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A quantum-like entertainement

Martin Devaud, Thierry Hocquet

Published Sep 13, 2018• 0 citations
OpenAlex

Abstract

We propose an intuitive introduction to some ideas of quantum mechanics, by means of a toy-model: a chain of coupled classical pendula. Quasi-particles are associated with the eigenmodes of the chain, ruled by a Schrödinger-like motion equation when using the classical Glauber variables. The set of the physical states of the system is shown to be a Hilbert C-vector space, in which observables can be defined. A continuous description of the chain leads to a Klein-Gordon motion equation. At last, a 3D extension is proposed, accompanied with a few speculative remarks, inter alia concerning the link with Special Relativity and the covariance of the Schrödinger-like equation.

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A connection between quantum Hilbert-space and classical phase-space operators

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We develop a general `classicalization' procedure that links Hilbert-space and phase-space operators, using Weyl's operator. Then we transform the time-dependent Schrödinger equation into a phase-space picture using free parameters. They include position Q and momentum P . We expand the phase-space Hamiltonian in an ℏ-Taylor series and fix parameters with the condition that coefficients of ℏ 0 , -iℏ 1 ∂/∂ Q and iℏ 1 ∂/∂ P vanish. This condition results in generalized Hamilton equations and a natural link between classical and quantum dynamics, while the quantum motion-equation remains exact . In this picture, the Schrödinger equation reduces in the classical limit to a generalized Liouville equation for the quantum-mechanical system state. We modify Glauber's coherent states with a suitable phase factor S ( Q , P , t ) and use them to obtain phase-space representations of quantum dynamics and quantum-mechanical quantities.

