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CRC Press eBooks

Schrödinger's Equation

F.J. Duarte

Published Jan 15, 2024• 0 citations
Publisher

Abstract

The Schrödinger equation is derived from a heuristic approach using the complex wave equation and from a more rigorous approach, á la Feynman, utilizing Dirac’s notation. The emergent nature of the Schrödinger equation is emphasized in this chapter. Applications of this iconic equation to semiconductor laser emission are illustrated in detail.

DOI
10.1201/9781003398707-9

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Schrödinger's Equation

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As its name suggests, Schrodinger’s equation is the brain child of Schrodinger and was first disclosed in 1926 in a paper entitled An Undulatory Theory of the Mechanics of Atoms and Molecules . Schrodinger begins by considering the dynamics of a particle moving through a force field. The Schrodinger’s equation is arrived at in an alternative heuristic path modeled after Haken. The Schrodinger’s equation is a wave equation that incorporates classical particle concepts, and classical wave function concepts, in its derivation. In other words, a heuristic approach to Schrodinger’s equation utilizes Max Planck’s quantum energy, classical kinetic energy of a free particle, and the classical “wave functions of ordinary wave optics.” Schrodinger’s equation is capable of explaining all atomic phenomena except those involving magnetism and relativity.

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Cambridge University Press eBooks•Dec 18, 2025•0 citations
The Schrödinger equation is derived from a heuristic approach using the complex wave equation and from a more rigorous approach, á la Feynman, utilizing Dirac’s notation. The emergent nature of the Schrödinger equation is emphasized in this chapter. Applications of this iconic equation to semiconductor laser emission are illustrated in detail.
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).
Chapter 4 shows that, just as Heisenberg’s matrix mechanics represented a break with classical physics and the beginning of a new physics, Schrödinger’s wave mechanics represented another, no less radical, break with classical physics and at the same time opened up another path to the quantum world. Schrödinger’s theory is shown to make substantial use of de Broglie’s waves and related phenomena. The Schrödinger equation, being wave-like, constitutes mathematically an eigenvalue problem for bounded systems, and thus naturally leads to quantization, as is illustrated with a couple of simple examples. The chapter ends with Dirac’s abstract formulation, which has the advantage of being suitable for any description, be it Heisenberg’s, Schrödinger’s or any other.
The Schrödinger equation is a beautiful wave equation that leads us directly to the widely used Hamiltonians in quantum mechanics. The Schrödinger equation was also there at the very beginning of quantum mechanics. Therefore it should not be surprising that when most people refer to quantum mechanics they do so from the vantage point of the Schrödinger equation. However, there are also several other versions of quantum mechanics. Most notable is the version of quantum mechanics introduced by Heisenberg that was transformed into a matrix formalism via a collaborative effort of Born, Heisenberg, and Jordan. Then around came our favorite, Dirac's bra ket notation. Dirac also inspired Feynman to introduce his path integrals. But there are also others. A brief description follows.
As its name suggests, Schrodinger’s equation is the brain child of Schrodinger and was first disclosed in 1926 in a paper entitled An Undulatory Theory of the Mechanics of Atoms and Molecules . Schrodinger begins by considering the dynamics of a particle moving through a force field. The Schrodinger’s equation is arrived at in an alternative heuristic path modeled after Haken. The Schrodinger’s equation is a wave equation that incorporates classical particle concepts, and classical wave function concepts, in its derivation. In other words, a heuristic approach to Schrodinger’s equation utilizes Max Planck’s quantum energy, classical kinetic energy of a free particle, and the classical “wave functions of ordinary wave optics.” Schrodinger’s equation is capable of explaining all atomic phenomena except those involving magnetism and relativity.
Chapter 7 is devoted to the free particle and the derivation of the time-dependent Schrödinger equation. The free particle might be thought to be the simplest quantum problem of all, but a detailed treatment of it, including both Born and Dirac normalization, serves to highlight some surprising features. Powerful tools are introduced that are of wider application, notably the Green function method and the Feynman propagator. This material paves the way for the study of the time-dependent Schrödinger equation, the introduction of the current density of particles and the distinction between pure states and mixtures; the equation is applied specifically to the study of coherent states. The chapter ends with an introduction to the propagator in the general case.