Quantum theory of a non-Hermitian time-dependent forced harmonic oscillator having PT symmetry

We discuss the quantum theory of an harmonic oscillator with time-dependent mass and frequency submitted to action of a complex time-dependent linear potential with PT symmetry. Combining the Lewis and Riesenfeld approach to time-dependent non-Hermitian Hamiltonians having PT symmetry and linear invariants, we solve the time-dependent Schrodinger equation for this problem and use the corresponding quantum states to construct a Gaussian wave packet solution. We show that the shape of this wave packet does not depend on the driving force. Afterwards, using this wave packet state, we calculate the expectation values of the position and momentum, their fluctuations and the associated uncertainty product. We find that these expectation values are complex numbers and as a consequence the position and momentum operators are not physical observables and the uncertainty product is physically unacceptable.