In the late nineteenth and early twentieth centuries, a great
deal of experimental evidence began to accumulate for which
classical mechanics could provide no explanation. It was the
consideration and explanation of these data which led to the
development of quantum mechanics.
One of the most significant failures of classical mechanics
was its inability to explain the distribution of energy emitted
by a black body. Any hot object emits electromagnetic radiation,
and the maximum in the emitted wavelength shifts to shorter
wavelengths as the temperature of the emitter is raised.
A black body is the name given to a theoretical ideal emitter,
an object capable of absorbing and emitting all wavelengths
of radiation equally. A black body emitter may be successfully
approximated by a small opening into a heated cavity. The
emission curves of a black body have the following form:
where T_{1} > T_{2} > T_{3}.
The energy distribution, or spectral
energy density, is the energy per unit volume of the
cavity that is emitted in the wavelength interval λ to
λ + dλ. Note the total area under the curve increases
as the temperature increases, indicating that the hotter an
object is, the more energy it radiates per unit volume.
Analysis of the data from black body emitters led to the formulation
of the Wien displacement law
relating the temperature and the wavelength of maximum energy
density:
The problem of the energy distribution may be
treated classically by considering the electromagnetic field
as a collection of oscillators of all possible frequencies.
The presence of radiation of frequency ν then signifies
that the oscillator of that frequency has been excited. The
classical equipartition principle may then be applied to determine
the average energy of each independent oscillator. From this
basis follows an expression known as the RayleighJeans
law after the physicists who originally formulated it:
where ρ is the spectral energy density.
This law is very successful at long wavelengths but fails badly
at short ones  the inverse dependence upon λ means that
as the wavelength gets shorter and shorter, the spectral energy
density tends to infinity. This result is patently absurd, as
it suggests that even at room temperature, objects should radiate
strongly in the high frequency portion of the spectrum (gamma
rays, xrays and the ultraviolet), which is clearly not the
case. This failing of the law is termed the ultraviolet
catastrophe:
The experimental observations may be accounted for by limiting
the energy of each electromagnetic oscillator to discrete values.
(This is quite contrary to the classical view, in which all
possible energy values are allowed.) The name given to this
limitation of possible values is quantisation
of energy.
This idea was originally proposed by the physicist Planck,
and he discovered that the observed data were reproduced if
he supposed that the energies of an oscillator of frequency
ν were limited to integer multiples of hν, where h
was a fundamental constant which is now known as the Planck
constant. This assumption allowed him to derive the Planck
distribution:
with ρ the spectral energy density.
This expression provides a good agreement with experimental
data, and the constant h, which is an undetermined parameter
in the original theory, may be adjusted to obtain the best fit.
This allows measurement of the value of h by experiment.
This expression is similar in form to the RayleighJeans law,
the main difference being in the exponential term in the denominator.
At short wavelengths the exponential term is large, and as the
wavelength tends to zero, the exponential term tends to infinity
faster than the λ^{5} term tends to
zero. The upshot of this is that as the wavelength tends to
zero, so does the spectral energy density. Thus the ultraviolet
catastrophe is avoided.
At long wavelengths, the Planck
distribution reduces to the RayleighJeans law.
This is because the exponential term in the distribution is
much smaller than one when the wavelength is large, so it may
be approximated by 1 + (hc / λkT).
(Recall that e^{x} ≈ 1+
x when x is very small.) Substitution of this approximation
into the Planck distribution immediately gives the RayleighJeans
law.
The Wien law may also be obtained from the Planck distribution.
In this case we are looking for the position of the maximum
in the distribution. To find this we set dρ/dλ = 0
, and make the approximation that the wavelength is so short
that hc/λ>>kT. This gives
and when the values of the fundamental constants are substituted
in, the Wien law is obtained.
The reasoning behind the success of the Planck distribution
is as follows:
Rayleigh's approach failed because it assumed that the thermal
motion of atoms in the walls of a black body would excite all
the electromagnetic oscillators equally; the ultraviolet catastrophe
is a result of the excitation of high frequency oscillators.
According to Planck's hypothesis, however, an oscillator cannot
be excited unless it receives an energy of at least hν
(as this the minimum amount of energy an oscillator of frequency
ν may possess above zero. It cannot have an amount of energy
which is a fraction of hν, so it cannot accept an amount
of energy less than hν). For high frequency oscillators
(large ν), the amount of energy hν is too large to
be supplied by the thermal motion of the atoms in the walls,
and so they are not excited.
Quantisation of energy reduces the contribution to the emission
curve of high frequency oscillators, as the energy available
is not sufficient to excite them.
