440 DOC. 45
QUANTUM
THEOREM
functions
of
the
qi.
Type
b):
There
are
infinitely many p.-systems
at the location under consideration.
In this
case
the
pi
cannot
be
represented
as
functions
of
the
qi.
One notices
immediately
that
type
b)
excludes
the
quantum
condition
we
formulated in
§2.
On the other
hand,
classical statistical mechanics deals
essentially
only
with
type
b);
because
only
in
this
case
is the microcanonic ensemble
of
one
system equivalent
to the time
ensemble.3
[p.
89]
In
summarizing we can
say:
The
application
of
the
quantum
condition
(11)
demands that there exist orbits such that
a
single
orbit determines the
pi-field
for
which
a potential
J*
exists.
{2}
§5.
The
"Rationalized Coordinate
Space."
It has
already
been noted that the
pi
are,
in
general,
multivalued functions
of
the
qi.
As
a simple example we
look
again
at the
planar
revolution of
a point
that is under the
attracting
force
of
a
fixed center.
The
point moves
such that its distance
r
from the
attracting
center oscillates
periodically
between
a
minimal value
ri
and
a
maximal
value
r2.
Looking
at
a point
of
the
space
of
the
qi,
i.e.,
a point
in the
ring-shaped
surface bounded
by
the radii
r1
and
r2,
one
realizes that the orbital
curve,
in the
course
of
time,
will
pass infinitely
often and
arbitrarily
close to this
point,
or-to
phrase
it
a
little
imprecisely-will
pass
through
it.
But the radial
component
of
velocity
has different
signs, depending upon
whether the
passing occurs on an
orbital
segment
of
increasing or decreasing
r;
and
thus,
the
pi
are
two-valued functions
of
the
qv.
The associated inconvenience for the
imagination
is best resolved
by
the well-
known method that
Riemann
has introduced to the
theory
of
functions.
Imagine
the
ring-shaped
surface
to
be doubled such that
two
congruent ring-shaped
leaves
are on
top
of
each other. We
imagine
the orbital
parts
with
positive
dr/dt
in the
top-most ring,
those with
negative
dr/dt
drawn onto the lower
ring,
with the associated vector
system
of
the
pv.
We think
of
the two
leaves
as
connected
along
the circular lines because
the orbit must
proceed
from
one
ring-shaped
leaf
to the other whenever the orbital
curve
touches
one
of the two
limiting
circles. It is
easily
understood that the
pv
from
both leaves
agree along
these circles.
Interpreted on
this double
surface,
the
pv
are
[7]
not
only
continuous functions of the
qv
but also univalent
functions-this
represents
the value
of
this
concept.
3The
microcanonic ensemble contains
systems
which
possess,
with
given
qi,
still
arbitrarily given pi (commensurable
with
the
energy values).