Damped quantum mechanical oscillators with thermal fluctuations

The solution for the relevant Heisenberg operators of a quadratic time-dependent quantum mechanical Hamiltonian is presented in convolution form suitable for the inclusion of thermal fluctuations. The approach to equilibrium of an oscillator in contact with a thermal bath is studied. An expression is obtained for the reduction of the random fluctuation width of the position coordinate as a consequence of the non-Markovian nature of a quantized thermal bath.