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Nonlinearity

Mapping of the classical kinetic balance equations onto the Schrödinger equation

Adriana I. Pesci, Raymond E. Goldstein

Published Oct 9, 2004• 11 citations
Publisher

Abstract

Here we find a mapping onto the Sturm–Liouville operator of the first two balance equations derived from Boltzmann's equation. This mapping, which is irreversible and valid only for a subclass of solutions, is achieved by applying a Fourier transform to the momentum coordinate. In light of this irreversibility, it is necessary to develop a set of consistent prescriptions to find the probability of any physical quantity in the p -conjugate space such that it will coincide with the average over the momentum of the true probabilities obtained from the original Boltzmann equation. The one drawback of this prescription is that it is impossible to predict exactly the precise values of the position x and the momentum p at the same time. This uncertainty is limited by the relationship that all conjugate variables in a Fourier transform should obey, namely Δ x Δ p = η/2, where η is a free parameter of the theory. The prescriptions we have found appear to coincide with the postulates of quantum mechanics, when η is set equal to ℏ. This procedure seems to provide a statistical representation of quantum mechanics.

DOI
10.1088/0951-7715/18/1/011

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In previous work we have shown that the quantum potential can be derived from the classical kinetic equations both for particles with and without spin. Here, we extend these mappings to the relativistic case. The essence of the analysis consists of Fourier transforming the momentum coordinate of the distribution function. This procedure introduces a natural parameter η with units of angular momentum. In the non-relativistic case the ansatz of either separability, or separability and additivity, imposed on the probability distribution function produces mappings onto the Schrödinger equation and the Pauli equation, respectively. The former corresponds to an irrotational flow, the latter to a fluid with non-zero vorticity. In this work we show that the relativistic mappings lead to the Klein–Gordon equation in the irrotational case and to the second-order Dirac equation in the rotational case. These mappings are irreversible; an approximate inverse is the Wigner function. Taken together, these results provide a statistical interpretation of quantum mechanics.

