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American Journal of Physics

The Schrödinger equation in classical physics

Stanley M. Flatté

Published Dec 1, 1986• 16 citations
Publisher

Abstract

The use of the Schrödinger equation in classical physics was begun in 1950; in the classical context it is known as the parabolic wave equation. The derivation of the parabolic wave equation from the full classical wave equation is analogous to deriving the Schrödinger equation from the Klein–Gordon equation. An introduction to the Schrödinger equation by this route could be of help in introductory quantum-mechanics classes.

DOI
10.1119/1.14720

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The use of the Schrödinger equation in classical physics was begun in 1950; in the classical context it is known as the parabolic wave equation. The derivation of the parabolic wave equation from the full classical wave equation is analogous to deriving the Schrödinger equation from the Klein–Gordon equation. An introduction to the Schrödinger equation by this route could be of help in introductory quantum-mechanics classes.
In this study, both the Klein-Gordon equation and time-dependent Schrodinger equation are directly derived from De Broglie's (\lamda=h/p). Besides that, a new formula of the Klein-Gordon equation was obtained by using two equations, i.e., the classical wave function and relativistic total energy relation [[hw] = [(h^2 k^2)/2m] + V]. Moreover, a new equation of time-dependent Schrodinger (TDSE) in three-dimension 3D was derived and evidently obtained. Thus, the previous result corresponds to (TDSE) in 3D calculated using two equations, (i.e., Einstein energy and Newton's second law of mechanics).
In this paper the relationship of classical physics and quantum physics is studied by introducing a partial differential equation, which describes classical mechanics, but looks very similar to the Schrodinger wave equation of quantum mechanics. This work is largely based on David Bohm's causal interpretation of quantum mechanics, but is in some sense complementary to it. In Bohm's theory the Schrodinger wave equation is used to derive classical looking equations of motion for quantum physics. Here exactly the opposite is done. The equations of classical physics are put into a form resembling the Schrodinger equation of quantum physics.
Abstract A new derivation of Schrödinger’s equation is presented, based on Schrödinger’s original discussions on refraction and the optical-mechanical analogy, but adopting a much simpler formalism: Newtonian mechanics and some basic elements of classical wave theory (such as Snell’s law). We compare how particles and waves refract and show that the ‘law of particle refraction’ and the ‘law of wave refraction’ may become consistent if one assumes that a particle can be represented by a wave group. In this case, the differential equation whose solutions represent the waves forming such wave group is the Schrödinger equation. Due to the simplicity of the adopted mathematical formalism, we argue that this derivation can be used in quantum mechanics courses at introductory level to give students an idea of Schrödinger’s original path to his wave equation.
Abstract A heuristic derivation of the one-dimensional Schrödinger equation is presented based on an energy interpretation of the wave function. The spatial derivative naturally produces a term proportional to the kinetic energy, while the temporal derivative yields a term proportional to the total energy. By relating both contributions through the classical relation for total mechanical energy, the Schrödinger equation is obtained. The proposed approach provides a simple and intuitive framework to introduce this equation in modern physics or introductory-level quantum mechanics courses.
Physics and the natural world have been studied by humans since the first person wondered ‘Why?’ The Schrödinger equation is important because it attempts to describe a relatively new (to us) and unfamiliar part of our universe, behavior of subatomic particles, and helps modern scientists uncover secrets that have proven useful in technology and explaining our origins. The time dependent Schrödinger equation can be derived from various starting points in Classical such as the equation of a wave, the time independent Schrödinger equation, and Hamilton-Jacobi equations. Why Quantum mechanics exists, useful applications, and cases when it fails or is inaccurate are often not understood or even easy to explain to people uninvolved in physical studies. Not only do I hope for someone to be more educated on the topic after encountering this project, but I, myself, have learned a great deal more about the advanced topics I am interested in. Class: Math 488 – Senior Capstone