It is a rather curious fact that the exact quantal treatment of forced oscillation has only recently been discovered to be feasible. There may be a lot of general arguments which make the possibility of such a treatment very convincing, but I myself find the ground for it in an elementary observation that the wave equation of forced harmonic oscillation, including the case of subharmonic resonance, admits a solution of Gaussian type, i.e. of the form: an exponential of a quadratic. On this basis the complete solution and the related problems are analyzed here.
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Kôdi Husimi (1953) studied this question.