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CRC Press eBooks

Alternative Hamiltonians and Wigner Quantization

A. Horzela

Published Nov 30, 2003• 0 citations
Publisher

Abstract

Alternative Hamiltonians of classical mechanics, i.e. , canonically inequivalent Hamiltonians leading to the same Newton equations of motion, generate different phase space descriptions of the same mechanical systems. Relation of such descriptions to different algebraic descriptions of quantum systems consistent with the same time evolution - so called Wigner quantization - is presented and illustrated on example of two particles system moving on a line.

DOI
10.1201/9781482269079-105

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Alternative Hamiltonians of classical mechanics, i.e. , canonically inequivalent Hamiltonians leading to the same Newton equations of motion, generate different phase space descriptions of the same mechanical systems. Relation of such descriptions to different algebraic descriptions of quantum systems consistent with the same time evolution - so called Wigner quantization - is presented and illustrated on example of two particles system moving on a line.
The Wigner problem, i.e. the investigation of general quantum mechanical commutation relations consistent with the Heisenberg evolution equations of a given shape, is studied. We follow a recently proposed generalization of this idea within which the classical analogy is postulated only for the shape of the time evolution equations and not for a Hamiltonian itself. This links our investigation to the problem of alternative Hamiltonians of classical mechanics and to canonically inequivalent phase-space descriptions of physical systems governed by the same Newton equations of motion. Warned that the time evolution and the other symmetry generators may be given ambiguously even in the formalism of classical mechanics, we do not a priori assume the shape of their quantum analogues. Instead we only require that the set of basic algebraic relations, which quantum mechanical observables are to obey, has a Lie algebra structure. Such a requirement appears to be sufficient to find solutions for simple oscillator-like dynamics. New algebras of quantum mechanical observables are not constructed as a linear envelope of the Heisenberg algebra, and their representations reflect physical results unexpected in the framework of the canonical approach. We illustrate our approach in detail for the example of the one-dimensional harmonic oscillator using the representation of the generalized coherent states.
Noncanonical quantization schemes have been proposed in quantum physics long time ago. Recalling some of them I associate these investigations to the quantum mechanical im-plications of the canonically inequivalent Hamilton formula-tions of classical Newtonian dynamics. As the simplest illus-tration of such alternative Hamiltonian descriptions I consider one dimensional system of two particles interacting harmoni-cally. Description of the motion of their center of mass is stan-dard and guarantees the Galilean covariance but the internal motion is described as a noncanonical oscillator for which the Hamiltonian and the total energy are not related by a simple proportionality. Alternative Hamiltonians lead to alternative Poisson structures which may be considered as classical ana-logues of the noncanonical quantum commutation rules. Gen-Concepts of Physics, Vol. I (2004) 11 erated alternative quantizations are consistent with the same Heisenberg equations of motion, i.e., they satisfy the Wigner principle of quantization. For the one dimensional harmonic oscillator the simplest noncanonical algebraic description is given in terms of the Lie algebra so(p, q), p+ q = 3. In such a case the Hamiltonian is still a linear function while the energy is a quadratic polynomial of the main quantum number. This leads to nonstandard termodynamical properties obeyed by such a noncanonical oscillator. The 3-dimensional harmonic oscillator also has a noncanonical description given in terms of the simple Lie algebra, so(p, q), p+ q = 5 in the case under consideration. The most important physical consequence of such a description is the noncommutativity of the coordinate operators, interpreted to be an implication of the noncommu-tative geometry of the underlying spacetime.
One classical theory, as determined by an equation of motion or set of classical trajectories, can correspond to many unitarily {\em in}equivalent quantum theories upon canonical quantization. This arises from a remarkable ambiguity, not previously investigated, in the construction of the classical (and hence the quantized) Hamiltonian or Lagrangian. This ambiguity is illustrated for systems with one degree of freedom: An arbitrary function of the constants of motion can be introduced into this construction. For example, the nonrelativistic and relativistic free particles follow identical classical trajectories, but the Hamiltonians or Lagrangians, and the canonically quantized versions of these descriptions, are inequivalent. Inequivalent descriptions of other systems, such as the harmonic oscillator, are also readily obtained.
It is well known that to quantize any dynamical system it is necessary for a generator of the classical motion to exist in the form of a Hamiltonian function. It is shown, by using the example of a damped harmonic oscillator, that a particular class of inequivalent classical Hamiltonians exist which make quantization of the system ambiguous. Hence it is conclued that it is not sufficient for the Hamiltonian to merely generate the motion, but it must also be necessarily related via a canonical transformation to the total energy of the system.
Following the ideas of Wigner, we quantize noncanonically a system of two nonrelativistic point particles, interacting via a harmonic potential. The center of mass phase-space variables are quantized in a canonical way, whereas the internal momentum and coordinates are assumed to satisfy relations, which are essentially different from the canonical commutation relations. As a result, the operators of the internal Hamiltonian, the relative distance, the internal momentum, and the orbital momentum commute with each other. The spectrum of these operators is finite. In particular, the distance between the constituents is preserved in time and can take at most four different values. The orbital momentum is either zero or one (in units ℏ/2). The operators of the coordinates do not commute with each other and, therefore, the position of any one of the constituents cannot be localized; the particles are smeared with a certain probability in a finite space volume, which moves together with the center of mass. In the limit ℏ→0 the constituents ‘‘fall’’ into their center of mass and the composite system behaves as a free point particle.