Planck's Radiation Law for Thermal Sources:[1]
Reference: Radiation
Laws
To set the stage for subsequent discussions of laser physics and quantization
of the electromagnetic field we briefly explore the earliest, seminal notions
in the quantum theory of light. As we all remember, in 1900 Planck found
that he could account for the measured spectral distribution of radiation
from a thermal source by postulating that the energies of a
certain
set of harmonic oscillators are quantized! Drawing on Planck's
success, Einstein in 1905 was able to show that the extraordinary features
of photoelectric effect could be explained by hypothesizing the
corpuscularity
of the electromagnetic field. The crowning triumph of early quantum optics
is Einstein's amazingly simple, phenomenological theory of 1917 which provided
a quantitative basis for analyzing the absorption and emission of light
by atoms.
In treating thermal sources the basic assumption is that emitted radiation
is a sample of the total electromagnetic field -- viz. all of the
modes of a resonator -- in thermal equilibrium with its material environment
-- viz.
the walls of the resonator. In the traditional treatment
of the theory of
black-body radiation, a particular thermodynamic
system is assumed -- viz., a hollow resonator which is a cube of
length L with has perfectly conducting walls.[2]
Thus, to satisfy boundary conditions at the walls, the electric field associated
with a particular mode of the cavity is given by
 |
[ IV-1 ] |
where 
is a set of positive integers. To satisfy the homogeneous Helmhotz
equation we must have
 |
[ IV-2 ] |
The cycle-averaged value of the stored energy
density
associated
with the particular mode is given by
 |
[ IV-3 ] |
For a thermal source, the most significant experimentally measurable
object is the noise spectrum -- i.e., the frequency distribution
of the stored energy density. To obtain this distribution, we take the
energy density in the frequency range between 
and
-- viz.
 |
[ IV-4 ] |
Thus the density of states is defined as
 |
[ IV-5a ] |
In the frequency range where 
,
the following k-space argument holds:
 |
[ IV-5b ] |
This extremely important density-of-states construction may
be visualized most elegantly in two dimension. [3] |
 |
Following the traditional (Rayleigh-Jeans) argument, we identify
a resonator mode as a harmonic oscillator [4]
and take its average energy to be the classical thermodynamic value for
a system with two degrees of freedom -- i.e. 
Thus, we obtain the famous Rayleigh-Jeans radiation law (which impels
us to worry about an ultra-violet catastrophe.)
 |
[ IV-6 ] |
Planck's quantization hypothesis:
Following Planck we set
 |
[ IV-7 ] |
where n = 0, 1, 2,...... Assuming that the resonator mode is
in thermal equilibrium with its environment, we may use the Boltzmann probability
factor
to find the probability 
that the mode (read oscillator) is thermally excited to an energy 
 |
[ IV-8 ] |
The thermal mean value of n is obtained in a similar manner
[5]
-- viz.
 |
[ IV-9 ] |
which may be plotted
Equation [ IV-9 ] taken in conjunction with Equations [ IV-5b
] and [ IV-7 ] leads then directly to the Planck radiation law
 |
[ IV-10 ] |
From this expression we see that the Rayleigh-Jeans radiation
law is correct in the limit 
,
but, to our great relief, Planck has staved off the ultraviolet catastrophe.
We may then plot this famous and an exceedingly important result. (From
the particular point view of laser physics, the Planck radiation law gives
a measure of the background radiative thermal noise spectrum.)
Einstein's Phenomenological Theory of Radiation Processes
--
Einstein's "A" and "B" coefficients:
We return to the two-level system and, with Einstein, make
some physically reasonable (early quantum mechanical) postulates concern
the absorption and emission of light. Consider the two-level system shown
below. Einstein's postulates[6] are
embodied in a set of rate equations for the level populations- viz.
 |
[ IV-11 ] |
It particularly useful to examine the results of these equations
under conditions -- viz.
General equilibrium or steady-state condition:
 |
[ IV-12a ] |
Thermal equilibrium condition:
 |
[ IV-12b ] |
Of course, in the latter case we use fact that the level populations 
and 
at thermal equilibrium are related by Boltzmann's law -- i.e.
 |
[ IV-13 ] |
so that
 |
[ IV-14 ] |
Thus, by comparison with Equation [ IV-10 ], we see that
 |
[ IV-15a ] |
and
 |
[ IV-15b ] |
and, by comparison with Equation [ IV-9 ], we see that
 |
[ IV-16a ] |
or
 |
[ IV-16b ] |
[1] This section and some parts of the following
section draw heavily upon discussions in Rodney Loudon's The Quantum
Theory of Light (2nd edition), Oxford (1983).
[2] The detail results of black-body radiation theory
are indeed sensitive to the assumed character of the thermodynamic systemas
will be demonstration presently.
[3] For use in later discussions, we note here that
by this same argument the 2D density-of-states is found to be 
and
the 1D value 
.
[4] We justify this assertion in detail later: it
suffices to note here, in anticipation of that later discussion, that a
harmonic oscillator and an electromagnetic mode have analogous Hamiltonians.
[5] It is also easy to show that thermal mean-square
fluctuation in n is given by 
.
[6] Using our semiclassical theory of matter-radiation
interaction, we can easily show that
but, of course, Einstein did not have that well-developed theory at hand.
While the stimulated emission and absorption processes are reasonably intuitive,
it is the introduction of the spontaneous emission process which flows
from a profound understanding of the processes by which thermal equilibrium
is achieved.
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This page was prepared and is maintained
by R. Victor Jones, jones@deas.harvard.edu
Last updated March 9, 2000
