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Journal of Optics B Quantum and Semiclassical Optics

Operator algebras for general one-dimensional quantum mechanical potentials with discrete spectrum

Alfred W nsche

Published Sep 20, 2002• 12 citations
Publisher

Abstract

We define general lowering and raising operators of the eigenstates for one-dimensional quantum mechanical potential problems leading to discrete energy spectra and investigate their associative algebra. The Hamilton operator is quadratic in these lowering and raising operators and corresponding representations of operators for action and angle are found. The normally ordered representation of general operators using combinatorial elements such as partitions is derived. The introduction of generalized coherent states is discussed. Linear laws for the spacing of the energy eigenvalues lead to the Heisenberg–Weyl group and general quadratic laws of level spacing to unitary irreducible representations of the Lie group SU (1, 1) that is considered in detail together with a limiting transition from this group to the Heisenberg–Weyl group. The relation of the approach to quantum deformations is discussed. In two appendices, the classical and quantum mechanical treatment of the squared tangent potential is presented as a special case of a system with quadratic level spacing.

DOI
10.1088/1464-4266/4/6/301

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We define general lowering and raising operators of the eigenstates for one-dimensional quantum mechanical potential problems leading to discrete energy spectra and investigate their associative algebra. The Hamilton operator is quadratic in these lowering and raising operators and corresponding representations of operators for action and angle are found. The normally ordered representation of general operators using combinatorial elements such as partitions is derived. The introduction of generalized coherent states is discussed. Linear laws for the spacing of the energy eigenvalues lead to the Heisenberg–Weyl group and general quadratic laws of level spacing to unitary irreducible representations of the Lie group SU (1, 1) that is considered in detail together with a limiting transition from this group to the Heisenberg–Weyl group. The relation of the approach to quantum deformations is discussed. In two appendices, the classical and quantum mechanical treatment of the squared tangent potential is presented as a special case of a system with quadratic level spacing.
The conventional quantization of the harmonic oscillator in terms of operators Q and P can be implemented with the help of irreducible unitary representations of the Heisenberg-Weyl group which acts transitively and effectively on the simply connected classical phase space S q,p ≊ 2 . In the description of the harmonic oscillator in terms of angle and action variables ϕ and I the associated phase space S ϕ,I corresponds to the multiply connected punctured plane 2 – {0}, on which the 3-dimensional symplectic group Sp(2, ) acts transitively, leaving the origin invariant. As this group contains the compact subgroup U (1) it has infinitely many covering groups. In the here relevant irreducible unitary representations (positive discrete series) the self-adjoint generator K o of U (l) represents the classical action variable I . It has the possible spectra n + k, n = 0,1,...; k > 0, where k depends on the covering group. This implies different possible spectra for the action variable Hamiltonian ℏωK 0 of the harmonic oscillator. On the other hand, expressing the operators Q and P (non-linearly) in terms of the three generators K 0 etc. of Sp (2, ) leads to the usual framework. Possible physical (experimental) implications and generalizations to higher dimensions are discussed briefly.
This chapter examines the linear harmonic oscillator in the energy (Fock) representation, emphasizing its algebraic structure and spectral properties. It is shown that the energy spectrum is discrete and bounded from below. The energy raising and lowering (ladder) operators are introduced, and the energy eigenvalues are derived. The eigenstates are constructed algebraically from the ground state by repeated application of the raising operator, forming the Fock basis. These energy eigenstates are then transformed into the coordinate representation, where the corresponding wavefunctions are expressed in terms of Hermite polynomials. This establishes a connection between the algebraic and coordinate formulations of the quantum harmonic oscillator. Metadata/Keywords: linear harmonic oscillator, Fock representation, ladder operators, energy spectrum, energy eigenstates, ground state, Hermite polynomials, raising and lowering operators
On a one-dimensional lattice without uniform interval, the Hermitian conjugation of a q-differential operator is discussed. Then a deformation of quantum mechanics in one dimension is presented. As an application, the harmonic oscillator is discussed. The energy spectrum and the eigenfunctions are shown to depend on an arbitrary deformation function. The deformed coherent states are also discussed. It is found that the completeness relation of coherent states holds for the case of q-coherent states, i.e. the deformation of the Heisenberg-Weyl algebra is a q-analogue Hopf algebra.
In this article a study was made of the conditions under which a Hamiltonian which is an element of the complex $ \left\{ h (1) \oplus h(1) \right\} \uplus u(2) $ Lie algebra admits ladder operators which are also elements of this algebra. The algebra eigenstates of the lowering operator constructed in this way are computed and from them both the energy spectrum and the energy eigenstates of this Hamiltonian are generated in the usual way with the help of the corresponding raising operator. Thus, several families of generalized Hamiltonian systems are found, which, under a suitable similarity transformation, reduce to a basic set of systems, among which we find the 1:1, 2:1, 1:2, su(2)su(2)su(2) and some other non-commensurate and commensurate anisotropic 2D quantum oscillator systems. Explicit expressions for the normalized eigenstates of the Hamiltonian and its associated lowering operator are given, which show the classical structure of two-mode separable and non-separable generalized coherent and squeezed states. Finally, based on all the above results, a proposal for new ladder operators for the p:qp:qp:q coprime commensurate anisotropic quantum oscillator is made, which leads us to a class of Chen SU(2)SU(2)SU(2) coherent states.
"The possibility of describing noncommuting operators in quantum mechanics by classical type functions, and the associated expression of operator multiplication, is of considerable interest. The well known Wigner-Weyl-Moyal theory is an important example. Another is the diagonal representation of operators using standard coherent states. We develop general necessary and sufficient conditions for the existence of the diagonal representation in the context of a family of generalised coherent states associated with any unitary irreducible Lie group representation. Several examples illustrating these conditions, and interesting results in the Heisenberg-Weyl case, are presented".
In quantum mechanics with minimal length uncertainty relations the Heisenberg-Weyl algebra of the one-dimensional harmonic oscillator is a deformed SU(1,1) algebra. The eigenvalues and eigenstates are constructed algebraically and they form the infinite-dimensional representation of the deformed SU(1,1) algebra. Our construction is independent of prior knowledge of the exact solution of the Schrödinger equation of the model. The approach can be generalized to the DDD-dimensional oscillator with non-commuting coordinates.