Chapter 9
Light Quanta: Radiation and Absorption
So far we discussed propagation of light and its reflection and transmission
(or refraction) at an interface of dielectric media. We described characteristics of
light from the point of view of an electromagnetic wave. In this chapter, we describe
properties of light in relation to quantum mechanics. To this end, we start with
Planck’s law of radiation that successfully reproduced experimental results related to
a blackbody radiation. Before this law had been established, Rayleigh–Jeans law
failed to explain the experimental results at a high frequency region of radiation (the
ultraviolet catastrophe). The Planck’s law of radiation led to the discovery of light
quanta. Einstein interpreted Planck’s law of radiation on the basis of a model of twolevel atoms. This model includes so-called Einstein A and B coefficients that are
important in optics applications, especially lasers. We derive these coefficients from
a classical point of view based on a dipole oscillation. We also consider a close
relationship between electromagnetic waves confined in a cavity and a motion of a
harmonic oscillator.
9.1 Blackbody Radiation
Historically, the relevant theory was first propounded by Max Planck and then
Albert Einstein as briefly discussed in Chap. 1. The theory was developed on the
basis of the experiments called cavity radiation or blackbody radiation. Here,
however, we wish to derive Planck’s law of radiation on the assumption of the
existence of quantum harmonic oscillators.
As discussed in Chap. 2, the ground state of a quantum harmonic oscillator has an
energy
1
2 ħω. Therefore, we measure energies of the oscillator in reference to that
state. Let N 0 be the number of oscillators (i.e., light quanta) present in the ground
state. Then, according to Boltzmann distribution law the number of oscillators of the
first excited state N 1 is
© Springer Nature Singapore Pte Ltd. 2020
S. Hotta, Mathematical Physical Chemistry,
https://doi.org/10.1007/978-981-15-2225-3_9
339
Light Quanta: Radiation and Absorption
So far we discussed propagation of light and its reflection and transmission
(or refraction) at an interface of dielectric media. We described characteristics of
light from the point of view of an electromagnetic wave. In this chapter, we describe
properties of light in relation to quantum mechanics. To this end, we start with
Planck’s law of radiation that successfully reproduced experimental results related to
a blackbody radiation. Before this law had been established, Rayleigh–Jeans law
failed to explain the experimental results at a high frequency region of radiation (the
ultraviolet catastrophe). The Planck’s law of radiation led to the discovery of light
quanta. Einstein interpreted Planck’s law of radiation on the basis of a model of twolevel atoms. This model includes so-called Einstein A and B coefficients that are
important in optics applications, especially lasers. We derive these coefficients from
a classical point of view based on a dipole oscillation. We also consider a close
relationship between electromagnetic waves confined in a cavity and a motion of a
harmonic oscillator.
9.1 Blackbody Radiation
Historically, the relevant theory was first propounded by Max Planck and then
Albert Einstein as briefly discussed in Chap. 1. The theory was developed on the
basis of the experiments called cavity radiation or blackbody radiation. Here,
however, we wish to derive Planck’s law of radiation on the assumption of the
existence of quantum harmonic oscillators.
As discussed in Chap. 2, the ground state of a quantum harmonic oscillator has an
energy
1
2 ħω. Therefore, we measure energies of the oscillator in reference to that
state. Let N 0 be the number of oscillators (i.e., light quanta) present in the ground
state. Then, according to Boltzmann distribution law the number of oscillators of the
first excited state N 1 is
© Springer Nature Singapore Pte Ltd. 2020
S. Hotta, Mathematical Physical Chemistry,
https://doi.org/10.1007/978-981-15-2225-3_9
339