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).