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Acta Physica Sinica

NOTES ON THE QUANTUM INVARIANT AND ON THE-RELATION BETWEEN IT AND QUANTUM PHASE

Bo‐Zang Li, ZHANG LING-YUN, Zhang Xiang-dong, 中国科学院物理研究所和凝聚态物理中心;山西大学理论物理研究所

Published Jan 1, 1997• 0 citations
Publisher

Abstract

Based on the construction theorem of quantum invariant,the results of Lewis and Riesenfeld are rededuced in a much simpler way and extended significantly.Meanwhile,it is pointed out that the Lewis Riesenfeld phases are in general not of physical meaning unless the invariants take specific forms which have been worked out.As an example,the spin system in a magnetic field is discussed in detail.

DOI
10.7498/aps.46.2080

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Based on the construction theorem of quantum invariant,the results of Lewis and Riesenfeld are rededuced in a much simpler way and extended significantly.Meanwhile,it is pointed out that the Lewis Riesenfeld phases are in general not of physical meaning unless the invariants take specific forms which have been worked out.As an example,the spin system in a magnetic field is discussed in detail.
An alternative concept of the basic invariants is introduced. The Lewis-Riesenfeld invariant theory is extended to obtain a generalized invariant formulation. The formulation is then used to establish four facts: (i) Any invariant for a quantum system can be constructed in terms of the basic invariants. (ii) It is possible to introduce a solution-generating technique by making use of the basic invariants. (iii) The path integral in the generalized invariant formulation reduces to an ordinary integral. (iv) The study of noncyclic evolution of a quantum system reduces explicitly to the study of the cyclic evolution. Finally, phase factors and general solution for the driven generalized time-dependent harmonic oscillator are studied as an illustrative example.
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.
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)].
This paper is concerned with the connection between density matrix method, supersymmetric quantum mechanics and Lewis-Riesenfeld invariant theory. It is shown that these three formulations share the common mathematical structure: specifically, all of them have the invariant operators which satisfies the Liouville-Von Neumann equation and the solutions to the time-dependent Schrödinger equation and/or Schrödinger eigenvalue equation can be constructed in terms of the eigenstates of the invariants.