Journal of Physics A Mathematical and General•Aug 25, 2000•5 citations
We propose an intuitive introduction to some ideas of quantum mechanics, by means of a toy-model: a chain of coupled classical pendula. Quasi-particles are associated with the eigenmodes of the chain, ruled by a Schrödinger-like motion equation when using the classical Glauber variables. The set of the physical states of the system is shown to be a Hilbert C-vector space, in which observables can be defined. A continuous description of the chain leads to a Klein-Gordon motion equation. At last, a 3D extension is proposed, accompanied with a few speculative remarks, inter alia concerning the link with Special Relativity and the covariance of the Schrödinger-like equation.
The Schrodinger equation for a single spinless particle is formally obtained via a classical phonon model, namely the Frenkel-Kontorova model. Starting from a one-dimensional lattice of coupled harmonic oscillators, we show that the continuous limit of the corresponding Newtonian equation of motion yields the Klein-Gordon equation for a real-valued field. By introducing a complex-valued change of variables mixing the real-valued displacement and velocity fields, and by separating fast and slow time scales, the Klein-Gordon equation is written as the Schrodinger equation within the non-relativistic limit. This complex change of variable also allows to rewrite classical global observables of the phonon field, such as the total energy or momentum, as the corresponding quantum observables. Additionally, we show that when a friction force is incorporated into the classical model, the corresponding Klein-Gordon equation can be rewritten as a Schrodinger equation with a non-Hermitian Hamiltonian. While the global approach is limited here to the non-relativistic regime and does not address the measurement problem, quantization or relativistic effects, it nonetheless illustrates how quantum algebra and complex-valued wave functions can be exactly reproduced using classical dynamics. The relativistic regime for a spinless particle and the link between commutators and Poisson brackets is addressed in the second part of this series.
Classical mechanics is usually valid at the macro scale. The changes in the position and the momentum of the object over time are described by the Hamiltonian equation of motion. The wave function in quantum mechanics, if unobserved, evolves in a smooth and continuous way according to the Schrodinger equation, which is related to the Hamiltonian equation of motion. This equation describes a linear superposition of different states at time. A quantum mechanical description of a physical system is related to a probabilistic representation; it is described by a vector in Hilbert space. This description extends the two- or three-dimensional Euclidean space into spaces that have any finite or infinite number of dimensions. The state of a compound system is projected to the subspace that corresponds to the observed state and the vector representing the state is renormalized to the unit length. An observable describes a subspace for some dimensions with a special case of one dimension.
Stemming from the similar linearities of the Schrödinger equation in quantum mechanics on the one hand and of the harmonic oscillations in classical mechanics on the other hand, the idea that any N-degree-of-freedom harmonic oscillator (HON) is formally equivalent to a N-level quantum system is put forward. It is shown that the complex dynamic variables α introduced by R. J. Glauber can be regarded as the components of a state vector belonging to some N-dimension complex Hilbert space, and whose time-evolution is ruled by a Schrödinger-like equation. In case the classical HON is parametrically excited, the unitarity of the time-evolution of the associated quantum system is related to the Ehrenfest adiabaticity of the parametric excitation.
The formulation of quantum mechanics in terms of the picture of trajectory ensemble or of the hydrodynamical picture, developed in a previous article, is further supplemented for some points. Individual trajectories for some typical non-stationary state are explicitly obtained, illustrating that they generally show complicated fluctuations due to the action of quantum potential. Next, emphasis is placed on the fact that the Schrödinger field admits a mechanical model. Also the Klein-Gordon field is represented with a quasi-irrotational flow under some additional scalar field. This representation of Klein-Gordon field suggests its possible generalization to include the vortical motion. Such treatments of interacting Klein-Gordon and electromagnetic fields involve the formalism of Dirac's recent classical electrodynamics as a limiting case. Some remarks are given on this Dirac's theory. Finally, the picture of trajectory ensemble is generalized so as to represent wave equations with sources. In an appendix criticism is given on the attempt to interpret quantum mechanics in terms of Markoff process.
In the first part of this series we have shown how the Schrodinger equation for a single particle and the corresponding non relativistic quantum observables can be obtained from a purely classical phonon model through the Newtonian equations of motion. In this work we focus instead on how the classical Hamiltonian formalism applied to the same phonon system allows to recover the feature of relativistic quantum mechanics for a single spinless particle. Using the classical nature of the phonon model, we naturally define continuous Poisson brackets between classical observables, which allows to recover the dynamics of such observables, i.e. the Ehrenfest relations associated to real-valued Klein-Gordon fields. The Poisson brackets also permits to obtain the generic form of constants of motions, thus generalizing the concept of inner products and momentum on Klein-Gordon fields. We then connect the formalism of real-valued classical functionals with that of hermitian operators and complex-valued wave functions. This is done through the introduction of a non-local complex-valued change of variables which allows to rewrite the real-valued Klein-Gordon equation in a form akin to the Schrodinger equation, and the classical observables as quantum expectation values. Then, we show how this change of variables allows to rewrite the classical Poisson brackets as commutators of hermitian operators. This points out the strict equivalence between the Heisenberg formalism and the formalism of classical Poisson bracket. Eventually, we illustrate how the Poisson brackets allows to recover the transformations of Poincare group in 1+1 dimension together with its algebra. The latter makes the link between the Lorentz invariant inner product of Mostafazadeh and the Casimir invariant associated to the mass of particle.
We develop a general `classicalization' procedure that links Hilbert-space and phase-space operators, using Weyl's operator. Then we transform the time-dependent Schrödinger equation into a phase-space picture using free parameters. They include position Q and momentum P . We expand the phase-space Hamiltonian in an ℏ-Taylor series and fix parameters with the condition that coefficients of ℏ 0 , -iℏ 1 ∂/∂ Q and iℏ 1 ∂/∂ P vanish. This condition results in generalized Hamilton equations and a natural link between classical and quantum dynamics, while the quantum motion-equation remains exact . In this picture, the Schrödinger equation reduces in the classical limit to a generalized Liouville equation for the quantum-mechanical system state. We modify Glauber's coherent states with a suitable phase factor S ( Q , P , t ) and use them to obtain phase-space representations of quantum dynamics and quantum-mechanical quantities.