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Miloslav Znojil
Reviewer number:
Item's zbl-Number:
DE 016 788 279
Mista, Ladislav jun.; Filip, Radim:
Non-perturbative solution of nonlinear Heiseberg equations
J. Phys. A: Math. Gen 34, No. 27, 5603 - 5612 (2001).
82C10Quantum dynamics and nonequilibrium statistical mechanics general
Primary Classification:
81Q05Closed and approximate solutions to the Schroedinger, Dirac, Klein-Gordon and other quantum-mechanical equations
Secondary Classification:
81V80Quantum optics
quantum mechanics in Heisenberg picture, Heisenberg equations for creation/annihilation operators, expansions in the initial-time basis

Although a vast majority of the practical problems in quantum
mechanics is being solved in the Schroedinger picture, there still
exist exceptional situations where one could switch to the
Heisenberg picture. A partial review is provided by the two papers
in Physical Review D 40, 2739 and 3504 (1989) on the operator
differential equations by C. M. Bender and G. V. Dunne. These
authors also inspired my own, fairly discouraging experience with
the direct work in Heisenberg picture which was reported in Cz. J.
Phys. 41, 201 (1991).
The overall scepticism is significantly weakened in the many-body
context where the use of the creation and annihilation (i.e., in
effect, occupation-number) operators offers a definite advantage.
The reasons were recently summarized by D. F. Styer et al in
section II F of their review paper Am. J. Phys. 70, 288 (2002)).
In this context the present authors feel inspired by a very
specific model (viz., two oscillators with the 1:2 ratio between
their frequencies) and propose its study via an expansion of the
respective time-dependent creation and annihilation operators in
terms of their t=0 initial values. In this basis (of what they
call ``elementary processes") they (1) demonstrate the equivalence
of results of working in the Schroedinger and Heisenberg picture,
(2) emphasize specific merits of the latter approach, (3) outline
a general truncation scheme suitable, hopefully, for analogous
treatment of more complicated systems.
I must admit that my own, private scepticism still, in a slightly
weakened form, lasts. Before making recommendations I would like
to see the merits of this method in its application to a less
schematic example.
Remarks to the editors:

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