Abstract We derive the equations of quantum mechanics and quantum thermodynamics from the assumption that a quantum system can be described by an underlying classical system of particles. Each component φ j of the wave vector is understood as a stochastic complex variable whose real and imaginary parts are proportional to the coordinate and momentum associated with a degree of freedom of the underlying classical system. From the classical stochastic equations of motion, we derive a general equation for the covariance matrix of the wave vector, which turns out to be of the Lindblad type. When the noise changes only the phase of φ j , the Schrödinger and the quantum Liouville equations are obtained. The component ψ j of the wave vector obeying the Schrödinger equation is related to the stochastic wave vector by | ψ j | 2 = ⟨ | ϕ j | 2 ⟩ .
Abstract We derive the equations of quantum mechanics and quantum thermodynamics from the assumption that a quantum system can be described by an underlying classical system of particles. Each component φ j of the wave vector is understood as a stochastic complex variable whose real and imaginary parts are proportional to the coordinate and momentum associated with a degree of freedom of the underlying classical system. From the classical stochastic equations of motion, we derive a general equation for the covariance matrix of the wave vector, which turns out to be of the Lindblad type. When the noise changes only the phase of φ j , the Schrödinger and the quantum Liouville equations are obtained. The component ψ j of the wave vector obeying the Schrödinger equation is related to the stochastic wave vector by | ψ j | 2 = ⟨ | ϕ j | 2 ⟩ .
We derive the equations of quantum mechanics and quantum thermodynamics from the assumption that a quantum system can be described by an underlying classical system of particles. Each component ϕj of the wave vector is understood as a stochastic complex variable whose real and imaginary parts are proportional to the coordinate and momentum associated to a degree of freedom of the underlying classical system. From the classical stochastic equations of motion, we derive a general equation for the covariance matrix of the wave vector which turns out to be of the Lindblad type. When the noise changes only the phase of ϕj, the Schrödinger and the quantum Liouville equation are obtained. The component ψj of the wave vector obeying the Schrödinger equation is related to stochastic wave vector by ∣ψj∣2=⟨∣ϕj∣2⟩.
A wave dynamics of fields φ(p, q; t) ∈ L2(Γ) over the phase space Γ(p, q) of a classical system 𝒮 is derived from the Liouville theorem. We define the energy contained in a given field φ(p, q; t). We show that for a special class of fields, selected on physical grounds, the energy spectrum is given by a time-independent Schrödinger equation. This allows us to associate with 𝒮 an ordinary quantum system Q such that the values of the quantized energy coincide for the fields in the phase space of 𝒮 and for Q. Then we make use of Wiener's stochastic integral based on the theory of Brownian motion to derive probabilities which are the same as those one would obtain through Born's statistical postulate of quantum theory. From this it follows that we can regard normalized fields φ(p, q; t) as ``probability amplitudes'' leading to a probability density function ρ(p, q; t) = φφ* in the sense of Gibbs' statistical mechanics. Our work therefore appears as a bridge between a statistical theory (in the sense of Gibbs) of a mechanical system 𝒮 and the usual quantum theory of the related quantum system Q.
It is shown that the Bohm equations for the phase S and squared modulus ρ of the quantum mechanical wave function can be derived from the classical ensemble equations admiting an aditional momentum ps of the form proportional to the osmotic velocity in the Nelson stochastic mechanics and using the variational principle with appropriate change of variables. The possibility to treat gradS and ps as two parts of the momentum of quantum ensemble particles is considered from the view point of uncertainty relations of Robertson - Schroedinger type on the examples of the stochastic image of quantum mechanical canonical coherent and squeezed states.
We first point out that nonrelativistic quantum mechanics formulated at imaginary time s = it can formally be recognized as the Fokker-Planck description of a classical Brownian motion, which occurs (in general) in a medium where particles are absorbed and emitted. We next offer a new formulation of quantum dynamics, which is basically the Langevin treatment of that Brownian motion. More explicitly, we derive a noise average representation of the transition probability W ( X", s"|X', s' ) in terms of the solutions to a Langevin equation with a Gaussian white noise. Upon analytic continuation back to real time W ( X", s"|X', s' ) becomes the propagator of the original Schrödinger equation, namely we have ⟨ X", t"|X', t' ⟩ = W ( X", it"|X', it' ). The main advantage of this approach is to permit a systematic application to quantum-dynamical problems of the mathematical techniques developed in the theory of classical stochastic processes, even when vector potentials and/or time-dependent potentials are present.
Introduction. There exists a very comprehensive theory concerning the time development of probabilistic systems. This is the Theory of Stochastic processes (1). Quantum mechanics as described by the Schrödinger equation rests very heavily on probabilistic interpretation, and yet, after forty years of remarkably rapid advances and successes, it is not clear how it is related to stochastic theory. The formal similarity of certain aspects of, and equations used in these two theories are well known. The diffusion equation corresponds to the Schrödinger equation, a correspondence only marred by a square root of minus unity appearing with the time derivative in the latter equation. The Kolmogoroff–Chapman relation for probabilities in Brownian motion has an analogue (2) in quantum mechanics, not for probabilities though, but for the complex probability amplitudes. The uncertainty relation also has a stochastic analogue (3). Some of these analogous characteristics are actually symptomatic of the gulf which separates these two theories. There are solutions to the diffusion equation which have a very clear physical probabilistic meaning. This is not the case with the corresponding and formally similar solutions to the Schrödinger equation and is partly due to the √ − 1 mentioned earlier. The probabilistic interpretation of the solutions to the diffusion equation is direct and immediate. On the other hand, the probabilistic interpretation of the solutions, ψ of the Schrödinger equation involve the introduction of the probability density, p = ψ*ψ. One great difference between these two theories lies in their each having distinct principles of linear superposition. In Stochastic Theory probabilities are superposed and in quantum mechanics amplitudes or wave functions are superposed.
We derive the classical Hamilton-Jacobi equation from first principles as the natural description for smooth stochastic processes when one neglects stochastic velocity fluctuations. The Schrödinger equation is shown to be the natural exact equation for describing smooth stochastic processes. In particular, processes with up to quadratic stochastic fluctuations are electromagnetically coupled quantum point particles. The stochastic derivation offers a clear geometric picture for Quantum Mechanics as a locally realistic hidden variable theory. While that sounds paradoxical, we show that Bell's formula for local realism is incomplete. If one includes smooth stochastic fluctuations for the hidden variables, local realism is preserved and quantum mechanics is obtained. Quantum mechanics should therefore be viewed as a "nondeterministic, non-Bell locally realistic hidden variable theory". Since the description is simply a stochastic process, it should be relatively straightforward to create mesoscopic analogue systems that show all the hallmarks of Quantum Mechanics, including super-Bell correlations. In fact, any system that can be described by the linear time evolution of a density matrix is both a stochastic and a Quantum system from our point of view, since we show that the existence of a transition probability density directly implies the existence of a density matrix. Systems for which the stochastic degrees of freedom vary smoothly over time have quantum Hamiltonians with standard kinetic terms.