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The Wigner-Heisenberg Algebra in Quantum Mechanics

R. de Lima Rodrigues

Published Apr 3, 2013• 1 citation
Publisher

Abstract

ly R is the Klein operator ±exp[iπ(Ĥ − E0)] while in Schrodinger coordinate representation, first investigated by Yang, R is realised by ±P where P is the parity operator: P|x >= ±|x >, P−1 = P, P = 1, PxP−1 = −x. (7) The basic (anti-)commutation relation (1) and (3) together with their derived relation (4) will be referred to here as constituting the WH algebra which is in fact a parabose algebra for one degree of freedom. We shall assume here in after, without loss of generality, that c is positive, i.e. c =| c |> 0. Thus, in coordinate representation the generalized quantization a la Wigner requires that x = x, ∞− < x < ∞, p = −i d dx + ic 2x P, R = P. (8) Indeed, following Yang representation [6] we obtain the coordinate representation for the ladder operators as given by â± −→ a±c 2 = 1 √ 2 ( ± d dx ∓ c 2x P − x ) . (9) Yang’s wave mechanical description was further investigated in [7, 8]. The Wigner-Heisenberg Algebra in Quantum Mechanics http://dx.doi.org/10.5772/55994 479 4 Advances in Quantum Mechanics The present author have applied a super-realization so that R = Σ3 to illustrate the first application of our operator method to the cases of the Hamiltonian of an isotonic oscillator (harmonic plus a centripetal barrier) system. To obtain a super-realisation of the WH algebra, we introduce, in addition to the usual bosonic coordinates (x,−i d dx ), the fermionic ones b∓(= (b±)†) that commute with the bosonic set and are represented in terms of the usual Pauli matrices Σi, (i = 1, 2, 3). Indeed, expressing a ±( c 2 ) in the following respective factorised forms: a( c 2 ) = 1 √ 2 Σ1x (1/2)cΣ3 exp( 1 2 x)( d dx )exp( −1 2 x2)x−(1/2)cΣ3 (10) a−( c 2 ) = 1 √ 2 Σ1x (1/2)cΣ3 exp( −1 2 x)( −d dx )exp( 1 2 x2)x−(1/2)cΣ3 , (11) where these lader operators satisfy the algebra of Wigner-Heisenberg. From (1), (11) and (10) the Wigner Hamiltonian becomes H( c 2 ) = 1 2 [ a( c 2 ), a−( c 2 ) ] + = ( H−( c 2 − 1) 0 0 H+( c 2 − 1) = H−( c 2 ) ) , (12) where the even and odd sector Hamiltonians are respectively given by H−( c 2 − 1) = 1 2 { − d 2 dx2 + x + 1 x2 ( c 2 )( c 2 − 1) }

DOI
10.5772/55994

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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.

