Non-perturbative solution of nonlinear Heisenberg equations (Q2757879)

Although a vast majority of the practical problems in quantum mechanics is being solved in the Schrödinger 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 \textit{C. M. Bender} and \textit{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). NEWLINENEWLINENEWLINEThe 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 \textit{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 Schrödinger 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. NEWLINENEWLINENEWLINEI 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.