142
DOC.
11
ARGUMENTS FOR MOLECULAR AGITATION
rigorous way,
and without
assuming any
discontinuities. The road
we
take
to
this end
is
essentially
the
same
road that Einstein and
Hopf5
used
in
a paper
that
appeared
2
years ago.
We consider the translational motion of
a
freely
movable
resonator,
which
is
firmly
attached
to, say,
a
gas molecule,
under the influence of
an
incoherent
radiation field. In that
case, at
thermal
equilibrium
the
mean
kinetic
energy
that
the
gas
obtains
as a
result
of the
radiation
must
be the
same as
that which it would obtain
as a
result
of
collisions with other molecules. In that
way
one
obtains the connection
between the
density
of
black-body
radiation and the
mean
kinetic
energy
of
a gas
molecule, i.e.,
the
temperature.
Einstein and
Hopf
find
in this
way
the
Rayleigh-Jeans
law.
We shall
now carry
out
the
same
analysis
on
the
assumption
of
a
zero-point
energy.
The influence exerted
by
the radiation
can
be broken down into
two
different
effects
according
to
Einstein and
Hopf.
First,
the rectilinear translational motion of
the
resonator molecule
experiences
a
kind
of
friction,
caused
by
the radiation
pressure
on
the
moving
oscillator.
This
force K
is
proportional
to
the
velocity
v, so
that K
=
-Pv,
at
least
if
v
is
small
compared
with the
velocity
of
light.
The momentum
imparted
to
the
resonator
molecule
during
the short time
t, during
which
v
should
not
change noticeably,
is thus
-Pvt.
Second,
the radiation
imparts
momentum
fluctuations
A
to the
resonator
molecule,
which
are
independent
of the motion
of
the
molecule
to
a
first
degree
of
approximation
and
are
the
same
in all
directions,
so
that
only
their
mean square
value
A2 during
the time
t
determines
the
kinetic
energy.
If
the latter
possesses
the value
k(T/2)
demanded
by
statistical mechanics
(for
the sake
of
simplicity,
let
the oscillator
move
only
in the x-direction and oscillate
only
in the
z-direction),
then the
following equation
must hold
according
to
Einstein and
Hopf
(l.c.
p. 1107):
Ä2
=
2kTPr.
As for the calculation of
P,
we can assume
that
only
the oscillations induced
by
the
radiation itself
must be taken into
account,
and that these
can
be calculated
as
if the
zero-point
energy were
not
present.
Hence
we can use
the value calculated
by
Einstein and
Hopf
(l.c.
p.
1111):
P
=
3co
(n
_
_v
dp
IO77-v
{
3
d\
,
To calculate
A2, we
take
(l.c. p. 1111) as
the
momentum
experienced by
the
oscillator
during
time
t
in the x-direction:
; ;
dEr
J
= !k*dt =
0 0
ux
5
A. Einstein and
L.
Hopf,
Ann.
d.
Phys.
33
(1910):
1105-1115.
[15]