144
DOC.
11
ARGUMENTS FOR MOLECULAR AGITATION
~Kf =
hvorf0=
"
2
2hv
=
3
hac3.7
K 8i'r4v2
Consequently:
A2- 1
hcapr.
5ir
If
one
substitutes this in
the equation
2kTPr,
one arrives at Wien's radiation
law.
Nevertheless, we will now give up the
assumption that the oscillation of the resonator that
is
induced by the radiation can
be neglected.
If
we now assume that the energy of the oscillations imparted to the
resonator by radiation produces momentum fluctuations that are independent of the
fluctuations corresponding to the zero-point energy, then we can sum the mean square
values of the two momentum fluctuations.8 Thus, we have
to
add to the value of
A2
calculated above the value calculated
by
Einstein and Hopf
(l.c. p. 1114,
equation
(15)),
and we obtain
A2
= 1
hco-p'r
+
c4o"r
5W
40ir2v3
On the other hand,
3cci I
v~dp\
A2
=
2kTP'r
=
2kTi"10
(,~P
-
3
dv)
From this one gets the differential equation for p:
___
_vdp~
hp+
C3
3
cvj.
8irv3
Solving this equation we get
8~rv2
hv
3
1w ekT_1
Planck's radiation law, and the energy of the resonator comes
to
hv +Jzv.
hv
Summary
7
M.
Planck, Warmestrahlung (6th ed.),
p. 112
(equation (168)).
8It
need hardly be emphasized that this kind of procedure can only be
justified
by our
ignorance
of
the actual resonator laws.
[18]
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