Introduction to Quantum Ideas
35
This is known as Rayleigh-Jeans law and provides a good fit to the
experimental results in the region of long wavelengths. However, it is
unacceptable since it implies that the energy density increases indefinitely as
v increases and therefore the total energy per unit volume is infinite, contrary to
physical observations. This is known as the ultraviolet catastrophe.
Max Planck re-examined the basic assumptions in the theoretical approach
to black-body radiation. He introduced (1900) a new and revolutionary assumption
in the basic framework which allowed him to describe the experimental
observations with great accuracy. He associated each mode of electromagnetic
oscillation with atomic oscillators embedded in the walls of the cavity. However,
these oscillators were allowed to have only discrete energies which are integral
multiples of hv, and the energy distribution of these oscillators is according to
the Maxwell-Boltzmann distribution. Here h is a small number whose value is
given in Eq. (2.2), and is called Planck’s constant. Since the Maxwell-Boltzmann
distribution is given by exp (– E/kT), the average energy for each mode
oscillation is
0
0
exp (
/ )
exp (
/ )
n
n
nh v
nh v kT
nh v kT
∞
=
∞
=
−
ε =
−
∑
∑
exp ( / ) 1
hv
hv kT
=
−
(2.11)
Using this expression in Eq. (2.10) in place of kT, the following expression
is obtained
3
3
8
1
( )
exp ( / ) 1
h v
u v
hv kT
c
π
=
−
(2.12)
It is easy to see that in the region of long wavelengths, this equation
reduces to the Rayleigh-Jeans relation in Eq. (2.10). It gives an exponential
damping in the short-wavelength limit. Overall, it is in excellent agreement
with experimental observations over a wide range of temperatures. It also
leads to many of the special relations for black-body radiation. For example,
the energy radiated by a unit area of a black-body, per unit time is given by the
Stefan-Boltzmann law:
0
( ) /
4
∞
= ∫
c
U
u v d v
(2.13)
35
This is known as Rayleigh-Jeans law and provides a good fit to the
experimental results in the region of long wavelengths. However, it is
unacceptable since it implies that the energy density increases indefinitely as
v increases and therefore the total energy per unit volume is infinite, contrary to
physical observations. This is known as the ultraviolet catastrophe.
Max Planck re-examined the basic assumptions in the theoretical approach
to black-body radiation. He introduced (1900) a new and revolutionary assumption
in the basic framework which allowed him to describe the experimental
observations with great accuracy. He associated each mode of electromagnetic
oscillation with atomic oscillators embedded in the walls of the cavity. However,
these oscillators were allowed to have only discrete energies which are integral
multiples of hv, and the energy distribution of these oscillators is according to
the Maxwell-Boltzmann distribution. Here h is a small number whose value is
given in Eq. (2.2), and is called Planck’s constant. Since the Maxwell-Boltzmann
distribution is given by exp (– E/kT), the average energy for each mode
oscillation is
0
0
exp (
/ )
exp (
/ )
n
n
nh v
nh v kT
nh v kT
∞
=
∞
=
−
ε =
−
∑
∑
exp ( / ) 1
hv
hv kT
=
−
(2.11)
Using this expression in Eq. (2.10) in place of kT, the following expression
is obtained
3
3
8
1
( )
exp ( / ) 1
h v
u v
hv kT
c
π
=
−
(2.12)
It is easy to see that in the region of long wavelengths, this equation
reduces to the Rayleigh-Jeans relation in Eq. (2.10). It gives an exponential
damping in the short-wavelength limit. Overall, it is in excellent agreement
with experimental observations over a wide range of temperatures. It also
leads to many of the special relations for black-body radiation. For example,
the energy radiated by a unit area of a black-body, per unit time is given by the
Stefan-Boltzmann law:
0
( ) /
4
∞
= ∫
c
U
u v d v
(2.13)