is very different from the corresponding equation of motion in classical mechanics. In order to illustrate the fundamental differences between the two theories, one of the simplest problems of particle dynamics is solved in both Newtonian and quantum mechanics. This simple example also helps to show that quantum mechanics is the fundamental theory and classical mechanics is an approximation, a remarkably good approximation, when considering macroscopic objects. The solution of the Schrödinger equation is presented for a particle inside a box and the quantization condition is derived. The amazing possibility of quantum tunneling when a particle is incident on a barrier of height larger than the energy of the incident particle is also discussed. Finally the three-dimensional Schrödinger equation is solved for the hydrogen atom.
During 1925 three different but equivalent versions of quantum theory were proposed, Schrodinger 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. Schrodinger set up a linear and time-dependent wave-like equation, called Schrodinger wave equation, to describe the wave aspect of a particle taking account of de Broglie's relation for wavelength. This chapter discusses the Schrodinger equation and explores the general properties of the solution of the Schrodinger equation. It verifies the Ehrenfest's theorem in certain interesting systems and solves equation for linear harmonic oscillator and shows that their solutions are identical to the solutions of classical equation of motion.
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.
The Eherenfest theorem states that Schrodinger representation of quantum mechanics (wave mechanics) reproduces Newton laws of motion in terms of expectation values. Remarkably, the contrary is considered elusive and, indeed, many authors have tried to obtain wave mechanics starting from other alternative frameworks of classical mechanics (for instance, Hamilton-Jacobi theory). Despite this common opinion, we present here a simple method to make Newtonian dynamics develop naturally into Schrodinger representation. The proof is based on the assumption of matter waves and is laid out in three fundamental steps. First, the role of classical density functions is underlined in view of their use to define constants of the motion for massive particles. Thanks to this preparatory step, density functions generate wave functions whose spatial and time variables obey Newton laws of motion. The resulting wave equation is defined in dependence on a parameter that plays the identical role of the constant K introduced by Schrodinger in the original formulation of his theory. In the final step, the classical wave equation is treated under the hypothesis of conservative forces common to the Eherenfest theorem and, after some algebra, the Schrodinger equation emerges by means of the identification of the classical momentum with de Broglie momentum of matter waves.
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.
Replace the potential energy in the Hamiltonian operator from electromagnetic interaction potential energy to gravitational interaction potential energy. Define that the kinetic energy of a moving object is equal to its de Broglie wave's hν. Successfully obtained the Schrödinger equation describing planetary motion. The Schrödinger equation itself is a combination of classical mechanics and quantum mechanics. Quantum mechanics based on it cannot completely exclude classical mechanics. Many examples have been introduced where classical mechanics and quantum theory can be combined to describe the same object (whether it is macroscopic or microscopic). Local realism and determinism still have a market in modern physics. Quantum mechanics and classical mechanics have a compatible and complementary relationship.