Gauge transformations for a driven quantum particle in an infinite square well

Quantum mechanics of a particle in an infinite square well under the influence of a time-dependent electric field is reconsidered. In some gauge, the Hamiltonian depends linearly on the momentum operator which is symmetric but not self-adjoint when defined on a finite interval. In spite of this symmetric part, the Hamiltonian operator is shown to be self-adjoint. This follows from a theorem by Kato and Rellich which guarantees the stability of a self-adjoint operator under certain symmetric perturbations. The result, which has been assumed tacitly by other authors, is important in order to establish the equivalence of different Hamiltonian operators related to each other by quantum gauge transformations. Implications for the quantization procedure of a particle in a box are pointed out. A. Introduction The behaviour of a classical particle in a one-dimensional infinite square-well (a box, for short) under the influence of a time-dependent electric field has been studied in [1]. The interaction of the charged particle with the field is described by a term linear in the position. A time-periodic modulation of this term is sufficient to render the motion of the particle chaotic. In quantum mechanics, the system is described by Schrödinger’s equation on an interval of finite length, with wave functions vanishing at the boundaries. In [2], it is proposed to apply a gauge transformation to the Hamiltonian operator which results in an interaction term depending linearly on the momentum operator. Since the Hamiltonian operator in this gauge no longer depends on the position, it is straightforward to solve the time-dependent Schrödinger equation of this problem. In the classical limit, however, the solutions obtained in this way do not give rise to the expected irregular behaviour. The inconsistency is due to the unjustified assumption that the operator of the kinetic energy commutes with the momentum operator [2].