DOC.
11
ARGUMENTS FOR MOLECULAR AGITATION
143
where
f
is
the
moment
of
the
oscillator. First
we
will consider
only
the
case
when the
energy
of the
oscillation induced
by
radiation is
negligible compared
with the
zero-point energy
of the
resonator,
which is
certainly permitted
at
sufficiently
low
temperatures.
If
we
denote the maximum
moment
of
the
resonator
by
f0,
we
obtain:
f
=
fOcos
2vn0t
T
'
where T is
a
long
time interval and
n0/T0
= v0
is the
frequency
of the resonator. We
set
up
d
(§Jdx
as a
Fourier series:
dE.
Z
=
_
£c"cOS
2irn-
-
ÖJ.
dx T
Then
J
=
/
£
cn
cos
2trn
±
- ü"]/ö
cos
j2i™0
0
=
/oEC"
sin
77"
n0-
n
cos
7t
n0-
n
T
~
0
27T(n0
-
n)
T
since the
term
with
1/n0
+
n
drops out,
because
n
+
n0
is
a
very large
number. Now
if
one
sets
n/T
=
v
and
squares, one
obtains
+
oo
J2
=
A2=/02C"2|
2-1
T
8
J
/
r
sinz7r(v0
[^(v0
-
-
v)]2
v)t
d\,
-
oo
or
A2
=
1
^2
Now
(l.c.
p. 1114):
c2r
=
64
7T^V2
15 c2
Hence:
A2
=
8
1T3V2p~.f~
15
c2
If
the
resonator
possesses
the
zero-point energy
hv,6
then:
6It
turned out
that,
with the method
of
calculation sketched
here,
the
zero-point energy
must
be set
equal
to hv in
order to arrive at the Planck radiation formula. Future
investigations
must
show whether the
discrepancy
between
this
assumption
and the
assumption underlying
the
investigation on hydrogen disappears
if
the
calculation
is
more
rigorous.
[16]