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A quantum particle in a quadrupole radio-frequency trap. (English) Zbl 0613.46064

The quantum motion of a charged particle in a quadrupole radio-frequency trap is solved exactly in terms of the classical trajectories. Thus the quantum stability regions are exactly given in terms of the stability regions for the associated Mathieu equations, and quantum trapping is demonstrated inside these stability area. These exact results enable us to test a commonly believed ”static effective potential approximation” for a particle in a rapidly oscillating time-periodic potential: our results exhibit serious limitations to this approximation. As a subproduct of our approach, we solve the eigenvalue problem for the Floquet operator of our system. This exactly solvable system should be a good starting point for the study of quantum instabilities under small perturbations.
The perturbative treatment of the quantum stability problem has been studied by the author in two subsequent publications: Ann. Physics 173, 210-225 (1987) (for a semi-classical approach) and Ann. Inst. Henri Poincaré 47, 63-83 (1987) (for a quantum treatment of the stability problem).

MSC:

46N99 Miscellaneous applications of functional analysis
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
70F25 Nonholonomic systems related to the dynamics of a system of particles
37A30 Ergodic theorems, spectral theory, Markov operators
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