DOC.
11
ARGUMENTS
FOR MOLECULAR AGITATION 137
Doc.
11
Some
Arguments
for the
Assumption
of
Molecular
Agitation
at
Absolute Zero
and
Remark Added
in
Proof
by
A.
Einstein and O.
Stern
[Annalen
der
Physik
40: 551-60,
1913]
[1]
According
to
Planck's first
formula,
the
expression
for the
energy
of
a
resonator is:
(1)
hv
E
=
hv
kT
-
1
and
according
to
the
second:
(2)
E
=
hv hv
hv
2
kT
-
1
[2]
The
limiting
value
for
high temperatures,
if
we
break off the
expansion
of
e
with
the
quadratic term,
becomes for
(1):
hv
kT
lim E
=
kT
hv/2,
T
=
oo
2
for
(2):
lim E
=
kT.
T
=
00
Thus,
according
to formula
(1),
the
energy as a
function
of
temperature,
as
represented
in
Fig.1,
starts
with
zero
for
T
=
0,
the value
required by
the classical
theory,
but remains
consistently
a
bit
smaller than
the
latter, by hv/2,
at
high temperatures.
Accord-
ing
to
formula
(2),
the
energy
of the
resonator at
absolute
zero
is
hv/2,
contrary
to
the classical
theory,
but
at
high
temperatures
it
approaches asymptotically
the
energy required
by
the latter. On the other
hand,
the derivative
of
the
energy
with
respect
to
temperature,
i.e.,
the
specific
heat, is
the
same
in
the two
cases.
Thus,
these
formulas
are
equivalent
for
structures
with
unchanging
v,
whereas
the
theory
of those structures whose
v
has different values for different
states is
substantially
affected
by
the
assumption
of
a
zero-point energy.
The ideal
case
would
be that of
a
system consisting
of monochromatic structures
whose v-value
can
be
Fig.
1.