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

The Dirac equation is commonly demonstrated under stringent hypotheses and after considerable math work made in relativistic quantum mechanics and quantum field theory. Here, a purely geometric approach free from hypotheses is suggested. The suggestion draws inspiration from the technique of stereographic projection that was developed before the quantum era to solve gyroscopic problems of classical mechanics. The projected variable is the generalized (or canonical) momentum vector. Its undetermined geometric orientations define a sphere in the momentum space and the projection onto the equatorial plane generates the Pauli matrices as soon as the conventional stereographic matrix is introduced. The associated eigenvalue problem results in the Dirac equation and the eigenvector (bispinor) has components that are related to geometric elements of the momentum space. The procedure has the advantage of revealing the correct form of the Dirac matrices without the mathematical effort that characterizes the presentation of the equation in traditional approaches. The other remarkable advantage is that, unlike the common reduction to the case of free space, the spatial inhomogeneity due to interaction potentials is included in the demonstration from the very beginning….

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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

A few years ago, one of the former Editors of this journal launched “a call to action” (E. F. Taylor, Am. J. Phys. 71, 423 (2003)) for a revision of teaching methods in physics in order to emphasize the importance of the principle of least action. In response, we suggest the use of Hamilton’s principle of stationary action to introduce the Schrödinger equation. When considering the geometric interpretation of Hamilton-Jacobi theory, the real part of the action 𝑆 defines the phase of the wave function 𝑒𝑥𝑝(𝑖𝑆/ℏ) and requiring the Hamilton-Jacobi wave function to obey wave-front propagation (i.e., Re(𝑆) is a constant of the motion) yields the Schrödinger equation…

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American Journal of Physics and may be found at  DOI: 10.1119/5.0083015

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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

Negative kinetic energies can either be nonsense or convey a profound truth about the physical reality (e.g., antimatter in relativistic QED). In quantum mechanics, they are associated with some relevant manifestations (weak measurements, quantum backflow, Glauber quantum oscillator, etc.) and have been recently discovered in optics (optical backflow). Here, building on the analogy between optical waves and wave propagation of Hamilton-Jacobi (HJ) wave fronts, new effects caused by the Hamilton’s principal function  … read more

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