  With the aim to solve the time-dependent Schr\"{o}dinger equation associated
to a time-dependent non-Hermitian Hamiltonian, we introduce a unitary
transformation that maps the Hamiltonian to a time-independent
$\mathcal{PT}$-symmetric one. Consequently, the solution of time-dependent
Schr\"{o}dinger equation becomes easily deduced and the evolution preserves the
$\mathcal{C(}t\mathcal{)PT}$-inner product, where $\mathcal{C(}t\mathcal{)}$ is
a obtained from the charge conjugation operator $\mathcal{C}$ through a time
dependent unitary transformation. Moreover, the expectation value of the
non-Hermitian Hamiltonian in the $\mathcal{C(}t\mathcal{)PT}$ normed states is
guaranteed to be real. As an illustration, we present a specific quantum system
given by a quantum oscillator with time-dependent mass subjected to a driving
linear complex time-dependent potential.

Quantum Physics