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arXiv:quant-ph/0210193 · International Journal of Modern Physics A

From a Mechanical Lagrangian to the Schrödinger Equation: A Modified Version of the Quantum Newton Law

A. Bouda

Published Jul 30, 2003• 20 citations

Abstract

In the one-dimensional stationary case, we construct a mechanical Lagrangian describing the quantum motion of a nonrelativistic spinless system. This Lagrangian is written as a difference between a function T, which represents the quantum generalization of the kinetic energy and which depends on the coordinate x and the temporal derivatives of x up the third order, and the classical potential V(x). The Hamiltonian is then constructed and the corresponding canonical equations are deduced. The function T is first assumed to be arbitrary. The development of T in a power series together with the dimensional analysis allow us to fix univocally the series coefficients by requiring that the well-known quantum stationary Hamilton–Jacobi equation be reproduced. As a consequence of this approach, we formulate the law of the quantum motion representing a new version of the quantum Newton law. We also analytically establish the famous Bohm relation [Formula: see text] outside the framework of the hydrodynamical approach and show that the well-known quantum potential, although it is a part of the kinetic term, plays really the role of an additional potential as assumed by Bohm.

DOI: 10.1142/s0217751x03015076

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Using the relativistic quantum stationary Hamilton-Jacobi equation within the framework of the equivalence postulate, and grounding oneself on both relativistic and quantum Lagrangians, we construct a Lagrangian of a relativistic quantum system in one dimension and derive a third order equation of motion representing a first integral of the relativistic quantum Newton’s law. Then, we plot the relativistic quantum trajectories of a particle moving under the constant and the linear potentials. We establish the existence of nodes and link them to the de Broglie’s wavelength. PACS: 03.65.Bz; 03.65.Ca; Key words: relativistic quantum law of motion, Lagrangian, relativistic quantum stationary Hamilton-Jacobi equation, relativistic velocities.

Despite the fact that it has been known since the time of Heisenberg that quantum operators obey a quantum version of Newton's laws, students are often told that derivations of quantum mechanics must necessarily follow from the Hamiltonian or Lagrangian formulations of mechanics. Here, we first derive the existing Heisenberg equations of motion from Newton's laws and the uncertainty principle using only the equations

F=\frac{dP}{dt}

,

P=m\frac{dV}{dt}

, and

\left[X,P\right]=i

. Then, a new expression for the propagator is derived that makes a connection between time evolution in quantum mechanics and the motion of a classical particle under Newton's laws. The propagator is solved for three cases where an exact solution is possible 1) the free particle 2) the harmonic oscillator 3) a constant force, or linear potential in the standard interpretation. Such a picture may be useful for students as they make the transition from classical to quantum mechanics and help solidify the equivalence of the Hamiltonian, Lagrangian, and Newtonian formulations of physics in their minds.

Using the relativistic quantum stationary Hamilton-Jacobi equation within the framework of the equivalence postulate, and grounding oneself on both relativistic and quantum Lagrangians, we construct a Lagrangian of a relativistic quantum system in one dimension and derive a third order equation of motion representing a first integral of the relativistic quantum Newton's law. Then, we plot the relativistic quantum trajectories of a particle moving under the constant and the linear potentials. We establish the existence of nodes and link them to the de Broglie's wavelength.

The Classical Newton-Lagrange equations of motion represent the fundamental physical law of mechanics. Their traditional Lagrangian and/or Hamiltonian precursors when available are essential in the context of quantization. However, there are situations that lack Lagrangian and/or Hamiltonian settings. This paper discusses a description of classical dynamics and presents some irresponsible speculations about its quantization by introducing a certain canonical two-form ?. By its construction ? embodies kinetic energy and forces acting within the system (not their potential). A new type of variational principle employing differential two-form ? is introduced. Variation is performed over “umbilical surfaces“ instead of system histories. It provides correct Newton-Lagrange equations of motion. The quantization is inspired by the Feynman path integral approach. The quintessence is to rearrange it into an “umbilical world-sheet“ functional integral in accordance with the proposed variational principle. In the case of potential-generated forces, the new approach reduces to the standard quantum mechanics. As an example, Quantum Mechanics with friction is analyzed in detail.