The time-dependent canonical formalism: Generalized harmonicoscillator and the infinite square well with a moving boundary

Author: Cerveró J. M.   Lejarreta J. D.  

Publisher: Edp Sciences

E-ISSN: 1286-4854|45|1|6-12

ISSN: 0295-5075

Source: EPL (EUROPHYSICS LETTERS), Vol.45, Iss.1, 2010-03, pp. : 6-12

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Abstract

In this paper a systematic formalism for dealing withnon-relativistic time-dependentquantum Hamiltonians is presented. The starting point is the well-knownLewis and Riesenfeld ideawhich involves the construction of an invariant operatorI(x,t) which defines both the dynamics of the physical system and thecanonical formalism that hasto be used in order to obtain a consistent theoretical framework. In orderto exhibit the full powerof the formalism we discuss two examples: the generalized harmonicoscillator and the infinitesquare well with a moving boundary. As the first example has already beenanalyzed by the presentauthors from other different points of view, we are able to compare theresults of the canonicalformalism with these other approaches and, as it was expected, we obtainidentical descriptions ofthis physical system. After this, we turn to the case of the square wellwith a moving boundary. Themain surprise is that in order to obtain consistency with the formalism aneffective interactionappears which seems to be due to the time dependence of the boundary. Alsoconsistency with theprinciple of minimal coupling and gauge invariance is obtained just byusing this canonical operatorformalism. Finally some interesting physical applications are suggested anddiscussed.