228
DOC.
8
ANALYSIS
OF
A
RESONATOR'S MOTION
J2
then
appears as
the fourfold
sum over n,
m
and two
further
variables
n'
and
m'.
If
we
calculate the
mean
value
J2,
we
must
take into
account
the
fact
that the
angles
6mn
and
5m'n'
are totally
independent
of each
other,
and, thus,
that
only
those
terms in
which this
independence
does not
occur are
to be
considered
in the
averaging.
Obviously,
this
is
only
the
case
if
m
=
m' and
n
=
n',
and
we
arrive at
the desired
mean
value:
J2
=
3c3T
E-
E
sinY"
sin2
it
(n
-
m)L.
32 if
n
(n
-
m)2
T
Since
0»
E-
sin2n(n
-
m)L
=
-fsin2(v
-/a)
m
•
dfi
=
-
T
(n-m)2
'
#
T
7-J
/J
(v
/%»
-
_
T
TC2
T
and
sin2y
n
=
1
°
r-0
T5J
v
f^v
/2
J2
becomes
3c3
ar
z
-I
___ I
__
(12)
J2
-
32it~J
5B~~
CeTT2
4v0
Now
J2(7÷A)J2+271+A2,
and
since
the mean
values
J
and
I
vanish, expression (12) itself gives
the
value of the
momentum fluctuations
I2.
It
only
remains
to express the
mean
values
of
the
amplitudes
B2v0T
and
C2v0T
through the radiation
density
pv'0.
To
that end
we must again
consider the radiation
coming from
different
directions
and, as above,
relate the amplitude of the radiation
coming from a specific
direction
to
the
energy density by means
of
the
equation
A~TT
=
The amplitude
v0T
=
EA
sinç
v0T
[20]
[21]