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