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BENTHAM SCIENCE PUBLISHERS eBooks

Equation of Motion

P. Enders

Published Feb 12, 2019• 0 citations
Publisher

Abstract

Having got the quantum-mechanical equation of change of stationary states, it is rather straightforward to derive the quantum-mechanical equation of motion. As anticipated in the foregoing chapter, this is the time-dependent Schrödinger equation. Moreover, Huygens’ principle is shown to apply to QM, as obtained by Feynman in another way.

DOI
10.2174/9781681084497119010014

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Having got the quantum-mechanical equation of change of stationary states, it is rather straightforward to derive the quantum-mechanical equation of motion. As anticipated in the foregoing chapter, this is the time-dependent Schrödinger equation. Moreover, Huygens’ principle is shown to apply to QM, as obtained by Feynman in another way.
The time-independent Schrödinger equation represents a stationary-state equation. The stationary wave functions obtained so far are time independent. Their time-dependence is obtained by means of rather general arguments. Then, stationarystate functions are found. The next step in Newton’s and Euler’s representations of classical mechanics is (to derive) the equation of change of stationary-states. Here, Euler’s principles of stationary-state change are generalized to quantum-mechanical systems. This enables us to derive the quantum-mechanical equation of change of stationary-states. The time-independent Schrödinger equation, i.e, the equation of motion will follow in the next chapter.
Abstract The equation of motion in the standard formulation of non-relativistic quantum mechanics, the Schrödinger equation, is based on the Hamiltonian. In contrast, in Feynman’s path-integral formulation of quantum mechanics, the equation of motion is the propagation equation, which is based on the Lagrangian. That these two different equations of motion are equivalent was shown by Feynman, who provided a derivation of the Schrödinger equation from the propagation equation. Surprisingly, however, while in classical mechanics there exists a simple relationship between the Hamiltonian and the Lagrangian, there is nothing in Feynman’s derivation that gives any indication of this relationship. Here I show that the equations of motion in the Hamiltonian and Lagrangian formulations of quantum mechanics are, in fact, simply related to each other.
It is shown how the time-dependent Schrödinger equation may be simply derived from the dynamical postulate of Feynman's path integral formulation of quantum mechanics and the Hamilton–Jacobi equation of classical mechanics. Schrödinger's own published derivations of quantum wave equations, the first of which was also based on the Hamilton–Jacobi equation, are also reviewed. The derivation of the time-dependent equation is based on an a priori assumption equivalent to Feynman's dynamical postulate. de Broglie's concepts of ‘matter waves’ and their phase and group velocities are also critically discussed.
Chapter 7 is devoted to the free particle and the derivation of the time-dependent Schrödinger equation. The free particle might be thought to be the simplest quantum problem of all, but a detailed treatment of it, including both Born and Dirac normalization, serves to highlight some surprising features. Powerful tools are introduced that are of wider application, notably the Green function method and the Feynman propagator. This material paves the way for the study of the time-dependent Schrödinger equation, the introduction of the current density of particles and the distinction between pure states and mixtures; the equation is applied specifically to the study of coherent states. The chapter ends with an introduction to the propagator in the general case.
A derivation of the time-dependent Schrödinger equation from the time-independent one is considered. Instead of time, the coordinate of an additional degree of freedom, the clock, is introduced into the original time-independent Schrödinger equation. It is shown that the standard time-dependent Schrödinger equation can be obtained for the semiclassical clock only. For elucidation of the physical meaning of the equation obtained in this way, various types of clocks are discussed. In addition, the corresponding equation for the density matrix and formulas for the mean values of operators are derived.
Non‐relativistic quantum systems are analyzed theoretically or by numerical approaches using the Schrödinger equation. Compared to the options available to treat classical mechanical systems this is limited, both in methods and in scope. However, based on Nelson's stochastic mechanics, the mathematical structure of quantum mechanics has in some aspects been developed into a form analogous to classical analytical mechanics. We show here that finding the Nash equilibrium for a stochastic optimal control problem, which is the quantum equivalent to Hamilton's principle of least action, allows to derive two things: i) the Schrödinger equation as the Hamilton‐Jacobi‐Bellman equation of this optimal control problem and ii) a set of quantum dynamical equations which are the generalization of Hamilton's equations of motion to the quantum world. We derive their general form for the non‐stationary and the stationary case. For the harmonic oscillator, the stationary equations lead to the coherent states, and we establish a numerical procedure to solve for the ground state properties without using the Schrödinger equation. image