434 DOC. 45
QUANTUM
THEOREM
[p.
82]
Doc.
45
On the
Quantum
Theorem of Sommerfeld and
Epstein
by
A.
Einstein
(Presented
at
the session
of
May 11)
(cf.
above,
p.
79)
§1.
Previous Formulation. There is
hardly any more
doubt that the
quantum
condition
for
periodic
mechanical
systems
with
one degree
of freedom is
(after
Sommerfeld
and Debye)
pgq
=
pdq/dtdt
=
nh.
(1)
The
integral
is to be extended here
over one
full
period
of
movement;
q
denotes the
coordinate,
p the associated coordinate
of
momentum of the
system.
Sommerfeld's
work
on
the
theory
of
spectra proves
with
certainty
that in
systems
with several
degrees
of
freedom several
quantum
conditions have to take the
place
of this
single
quantum
condition;
in
general as many (l) as
the
system
has
degrees
of
freedom.
These l conditions
are,
according
to
Sommerfeld,
pidqi
=
nih.
(2)
As this formulation is not
independent
of
the
choice
of
coordinates,
it
can only
be
correct for
a
distinct choice
of
coordinates.
System
(2)
represents
a
distinct
statement
about the movement
only
after this choice has been made and
if
the
qi
are periodic
functions
of
time.
Further
principal progress
is due to
Epstein
(and
Schwarzschild).
The former
based his
rule for the
selection
of
the
Sommerfeld
coordinates
qi
on
Jacobi's
theorem,
which states in
a
well-known
manner:
Let
H(H(qipi))
be the
HAMILTONIAN
function of the
qi
and
pi
and
t,
which
occurs
in the canonical
equations
pi
=
-
dH/dqi
(3)
qi
=
dH/dpi,
(4)
[1]
[2]
[3]
[4]