436 DOC. 45
QUANTUM
THEOREM
J*
=
Ziji(qi), (8a)
where
Ji
depends upon
qi
but is
independent
of
the other
q.
SOMMERFELD's
quantum
conditions
(2)
shall then be valid for these coordinates
qi
if
the
qi
are
periodic
functions
of
t.
[p. 84]
Notwithstanding
the
great successes
that
have
been
achieved
by
the
SOM-
merfeld-Epstein
extension
of the
quantum
theorem
for
systems
of
several
degrees
of
freedom,
it still remains
unsatisfying
that
one
has to
depend on
the
separation
of
variables,
due to
(8),
because it
probably
has
nothing
to do with the
quantum problem
per
se.
In the
following
we
would like
to
suggest
a
minor modification
of
the
Sommerfeld-Epstein condition and
thereby
avoid this
deficiency.
I
will
briefly point
[6]
out the basic idea and
explain
it later in
more
detail.
§2. Modified
Formulation.
While
pdq is
an
invariant
with
systems
of
one degree
of
freedom
(i.e.,
it
is
independent
of
the choice
of
the coordinates
q),
the individual
products
pidqi
in
a
system
of
several
degrees
of
freedom, on
the other
hand, are
not
invariants; therefore,
the
quantum
condition
(2)
has
no
invariant
meaning.
Invariant
is
only
the
sum
Zipidqi
extended
over
all l
degrees
of
freedom. In order to derive
from this
sum
a multiplicity
of
invariant
quantum
conditions, one can
proceed
as
follows. Look at the
pi
as
functions
of
qi.
Or
speaking geometrically,
one can
view
pi
as a
vector
(of
"covariant"
character)
within
an
l-dimensional
space
of
the
qi.
If
I
draw
any
closed
curve
inside this
qi,-space
(and
it need
by no means
be
an
"orbital
path"
of the
mechanical
system)
then its line
integral
fZpidqi
(9)
is
an
invariant.
If
the
pi are
any
functions
of
the
qi,
then each closed
curve will,
in
general, provide
a
different value for the
integral (9).
However,
if
the vector
pi
is
derivable from
a
potential
J*,
i.e.,
if
the
following
relations hold true
dpi
dpk
dqk
dqi
=
0, (10)
respectively,
pi
=
dJ*/dq,
(10a)
then the
integral (9)
has the
same
value for all closed
curves
that
can
be
continuously
transformed into
one
another. And the
integral
(9)
vanishes for all
curves
that
can
be
[p. 85]
contracted to
a
point by
a
continuous modification.
However,
if
the
space
of the
qi