Chapter 4
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 a 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.
4.1 Electric Dipole Transition
We have a time-dependent Schrödinger equation described as
Hψ ¼ ih
∂ψ
∂t
:
ð1:47Þ
Using the method of separation of variables, we obtained two equations expressed
below.
Hϕ x
ð Þ ¼ Eϕ x
ð Þ,
ð1:55Þ
© Springer Nature Singapore Pte Ltd. 2020
S. Hotta, Mathematical Physical Chemistry,
https://doi.org/10.1007/978-981-15-2225-3_4
125
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 a 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.
4.1 Electric Dipole Transition
We have a time-dependent Schrödinger equation described as
Hψ ¼ ih
∂ψ
∂t
:
ð1:47Þ
Using the method of separation of variables, we obtained two equations expressed
below.
Hϕ x
ð Þ ¼ Eϕ x
ð Þ,
ð1:55Þ
© Springer Nature Singapore Pte Ltd. 2020
S. Hotta, Mathematical Physical Chemistry,
https://doi.org/10.1007/978-981-15-2225-3_4
125