Time Dependence and Selection Rules

Up to this point, the wavefunctions considered do not evolve with time. In some cases, the Hamiltonian may have time-dqtendent terms indicating that the system changes with time. An important example is when electromagnetic radiation interacts with a system. Electromagnetic radiation consists of electric and magnetic fields that oscillate in space and time. When electromagnetic radiation interacts with a molecule (such as in spectroscopy), the oscillating fields will result in a time-dependent element in the complete Hamiltonian for the molecule. As already observed in the case of infrared spectroscopy, this interaction may result in a transition of states. 

If time, t, is a variable in a quantum mechanical system, then there must be an operator associated with time. The operator time, i, (just like position) consists of multiplication by t. 

Postulate 111 (see Section 2.2) can now be generalized for any system including the variable time (T is a wavefunction that includes time)  

The wavefunction F in Equation 6-64a and b is a function of time, and the Hamiltonian may also have time dependence. Based on the extension of Postulate III for the TDSE, the wavefunctions are eigenfunctions of both space and time. 

The TDSE is a generalization of the time independent Schroedinger equation, TISE (the type of Schroedinger equation considered up to this point). The TDSE does not, however, invalidate the TISE. Rather, the TISE is a case where the Hamiltonian is independent of time. For a time independent Hamiltonian, the wavefunction F is separable in terms of space and time. 

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