Simple derivation of Schrödinger equation from Newtonian dynamics
Simple derivation of Schrödinger equation from Newtonian dynamics Michele Marrocco Dipartimento di Fisica, Università di Roma ‘La Sapienza’ P.le Aldo Moro 5, I-00185 Rome, Italy & ENEA (Italian National Agency for New Technologies, Energies and Sustainable Economic Development) via Anguillarese 301, I-00123 Rome, Italy
Abstract
If Schrödinger representation of quantum mechanicsreproduces Newton’s laws of motion in terms of expectation values (Ehrenfest theorem), the contraryis considered elusive. Against this opinion, we present here a simple method to make Newtonian dynamics develop into Schrödinger representation. The proofis laid outin two steps. First, Newton’s laws of motion are used to determine a classical wave equation whose similarity with the Schrödinger equation is mediatedby a parameter that plays the identical role of the constant Kintroduced by Schrödinger in the original formulation of his theory. In the second step, the classical wave equation becomes exactly the Schrödinger equationthanks to the numerical value of the parameter obtained from the identification of the classical momentum with de Broglie momentum of matter waves.
I. INTRODUCTION
One of the main routes to quantum mechanics runs through the Schrödinger equationand the concept of wave function. Despite theundoubted importanceof this cornerstone of modern physics, the subject is ordinarily introduced in courses on quantum mechanics without much detail aboutits conceptual foundations, with the result that, according to prominent physicists, the derivation ofthe Schrödinger equation conveysa senseof dissatisfaction. In reality, the disappointmenthas something to do with the heuristic explanation reportedby Schrödingerin his original conjecture. Its weaknessis, for instance, underlinedby Feynmanin the third volume of the Lectureswhere one can read the following comment about the founder of wave mechanics: “some of the arguments he used were even false, but that does not matter; the only important thing is that the ultimate equation gives a correct description of nature”. Against this compliant attitude… read more