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The Origin of Randomness in Quantum Mechanics

Michail A. Zak

Published Jan 1, 2014• 4 citations
OpenAlex

Abstract

A mysterious problem of randomness in quantum mechanics is revisited. This problem being hidden in the Schrodinger equation becomes transparent in its Madelung version. It has been demonstrated that randomness in quantum mechanics has the same mathematical source as that in turbulence and chaos. Special attention is concentrated on equivalence between the Schrodinger and the Madelung equations in connection with the concept of stability in dynamics.

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A mysterious problem of randomness in quantum mechanics is revisited. This problem being hidden in the Schrodinger equation becomes transparent in its Madelung version. It has been demonstrated that randomness in quantum mechanics has the same mathematical source as that in turbulence and chaos. Special attention is concentrated on equivalence between the Schrodinger and the Madelung equations in connection with the concept of stability in dynamics.
Two unsolved problems in quantum mechanics are addressed: a source of randomness and an origin of entanglement. These problems being hidden in the Schrodinger equations became transparent in its Madelung version. Special attention is concentrated on equivalence between the Schrodinger and the Madelung equations. It has been demonstrated that randomness in quantum mechanics has the same mathematical source as that in turbulence and chaos, and the origin of entanglement is the global constraint imposed by the normalization constraint of the probability density that becomes an additional variable in the Madelung version of the Schrodinger equation.
There has been proven that mathematical origins of randomness in quantum and Newtonian physics are coming from the same source that is dynamical instability. However in Newtonian physics this instability is measured by positive finite Liapunov exponents averaged over infinite time period, while in quantum physics the instability is accompanied by a loss of the Lipchitz condition and represented by an infinite divergence of trajectories in a singular point. Although from a mathematical viewpoint such a difference is significant, from physical viewpoint it does not justify division of randomness into “deterministic “(chaos) and “true” (quantum physics). The common origin of randomness in Newtonian and quantum physics presents a support of the correspondence principle that is being searched by quantum chaos theory.
There has been proven that mathematical origins of randomness in quantum and Newtonian physics are coming from the same source that is dynamical instability. However in Newtonian physics this instability is measured by positive finite Liapunov exponents averaged over infinite time period, while in quantum physics the instability is accompanied by a loss of the Lipchitz condition and represented by an infinite divergence of trajectories in a singular point. Although from a mathematical viewpoint such a difference is significant, from physical viewpoint it does not justify division of randomness into deterministic (chaos) and true (quantum physics). The common origin of randomness in Newtonian and quantum physics presents a support of the correspondence principle that is being searched by quantum chaos theory.
The true dynamical randomness is obtained as a natural fundamental property of deterministic quantum systems. It provides quantum chaos passing to the classical dynamical chaos under the ordinary semiclassical transition, which extends the correspondence principle to chaotic systems. In return one should accept the modified form of quantum formalism (presented by the Schroedinger equation) which, however, does not contradict the ordinary form and the main postulates of quantum mechanics. It introduces the principle of the fundamental dynamic multivaluedness (redundance) extending the quantum paradigm to complex dynamical behaviour. Moreover, a causal solution to the well-known problems of the foundation of quantum mechanics, those of quantum indeterminacy and wave reduction, is also found using the same method. The concept of the fundamental dynamic uncertainty thus established is universal in character and provides a unified scheme of the complete description of arbitrary complex system of any origin (physics/9806002). This scheme incorporates, in particular, universal definitions of complexity, (non)integrability, and general solution, as well as the physically complete notion of probability. One obtains thus a self-consistent hierarchic picture of the world characterised by a (high) non-zero complexity and containing the intrinsic causal randomness, where the causally extended quantum mechanics can be consistently interpreted as several lowest levels of complexity (quant-ph/990215,16).
In light of a recently discovered general solution to the Schrödinger equation in an arbitrary integer-dimensional real domain with arbitrary initial conditions, boundary conditions, and quantum mechanical potential via a modified Cole-Hopf transform, a solution of the Madelung equations of equal generality is derived in this work. After a sufficient degree of mathematical context is established by a brief recapitulation of the solution of the Schrödinger equation, the derivation of the Madelung equations from first principles proves that a general Madelung flow field is implied by a general Schrödinger wave function through illustration of the mathematical essence of the quantities in the Madelung equations. The solution follows by a decomposition of the newly found general wave function into its polar form and its substitution into the definitions of the Madelung flow velocity and density, and a transfer from wave function initial and boundary conditions to density and velocity boundary conditions. The transfer between the complex scalar-valued wave function and the real vector-valued velocity field is achieved by a conditional inversion of the gradient operator. Plots of the velocity and vorticity take mere milliseconds to generate on consumer hardware via an enclosed GNU Octave script and show highly turbulent patterns consistent with the laws of quantum mechanics, showing that this analytical Madelung solution possesses unprecedented ability to deterministically replicate quantum turbulence without any numerical methods at all, effectively democratizing the study of quantum turbulence with its internal structure and mathematical origin. This Madelung solution paves the way to equally general solutions of unsolved problems in classical fluid mechanics including the Euler and Navier-Stokes equations, whose methods and analyses including existence and smoothness are expounded in works currently in production by the author.