The Forced Harmonic Oscillator, Decoherence, and Schrödinger Cat States

The harmonic oscillator is a very significant problem in quantum mechanics, because it is one of the very few, relatively non-trivial problems that has an exact analytic solution. In addition to this, the harmonic oscillator solution and algebraic formalism has applications throughout modern physics (e.g. many particle systems, QED, quantum field theory, approximations of real systems such as molecular bonds, etc.). As it turns out, adding a force term to the linear harmonic oscillator gives a quantum system that can also be solved for exactly. This is of great value because it means we can produce analytic solutions to systems that are not quite as idealized as the simple LHO and it allows us to know exactly how harmonic oscillator states, particularly coherent states, evolve in the presence of force. Although the first exact solution of the Schrödinger equation for the forced quantum harmonic oscillator was found by Husimi in 1953 (and independently by Kerner in 1957), it turns out that Caltech’s very own Richard Feynman was actually the first to treat the FHO problem in 1948, using the path integral method (of course). These first treatments of the FHO were done before the first comprehensive treatment of coherent states was given by Glauber in 1963, and consequently the early literature on the FHO problem utilizes a somewhat cumbersome treatment of the problem (i.e. solving the differential equations for displaced gaussians and Hermite polynomials with moving centers, or, in Feynman’s case, the solution being given in a largely unfamiliar formulation of quantum mechanics). The FHO has been solved for several different ways and for several different levels of generality (i.e. inclusion of damping, allowance for time-dependent frequency and/or mass, use of path integral method, use of the Heisenberg picture with invarients of motion, etc.), but I will present the constant mass, constant frequency, no damping problem and use the notation and techniques of the LHO and interaction picture that we learned in in class so as to make the lecture more accessible.