Time-dependent Schrodinger equation Subject

If we fix a realization of the path Q(t), then, when performing the path integration over q, the particle may be treated as if it were subject to a time-dependent potential Vint[<2(r), q]. From traditional quantum mechanics it is clear that this integration is equivalent to the solution of the time-dependent Schrodinger equation in imaginary time ... 

A bright state is prepared at t = 0. Lee and Heller (1979) derive a rigorous description of the wavepacket that is actually prepared when the excitation pulse is of finite duration. It is not a single eigenstate. It evolves in time subject to the spectroscopic time-independent Heff, as specified by the time-dependent Schrodinger equation Eq. (9.1.2). 

Our task is to find approximate solutions to the time-independent Schrodinger equation (Eq. (2)) subject to the Pauli antisymmetry constraints of many-electron wave functions. Once such an approximate solution has been obtained, we may extract from it information about the electronic system and go on to compute different molecular properties related to experimental observations. Usually, we must explore a range of nuclear configurations in our calculations to determine critical points of the potential energy surface, or to include the effects of vibrational and rotational motions on the calculated properties. For properties related to time-dependent perturbations (e.g., all interactions with radiation), we must determine the time development of the... 

Suppose that the atom (or nucleus) initially in an eigenstate 1 is subjected to a small time-dependent potential V (t) on top of the unperturbed Hamiltonian Ho2 It is then possible to treat the coefficients an in Eq. (A3.12) as functions of time, with ai 2(r) 1 being the probability that it is still in state 1 after a time x and a2 2(t) < C 1 the probability that it has undergone a transition to another eigenstate 2 . Substituting in Schrodinger s equation (A3.8),... 

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