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Here we find a mapping onto the Sturm–Liouville operator of the first two balance equations derived from Boltzmann's equation. This mapping, which is irreversible and valid only for a subclass of solutions, is achieved by applying a Fourier transform to the momentum coordinate. In light of this irreversibility, it is necessary to develop a set of consistent prescriptions to find the probability of any physical quantity in the p -conjugate space such that it will coincide with the average over the momentum of the true probabilities obtained from the original Boltzmann equation. The one drawback of this prescription is that it is impossible to predict exactly the precise values of the position x and the momentum p at the same time. This uncertainty is limited by the relationship that all conjugate variables in a Fourier transform should obey, namely Δ x Δ p = η/2, where η is a free parameter of the theory. The prescriptions we have found appear to coincide with the postulates of quantum mechanics, when η is set equal to ℏ. This procedure seems to provide a statistical representation of quantum mechanics.
In previous work we have shown that the quantum potential can be derived from the classical kinetic equations both for particles with and without spin. Here, we extend these mappings to the relativistic case. The essence of the analysis consists of Fourier transforming the momentum coordinate of the distribution function. This procedure introduces a natural parameter η with units of angular momentum. In the non-relativistic case the ansatz of either separability, or separability and additivity, imposed on the probability distribution function produces mappings onto the Schrödinger equation and the Pauli equation, respectively. The former corresponds to an irrotational flow, the latter to a fluid with non-zero vorticity. In this work we show that the relativistic mappings lead to the Klein–Gordon equation in the irrotational case and to the second-order Dirac equation in the rotational case. These mappings are irreversible; an approximate inverse is the Wigner function. Taken together, these results provide a statistical interpretation of quantum mechanics.
Here we present a study of the solutions and mathematical structure of the momentum-averaged Liouville (or Collisionless Boltzmann) equation in Fourier space. We show that the averaging procedure leads to a formalism identical to that of the density matrix of quantum mechanics. This mathematical mapping leaves the averages of all quantities unaltered and provides a unique way to construct the Hermitian version of a given operator. This seems to be the only method that resolves the ambiguity of Hermitization of operators that contain products of non-commuting variables. We also present a systematic perturbation scheme to evaluate correctly the classical solutions from the quantum ones and a formal proof of the approximate correspondence between the Poisson brackets and commutators.
We derive a classical Schrodinger type equation from the classical Liouville equation in phase space. The derivation is based on a Wigner type Fourier transform of the classical phase space probability distribution, which depends on an arbitrary constant ααα with dimension of action. In order to achieve this goal two requirements are necessary: 1) It is assumed that the classical probability amplitude Ψ(x,t)Ψ(x,t)Ψ(x,t) can be expanded in a complete set of functions Φn(x)Φ_n(x)Φn​(x) defined in the configuration space; 2) the classical phase space distribution W(x,p,t)W(x,p,t)W(x,p,t) obeys the Liouville equation and is a real function of the position, the momentum and the time. We show that the constant ααα appearing in the Fourier transform of the classical phase space distribution, and also in the classical Schrodinger type equation, has its origin in the spectral distribution of the vacuum zero-point radiation, and is identified with the Planck's constant ℏ\hbarℏ.
In non-relativistic quantum mechanics, the absolute square of Schrödinger's wave function for a particle in a potential determines the probability of finding it either at a position or momentum at a given time. In classical mechanics the corresponding problem is determined by the solution of Liouville's equation for the probability density of finding the joint position and momentum of the particle at a given time. Integrating this classical solution over either one of these two variables can then be compared with the probability in quantum mechanics. For the special case that the force is a constant, it is shown analytically that for an initial Gaussian probability distribution, the solution of Liouville's integrated over momentum is equal to Schrödinger's probability function in coordinate space, provided the coordinate and momentum initial widths of this classical solution satisfy the minimal Heisenberg uncertainty relation. Likewise, integrating Lioville's solution over position is equal to Schrödinger's probability function in momentum space. Quanta 2017; 6: 53–56.
The usual Heisenberg uncertainty relation, ΔX ΔP ≥ ℏ/2, may be replaced by an exact equality for suitably chosen measures of position and momentum uncertainty, which is valid for all wavefunctions. This exactuncertainty relation, δX ΔPnc ≡ ℏ/2, can be generalised to other pairs of conjugate observables such as photon number and phase, and is sufficiently strong to provide the basis for moving from classical mechanics to quantum mechanics. In particular, the assumption of a nonclassical momentum fluctuation, having a strength which scales inversely with uncertainty in position, leads from the classical equations of motion to the Schrödinger equation.
A Maxwell-Boltzmann gas consists of particles which undergo physical collisions which are associated with the idea of causality, but we argue that reaction balance: e1+e2=e3+e4 and n(e1)n(e2)= n(e3)n(e4) represent steady stream energy-probability conservation and so one need not actually think in terms of direct collisions and causality i.e. one may just as well write: n(e1)n(e2)=n(e1+e2). In the case of quantum mechanics, we argue, as we have done in previous notes, that the free particle form follows from the Lorentz invariant A= -Et+px. In a rest frame, there is energy, but no net momentum. We thus argue for a physical separation of the -Et and px terms. We further suggest that energy is associated with a resonance in time and p momentum with one in space i.e exp(-iEt) and exp(ipx). Given that A=-Et+px is also the relativistic/nonrelativistic classical action, we suggest that these represent probability resonances associated with constant p motion. As a result, given that p is associated with impulse, we suggest that an interaction means forming a new resonance i.e. p1->p2 implies exp(ip1 x)--> exp(ip2 x). As a result, interactions take place in length regions of hbar/p. If this value is very small, as it is in the classical world, then it appears that the resonances occur at points. In the quantum world, it is the opposite. If one wishes to treat quantum mechanics in a statistical manner, one needs to establish an energy-probability balance. Even in classical mechanics, impulses occur that change p, but now one has resonances which occur over a wavelength long piece of space. Furthermore a given p may be anywhere, so one thinks in terms of a(p)exp(ipx) probabilities. We try to show that a probability-energy balance is equivalent to the Schrodinger equation. Thus we argue that quantum mechanics uses a p-probability-energy balance for the time-independent Schrodinger equation and so does not necessarily address causality and physical dynamics.