Abstract

Raising and lowering operators are constructed for the harmonic oscillator with a time-dependent force constant and for the damped and linearly forced oscillator. The demand that the total derivative [ a ( x, p, t), H ] + i ħ ∂/∂ t a ( x, p, t) of the time-dependent operator a ( x, p, t) and its Hermitian adjoint a †( x, p, t) be zero makes these the lowering and raising operators for a set of eigenvectors to the dynamic harmonic oscillators. The lowest eigenvector can therefore be constructed solving a first-order differential equation in ∂/∂ x, whereafter a complete set can be generated by the application of a †( x, p, t). The functional time dependence of a ( t) and a †( t) is simply given as the solution to Newton's second law of motion.

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