Consider \(\hat{H}_{0}\) corresponding to the hydrogen atom, with the initial state \(i\) being the ground state, and \(\lambda H_{1, m i}=\lambda e^{-m A+\tilde{\lambda} t}\), for \(t<0\), where \(A\) is a constant. Calculate the probability of transition to the \(m\) th energy eigenstate as a function of time.