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American Journal of Physics

Suppose Newton had invented wave mechanics

Willis E. Lamb

Published Mar 1, 1994• 14 citations
Publisher

Abstract

It is difficult to understand quantum mechanics. A major reason for this is that people do not realize that classical mechanics can be very well described by quantum mechanics. It would all have been much easier if Newton had not developed classical mechanics from the equation relating force, mass, and acceleration of a particle, but had started from a simple form of the Schrödinger wave equation. The equation F=ma could just as well have waited for its derivation from quantum mechanics by Paul Ehrenfest in 1927.

DOI
10.1119/1.17597

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It is difficult to understand quantum mechanics. A major reason for this is that people do not realize that classical mechanics can be very well described by quantum mechanics. It would all have been much easier if Newton had not developed classical mechanics from the equation relating force, mass, and acceleration of a particle, but had started from a simple form of the Schrödinger wave equation. The equation F=ma could just as well have waited for its derivation from quantum mechanics by Paul Ehrenfest in 1927.
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.
Science has developed through a variety of investigations more or less over the time scale of human existence. On this scale, quantum mechanics is an extremely young field, existing essentially for only just over a century. Even our understanding of classical mechanics has existed for a comparatively long period—roughly having been formalized with Newton's equations published in his Principia Mathematica , in April 1686 (1687). Up until the very end of the nineteenth century, mechanics in the form of Newton's laws were comfortable, being understood and accepted by all who dealt with the motion of reasonable bodies. With the turn of the new century, however, mechanics would be shaken to its roots. The change began already on December 14, 1900, when Planck (1900) presented his new theory of black-body radiation to the German Physical Society, and this change would affect mechanics in physically small structures (and lead to quantum mechanics). Mechanics of large bodies would similarly be changed in 1905 with the arrival of Albert Einstein's (1905a) relativity theory. In this book, we will deal with only the first of these two great breakthroughs.
Classical mechanics and quantum mechanics have long been framed as binary opposites—but this dichotomy has yielded no real benefits, only persistent confusion. The fundamental equations of quantum mechanics cannot logically demarcate a clear boundary between the quantum and classical realms; in fact, the mass m and potential energy V in the Schrödinger equation seamlessly traverse the divide between the microscopic and macroscopic. The electromagnetic potential energy in the Schrödinger equation can be replaced with the potential energy of interactive forces, which can further be converted into force itself using the relation F = V/r. Derivations have now produced a force operator and quantum mechanical equations that explicitly incorporate force—including a quantum formulation of Newton’s second law. These results provide direct evidence that quantum and classical mechanics are not incompatible, but complementary. By resolving the historical absence of “force” as a formal concept in quantum mechanics, we bridge the conceptual gap between macroscopic classical frameworks and microscopic quantum theory, challenging long-held perceptions of their irreconcilability. This unified approach explains why nanoscale particles exhibit quantum properties, simplifies calculations of quantum forces, and lays the groundwork for a generalized wave mechanics that unifies mechanical descriptions across all scales.
The following equation pair shows the close connection between quantum and classical mechanics. F = m × a = ×( ) ( )average F m average a The first equation is the basis of classical mechanics. It summarizes Newton’s laws, where F is the force, m is the mass, and a is the acceleration. The second equation is the quantum-mechanical generalization. It looks almost the same except for the word average. Averaging is necessary because particles in quantum mechanics are associated with waves, and one cannot precisely determine the position or motion of a wave. Only the averages are known.