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Journal of Mathematical Physics

Geometry of the Schrödinger equation and stochastic mass transportation

Stephan Völlinger

Published Aug 1, 2005• 6 citations
Publisher

Abstract

The Schrödinger equation is deduced from a geometric principle. Lagrange, Hamilton-Jacobi, and Hamilton formalisms are defined almost analogously to the deterministic case, which can be identified as geometric optic. The form of these formalisms is identical to the deterministic formalisms. Furthermore, it will be shown how the deterministic case is “superficial” to the stochastic one if Planck’s constant is very small. An elementary proof of Heisenberg’s uncertainty relation finishes the paper.

DOI
10.1063/1.1998835

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The Schrödinger equation is deduced from a geometric principle. Lagrange, Hamilton-Jacobi, and Hamilton formalisms are defined almost analogously to the deterministic case, which can be identified as geometric optic. The form of these formalisms is identical to the deterministic formalisms. Furthermore, it will be shown how the deterministic case is “superficial” to the stochastic one if Planck’s constant is very small. An elementary proof of Heisenberg’s uncertainty relation finishes the paper.
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
The Hamilton–Jacobi equation in classical mechanics and the related eikonal equation in geometrical optics are often described as the “point of closest approach” between classical and quantum mechanics. Most textbook treatments of Hamilton–Jacobi theory are aimed at graduate students and derive the equation only after a long introduction to canonical transformations. Most treatments of the eikonal equation only emphasize its use in geometrical optics. We show that both the Hamilton–Jacobi equation and the eikonal equation can be derived by a common procedure using only elementary aspects of the Lagrangian and Hamiltonian formalisms introduced in undergraduate classical mechanics courses. Through this common approach, we hope to highlight to undergraduates the deep connections between classical mechanics, classical wave theory, and Schrödinger’s wave mechanics.
In (1), it is suggested that one may obtain the Schrodinger equation by using Hamilton’s equations as a starting point, i.e dp/dt = - dH/dq and dq/dt = dH/dp. (1) defines new variables phi(x,t) = ap+ibq and phi*(x,t) and writes: iC dphi/dt partial = {phi, H} and iC dphi*/dt partial ={ phi*, H}, where {} are the Poisson brackets. (1) then changes phi(x,t) into a stochastic variable which still follows these Hamilton-like relations iC dphi/dt = Integral dx H1(x1,x) phi(x) + b(x,t) where H= Integral dx1dx H1 phi*(x1)phi(x) and b(x,t is a random function. In other words, Hamilton’s theory with an added stochastic b(x,t) is applied to a stochastic variable in order to obtain the Schrodinger equation. The catch, however, is that (1) introduces a number of conditions and assumptions to simplify the problem and actually obtains iC dphi/dt = Integral dx H1(x1,x)phi(x) with b(x,t) gone. We suggest that this implies that one may apply the Hamilton-like form directly to a stochastic variable without b(x,t) present. To strengthen this argument, one may consider a bound state in which b(x,t) may be said to average to 0. Basically, the idea of (1) seems to be that quantum mechanics follows from introducing stochastic physics linked to usual Newtonian trajectory physics, in this case Hamilton’s theory. We argue that this is very similar to the classical action approach we used in (2) to obtain exp(-iEt+ipx) from the classical free particle action applied to stochastic motion and so we compare the two approaches.