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Journal of Mathematical Physics

Invariants and Lie algebraic solutions of Schrödinger equation

Franco Salmistraro, Riccardo Rosso

Published Sep 1, 1993• 8 citations
Publisher

Abstract

A representation of invariants in quantum mechanics for a general time-dependent Hamiltonian is provided. The main properties of these operators and their relevance for solving the Schrödinger equation are discussed in light of this representation. In particular, the well-known Lewis and Riesenfeld’s method is shown to be recovered quite directly by the present approach. A detailed analysis is concerned with Hamiltonians which at each instant are elements of a finite dimensional Lie algebra. Exploiting the Wei and Norman’s theory for such Hamiltonians, it is found that the evolution of a quantum system is always determined, in part or fully, by an invariant which belongs to the same algebra.

DOI
10.1063/1.530017

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International Journal of Quantum Chemistry•Jul 8, 2010•7 citations
A representation of invariants in quantum mechanics for a general time-dependent Hamiltonian is provided. The main properties of these operators and their relevance for solving the Schrödinger equation are discussed in light of this representation. In particular, the well-known Lewis and Riesenfeld’s method is shown to be recovered quite directly by the present approach. A detailed analysis is concerned with Hamiltonians which at each instant are elements of a finite dimensional Lie algebra. Exploiting the Wei and Norman’s theory for such Hamiltonians, it is found that the evolution of a quantum system is always determined, in part or fully, by an invariant which belongs to the same algebra.
The unitary operators U(t), describing the quantum time evolution of systems with a time-dependent Hamiltonian, can be constructed in an explicit manner using the method of time-dependent invariants. We clarify the role of Lie-algebraic techniques in this context and elaborate the theory for SU(2) and SU(1,1). We show that the constructions known as Magnus expansion and Wei-Norman expansion correspond with different representations of the rotation group. A simpler construction is obtained when representing rotations in terms of Euler angles. The many applications are postponed to Part II of the paper.
We discuss the Schr\"odinger equation with a time-dependent Hamiltonian that can be written as a linear combination of operators which span a finite-dimensional Lie algebra. The equations of motion for the operators in the Heisenberg representation are shown to be useful in calculating matrix elements and transition probabilities, as well as in obtaining the time-evolution operator. A general time-dependent one-dimensional bilinear Hamiltonian is considered as an illustrative example, and the product form for the time-evolution operator is shown to be global.
There exist a number of typical and interesting systems or models which possess three-generator Lie-algebraic structure in atomic physics, quantum optics, nuclear physics and laser physics. The well-known fact that all simple 3-generator algebras are either isomorphic to the algebra sl(2,C)sl(2,C)sl(2,C) or to one of its real forms enables us to treat these time-dependent quantum systems in a unified way. By making use of the Lewis-Riesenfeld invariant theory and the invariant-related unitary transformation formulation, the present paper obtains exact solutions of the time-dependent Schrödinger equations governing various three-generator quantum systems. For some quantum systems whose time-dependent Hamiltonians have no quasialgebraic structures, we show that the exact solutions can also be obtained by working in a sub-Hilbert-space corresponding to a particular eigenvalue of the conserved generator (i.e., the time-independent invariant that commutes with the time-dependent Hamiltonian). The topological property of geometric phase factors in time-dependent systems is briefly discussed.
Universal quantum integrals of motion are introduced, and their relation with the universal quantum invariants is established. The invariants concerned are certain combinations of the second- and higher-order moments (variances) of quantum-mechanical operators, which are preserved in time independently of the concrete form of the coefficients of the Schrödinger equation, provided the Hamiltonian is either a generic quadratic form of the coordinate and momenta operators, or a linear combination of generators of some finite-dimensional algebra (in particular, any semisimple Lie algebra). Using the phase space representation of quantum mechanics in terms of the Wigner function, the relations between the quantum invariants and the classical universal integral invariants by Poincaré and Cartan are elucidated. Examples of the `universal invariant solutions' of the Schrödinger equation, i.e. self-consistent eigenstates of the universal integrals of motion, are given. Applications to the physics of optical and particle beams are discussed.
By making use of the Lewis-Riesenfeld invariant theory, the solution of the Schrödinger equation for the time-dependent linear potential corresponding to the quadratic-form Lewis-Riesenfeld invariant Iq(t)I_{\rm q}(t)Iq​(t) is obtained in the present paper. It is emphasized that in order to obtain the general solutions of the time-dependent Schrödinger equation, one should first find the complete set of Lewis-Riesenfeld invariants. For the present quantum system with a time-dependent linear potential, the linear Il(t)I_{\rm l}(t)Il​(t) and quadratic Iq(t)I_{\rm q}(t)Iq​(t) (where the latter Iq(t)I_{\rm q}(t)Iq​(t) cannot be written as the squared of the former Il(t)I_{\rm l}(t)Il​(t), {\it i.e.}, the relation Iq(t)=cIl2(t)I_{\rm q}(t)= cI_{\rm l}^{2}(t)Iq​(t)=cIl2​(t) does not hold true always) will form a complete set of Lewis-Riesenfeld invariants. It is also shown that the solution obtained by Bekkar {\it et al.} more recently is the one corresponding to the linear Il(t)I_{\rm l}(t)Il​(t), one of the invariants that form the complete set. In addition, we discuss some related topics regarding the comment [Phys. Rev. A {\bf 68}, 016101 (2003)] of Bekkar {\it et al.} on Guedes's work [Phys. Rev. A {\bf 63}, 034102 (2001)] and Guedes's corresponding reply [Phys. Rev. A {\bf 68}, 016102 (2003)].