  It is in accordance with such experiments as single photon self-interference
that a photon, conveying one radiation energy quantum "$ h \times$ frequency",
is spatially extensive and stretches an electromagnetic wave train. A wave
train, hence an energy quantum, can only be emitted by its source gradually. In
both the two processes the wave and "particle" attributes of the radiation
field are simultaneously prominent, where an overall satisfactory theory has
been lacking. This paper presents a first principles treatment, in a unified
framework of the classical and quantum mechanics, of the latter process, the
emission of a single radiation quantum based on the dynamics of the
radiation-emitting source, a charged oscillator which is itself extensive
across its confining potential well. During the emission of one single
radiation quantum, the extensive charged oscillator undergoes a continuous
radiation damping and is non-stationary. This process is in this work treated
using a quasi stationary approach, whereby the classical equation of motion,
which directly facilitates the correspondence principle for a particle
oscillator, and the quantum wave equation are established for each sufficiently
brief time interval. As an inevitable consequence of the division of the total
time for emitting one single quantum, a fractional Planck constant $h$ is
introduced. The solutions to the two simultaneous equations yield for the
charged oscillator a continuously exponentially decaying Hamiltonian that is at
the same time quantised with respect to the fractional-$h$ at any instant of
time; and the radiation wave field emitted over time stretches a wave train of
finite length. The total system of the source and radiation field maintains at
any time (integer $n$ times) one whole energy quantum, $h \times$ frequency, in
complete accordance with the notion of quantum mechanics and experiment.