•Jan 1, 2004•0 citations
ly R is the Klein operator ±exp[iπ(Ĥ − E0)] while in Schrodinger coordinate representation, first investigated by Yang, R is realised by ±P where P is the parity operator: P|x >= ±|x >, P−1 = P, P = 1, PxP−1 = −x. (7) The basic (anti-)commutation relation (1) and (3) together with their derived relation (4) will be referred to here as constituting the WH algebra which is in fact a parabose algebra for one degree of freedom. We shall assume here in after, without loss of generality, that c is positive, i.e. c =| c |> 0. Thus, in coordinate representation the generalized quantization a la Wigner requires that x = x, ∞− < x < ∞, p = −i d dx + ic 2x P, R = P. (8) Indeed, following Yang representation [6] we obtain the coordinate representation for the ladder operators as given by â± −→ a±c 2 = 1 √ 2 ( ± d dx ∓ c 2x P − x ) . (9) Yang’s wave mechanical description was further investigated in [7, 8]. The Wigner-Heisenberg Algebra in Quantum Mechanics http://dx.doi.org/10.5772/55994 479 4 Advances in Quantum Mechanics The present author have applied a super-realization so that R = Σ3 to illustrate the first application of our operator method to the cases of the Hamiltonian of an isotonic oscillator (harmonic plus a centripetal barrier) system. To obtain a super-realisation of the WH algebra, we introduce, in addition to the usual bosonic coordinates (x,−i d dx ), the fermionic ones b∓(= (b±)†) that commute with the bosonic set and are represented in terms of the usual Pauli matrices Σi, (i = 1, 2, 3). Indeed, expressing a ±( c 2 ) in the following respective factorised forms: a( c 2 ) = 1 √ 2 Σ1x (1/2)cΣ3 exp( 1 2 x)( d dx )exp( −1 2 x2)x−(1/2)cΣ3 (10) a−( c 2 ) = 1 √ 2 Σ1x (1/2)cΣ3 exp( −1 2 x)( −d dx )exp( 1 2 x2)x−(1/2)cΣ3 , (11) where these lader operators satisfy the algebra of Wigner-Heisenberg. From (1), (11) and (10) the Wigner Hamiltonian becomes H( c 2 ) = 1 2 [ a( c 2 ), a−( c 2 ) ] + = ( H−( c 2 − 1) 0 0 H+( c 2 − 1) = H−( c 2 ) ) , (12) where the even and odd sector Hamiltonians are respectively given by H−( c 2 − 1) = 1 2 { − d 2 dx2 + x + 1 x2 ( c 2 )( c 2 − 1) }
In quantum mechanics, physical observables are represented by operators on a certain Hilbert space. The question of how such operators commute, has been a matter of discussion. In the standard perspective, the operators corresponding to the position and momentum of a system are assumed to satisfy the canonical commutation relations. It is known that these relations imply that the Hamilton and Heisenberg equations of motion are compatible as operator equations. However, Wigner showed that the inverse statement is not true. Therefore, it is a much weaker constraint to impose the compatibility of the equations of motion. For any physical system, this results in a set of compatibility conditions, which form the core of Wigner quantization. The key to finding operators that are subject to the compatibility conditions is provided by Lie superalgebras and their representations. Lie superalgebras can be defined as algebras generated by odd elements satisfying particular superbracket relations. Using these defining relations, Lie superalgebra generators can be found that obey the compatibility conditions. The Lie superalgebra elements act as operators on a vector space if we consider Lie superalgebra representations. Various physical systems are investigated in this thesis in the context of Wigner quantization. For each of these systems solutions are found in terms of Lie superalgebra generators, after which specific representations are examined. In such representations, the focus lies on determining particular physical properties of the system. Most of the studied systems are harmonic oscillator models. First, we examine a set-up of coupled harmonic oscillators, for which the interaction is represented by an interaction matrix. Then we focus on the angular momentum content of a 3N3N3N-dimensional harmonic oscillator. Finally, our attention goes to two one-dimensional systems, namely the free particle and the Berry-Keating-Connes Hamiltonian. The latter of these Hamiltonians is notorious for its possible connection with the Riemann hypothesis. All of the aforementioned Hamiltonians have been extensively investigated in the context of canonical quantization, so that our results can be compared to the well-known canonical case.
The authors elucidate the utility of a super-realised Wigner-Heisenberg oscillator algebra with its inherent built-in direct generalisation of the usual oscillator ladder operators, but satisfying a generalised quantum commutation rule, as an effective algebraic tool for the easier spectral resolution of general oscillator-related potentials. To illustrate the procedure they consider here the full 3D isotropic harmonic oscillator problem and also the problems of the 1D-isotonic and d-dimensional radial oscillator systems. They also point out the intimate connection between the Wigner-Heisenberg algebra and the quantum mechanical SUSY algebra associated with these systems.
The commutator of position and momentum operators is first considered in the coordinate and momentum representations, and then in an abstract Hilbert space. The reader is then introduced to canonical operators with the associated canonical quantization of commutation rules, which consists of maps defined on a symplectic space and taking values in the set of unitary operators on a Hilbert space. If the symplectic space is finite-dimensional, all of its irreducible representations are unitarily equivalent. The Weyl exponentiated form of the commutation relations is described and investigated. One then arrives at the definition of Weyl systems, and it is shown how to recover from them the formulation in terms of self-adjoint operators which satisfy a generalized form of commutation relations. Further topics are the Heisenberg representation for temporal evolution and the generalized uncertainty relations. Unitary operators associated with symplectic linear transformations are then considered, and, within this framework, translations, rotations and the harmonic oscillator are considered. The Weyl programme is then reassessed. Lastly, the basic postulates of modern quantum theory are eventually studied: the probabilistic nature of the predictions of results of measurements; the correspondence between observables and self-adjoint operators on Hilbert spaces; the expected result of measuring an observable quantity; and the evolution of the state vector in systems which are not affected by any external influence. The chapter ends with an outline of rigged Hilbert space formalism and its applications. The commutator in wave mechanics In the first part of this book we studied wave mechanics within the framework of the Schrödinger equation.
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.
A derivation of the coordinate representation of the q-deformed oscillator is given. The Hilbert space associated with it is defined on a finite coordinate space y: (-\ensuremath{\pi}/2s,\ensuremath{\pi}/2s), where s is related to the deformation q via q=exp(-s2{\mathit{s}}^{2}s2), with the respective ``coordinate'' and ``momentum'' operators given by X${\mathrm{^}}_{\mathit{s}}$=y-As{\mathit{A}}_{\mathit{s}}As​(y), P${\mathrm{^}}_{\mathit{s}}$=-id/dy-iAs{\mathit{iA}}_{\mathit{s}}iAs​(y), where As{\mathit{A}}_{\mathit{s}}As​(y)=1/2d ln${\mathrm{\ensuremath{\sigma}}}_{\mathit{s}}$(y)/dy, ${\mathrm{\ensuremath{\sigma}}}_{\mathit{s}}$ being a measure in this space. The commutation rule obeyed by them is X${\mathrm{^}}_{\mathit{s}}$P${\mathrm{^}}_{\mathit{s}}$-P${\mathrm{^}}_{\mathit{s}}$X${\mathrm{^}}_{\mathit{s}}$=i${\mathrm{\ensuremath{\Elzxh}}}_{\mathit{s}}$(y)=i{1-dAs{\mathit{dA}}_{\mathit{s}}dAs​ /dy}. After constructing a suitable representation in the momentum space, we also set up the Wigner distribution function in this phase space. A brief discussion of these results and their interpretation are given.
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.