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3-5: The Discovery of Energy Quanta |
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As learned
on the preceding page,
Planck found
Planck's formula
which can
well reproduce
the energy distribution
(spectrum)
of the cavity radiation
for all regions of frequency.
When we derived
the Rayleigh-Jeans formula,
we obtained the number
of the normal oscillations
in a vacuum
(cavity), i.e.
the number of the degrees
of freedom of
the oscillations of the
electromagnetic field in
a vacuum.
The number of the normal
oscillations
of the frequency between
and
per unit volume is
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If a constant energy kT
is partitioned
to all these degrees
of freedom,
then we obtain
the Rayleigh-Jeans formula.
Comparing Planck's formula
and the Rayleigh-Jeans formula,
we can easily see that,
in realistic cases,
the constant energy kT
is not equally partitioned,
but the energy partitioned
actually is
Namely,
the partitioned energy
is reduced
by a weight function
The form of this weight
function must be
whose value is
close to 1 when
x is small,
but becomes much smaller
than 1 when
x is large.
The energy distribution
would therefore be
severely reduced
for large x values.
Thus, we should realize
that, when
is large,
the law of equipartition
of energy
is not satisfied.
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[Energy Quanta]
Thus,
Planck discovered
Planck's formula
which is well fit
to the experimental results.
This discovery itself
must be great, but
Planck's greatness
is in having not
stayed there.
Planck pursued
the origin of Planck's formula,
and finally he reached
the epoch-making idea
of energy quanta.
Let us explain this below.
If we divide
the matter
into finer and finer portions,
we ultimately reach
molecules or atoms.
Thus the matter is
not continuous but
the aggregate
of a huge number
of fundamental particles.
We called this nature
of the matter
the "atomic nature".
Moreover, we learned that
there exists
the fundamental unit
of electric charge,
i.e. the elementary charge.
Namely, the electricity
has also an atomic nature.
Planck thought
that there would be
fundamental units of energy.
He called these energy quanta,
which is just
an atomic nature of energy.
Standing on this viewpoint,
he was successful
in deriving
Planck's formula
(1900).
This is a kind
of revolution
in the natural science,
which can never
be understood
with the classical theory.
This was the start
of the new physics
in the 20th century.
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[Energy Quanta and Planck's Formula]
It was previously
said that the cavity radiation
is equivalent
to a collection of
a large number
of harmonic oscillators.
The energy of
each oscillator
is written
in a form of
where
q is
the coordinate variable
of the oscillator and
p denotes
the momentum variable.
As discussed
in detail on
the page: 3-2-A,
we can get
the mean value
of the energy E
using
the Boltzmann distribution
as
where the normalization
constant
is given by
If the energy E
is a continuous quantity,
the mean value
of the energy would be
given by
This is the ordinary law
of equipartition of energy.
However,
if the energy E
is not a continuous quantity
and cannot take
any value freely,
then
p
and
q
cannot be treated
as continuous quantities.
Now, let the amount
of the energy quantum be
 .
Suppose that the energy
of the oscillator
must be an integral multiple
of the energy quantum,
 .
Then
p
and
q
should fulfill
the relation
Considering this,
the calculation of
the mean value of E
should be done
not by the integration but
by some other procedure.
For this purpose,
we need some mathematical
expressions, so that
we explain on the other
page,
3-5-A:
Derivation of Planck's formula.
The explanation
is not so difficult but,
if you feel somewhat
tedious, you may skip it.
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[The Discovery of Energy Quanta]
As explained
on the page
3-5-A,
if we consider
that the energy
of the harmonic oscillator,
E,
can take only
an integral multiple of
the energy quantum
then the mean value
would be given
by the formula
This is nothing but
Planck's formula
which is very well fit
to experiment.
This concludes that
the energy of a
cavity radiation
with a frequency
can take only
an integral multiple of
the energy quantum
 .
People have
usually thought
that the energy is
a continuous quantity,
but, since the discovery
of energy quanta,
the energy
would be discontinuous
and has an
elementary unit
 .
Thus,
an epoch-making idea
that the atomic nature
in the natural world
exists not only
in the matter and
in the electricity
but also in the energy
was brought us by Planck
at the beginning
of the 20th century
(1900).
This was just the opening
of the new science
of the new century.
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[Planck's Constant]
Precise measurements
of Planck's constant
have been done
and its currently
accepted value is
Planck's constant
has a dimension of action;
i.e., [energy] x [time] = [action],
so that it is
sometimes called
an action quantum.
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[Heat Capacities of Solids]
On the page
Molecular Motions and Specific Heats
we discussed
the heat capacities
of solids.
There we learned
that,
if one mole of solid
is regarded as
a collection of
3NA
harmonic oscillators,
the theoretical value
of the heat capacity
obtained by using
the law of equipartition
of energy
can well reproduce
the experimental value
at a temperature higher
than the room temperature.
However, if the temperature
goes down, the law
of equipartition of energy
does not work well
and theory is not fit
to experiment.
As soon as the idea
of energy quanta
was proposed
by Planck in 1900,
Einstein and
P. J. W. Debye
(the Netherlands, 1884 - 1966)
applied this idea
to the cases of solids.
They showed that,
if we consider that
the energy of an oscillation
of frequency
.
can take only
an integral multiple of
,
the theoretical heat capacities
can amazingly well reproduce
the corresponding
experimental data
all over the regions
of temperature (1907, 1912).
Thus, it was clarified
that the nature that
the energy consists of
"grains"
-- we call this
the atomic nature --
is held not only
for the cavity radiation
but also
for solids.
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