Optical Transition and Selection Rules

In Sect. 1.2 we showed the Schrödinger equation as a function of space coordinates and time. In subsequent sections, we dealt with the time-independent eigenvalue problems of a harmonic oscillator and hydrogen-like atoms. This implies that the physical system is isolated from the outside world and that there is no interaction between the outside world and physical system we are considering. However, by virtue of the interaction, the system may acquire or lose energy, momentum, angular momentum, etc. As a consequence of the interaction, the system changes its quantum state as well. Such a change is said to be a transition. If the interaction takes place as an optical process, we are to deal with an optical transition. Of various optical transitions, the electric dipole transition is common and the most important. In this chapter, we study the optical transition of a particle confined in a potential well, a harmonic oscillator, and a hydrogen using a semiclassical approach. A question of whether the transition is allowed or forbidden is of great importance. We have a selection rule to judge it.