A3) and

I

l ^r

^{1 (O)}

<'\t'J~jV\11-'f~n,k"l="m> l.,!J{O)>r2)

'l/lj±n,k:r:rn>

~

N'i±n,k=F'rn 1/l;±n,k'Fm>

+

(0) (0) 'l'J

~ E-

J±n, -.-m k""

-E·

J,,.. ^{1! '}

where

N'n 1,n

2

is a normalization factor (which differs by a factor of (nn)°""*

from eq 33), and the zeroth-order wave functions 11'-'A~~na> are taken to be WKB wave functions in action-angle variables.

¹²· 18

·21

·34

These wave func- lions are defined, in the angle representation, as

(A3) and

In eqs Al - A4, we consider only the zeroth-order wave functions ht}~>

and 11'-'J~n.Ji::i=rn.> with quantum numbers (n 1,n2) equal to (j ,k) and

(j ±n,k+m), respectively. These wave functions are assumed to be from a

nearly resonant manifold of states having a constant value of I

^{fJ 29}

but

also ·having weak couplings (Le., a weak resonance). These states are

moreover assumed to have the strongest mixing due to "direct", or

-52-

classical-like, first order couplings between the basis states with zeroth- order quantum numbers n

₁

and n

₂

differing byn and m, respectively (cf.

discussion in the Introduction and ref 30). For a given conserved action Ip.

²⁹

the center of the classical resonance is also assumed to be given by resonant "quantum numbers" (eq 20) close to the zeroth-order quantum numbers of the I '¢1}~> state.

For WKB basis functions, the matrix elements of the perturbation appearing in eqs Al and A2 have the form of Fourier components,

¹⁸

e.g.,

f ^{27ff 27f}

<""{O)

I VI

,,1,(0)

> = ¹ d 8 d 8 V(fint 8)e

-i(n61 - m8a)

't'J+n,k-m 'I'],~

(27r)2 a

^{0 1 2 '}

(A5)

where [int = ^{(Ifnt ,J~nt)} are some intermediate values of the actions

chosen, as in ref 18, to give good agreement between the Fourier com-

ponent and the quantum matrix element (and also to make the former

Hermitian

^{18 ) .}

If the zeroth-order states described by the wavefunctions

in eqs A3 and A4 are near the center of the classical resonance, and if the

width of the resonance contains only two or three quantum states, the

choice (lint .I~nt) = ^(!LI~) (Le., the resonant actions) gives good agree-

ment between the WKB and exact matrix elements.

¹⁸

The relationship of

the Fourier components evaluated in this way to the width of the classical

resonance is given in eqs 8 and 16 - 19. Using that relationship, and the

(exact) expression given in eq 36 for the zeroth-order energy differences

that appear in the denominators of the perturbation expansion terms,

the approximate wave functions may be rewritten as

-53-

- , rl (Q) qei7 (0)

l\llj,lc> - N j,k 1\11;,k> - ₄ [v(j) _ l] l\11;-n,k+m>

qe-i7 (O)

l

+ ₄ [v(j) + l] I it';

+n,k-m>

(A6) and

I

\llj±n,k:i=rn> -

- N ,

j±n,k:r:'rn \ll;±n,k?rn>

rll

(o) + -

4[v(j)

ge :.pj.7

^± l]1/l;,k> I

co)

l · ^(A?) The phase e

±i7

in eqs A6 and A? is defined as in ref 18, and the number v(j) is the order of the Mathieu equation (eq 29) for n

₁

= ^j (cf. discussion following eq 37).

The squared overlaps of interest in the present paper are hence given by

I

<\llJ~

^I

1/11

^,k

>

¹²

=

^I

N'j

,k 1

2

(AS)

and

I

^(O)

I"" 1

^{2 _}

^N'

^qe:.pj.7

I 1

2 '

<\11;,k 'f'j±n,k'Fm> -

1

j±n,k=Fm

4 [ll(i) ± l]

1

(A9)

which, upon expansion to. second order about q = 0, yield the same expressions as are given in the text (eqs 41 and 42). These overlap formu- las are valid as long as the two zeroth-order states 11/IJ?~> and 11/l}~n,k'Fm>

are, for a given I fJ• near the center of the classical resonance and not directly at their avoided crossing point. ³⁶ It was also assumed in the derivation given in this Appendix that the width 6./

₁

of the resonance was small ( < n) and therefore that q was small.

A final point with regard to the limits of validity of eqs 41 and 42 of the text concerns the contribution to these formulas from unphysical

"states" in the uniform semiclassical wavefunction (eq 32). These

~54-

unphysical states correspond to zeroth-order WKB wave functions (cf. eqs A3 and A4) with negative quantum numbers.

³⁹

For example, it is known for the n :m resonance that the constant action Ip equals mn

₁

+ nn

₂

+ (n+m)/2 for states in the nearly degenerate resonant manifold.

²⁹

Therefore, a resonance having resonant "quantum numbers"

(ni ,nz)

~

(j-ne-,O+mc), where c is a small parameter, would give

Ip~

mj + (n+m)/2. As a result, an a-motion state 1/ln

₁

(a) withn

₁

= ^j+n

would, for constant I 13, ^{require n}

²

to equal - m and would there by be unphysical. In this special case, the uniform serniclassical wave function would indeed have a contribution"" exp[2i(j + n)a/n] as can be seen by inspection of eq 32. A similar statement may also be made for the case in which (nLn2)

~

(O,k), and, in that instance, an unphysical state with a negative n

₁

would contribute to the uniform semiclassical wave function.

In both of the cases mentioned above, the unphysical states have a small contribution to the normalization factor Nn

₁

(eqs 33 and 34) and would therefore lead to slight errors in the overlap formulas (eqs 41 and

physical zeroth-order states that are nearest the center of the classical resonance, e.g., !1J.'}?J> ^and 11/IJ~.m>· This situation supposes that the resonant quantum numbers (ni ,n2) are some values intermediate between (j ,0) and (j-n,m), but not those giving an avoided crossing.

³⁶•37

The contribution from these physical states to the normalization factor

Nn

₁

(eqs 33 and 35) will then be much larger than that from any unphysi-

cal state(s). (The unphysical zeroth-order state in the present example

would be 11/l}~n.-m>.) More specifically, the factor [z;(j) - 1]-

²

in eqs 33

and 34 corresponding to the contribution from the physical zeroth-order

state tJ~(a) to the normalization factor for the eigenstate ti(a) will be

-55-

much larger than that for any unphyskal states. (The two interacting physical states are near, but not at, their avoided crossing point.

¹²·36

·37

Hence, the term ,... [v(j) - 1]-

²

in Ni corresponding to the contribution from the physical zeroth-order state I 1frJ~n.m> to the normalization of

\1' _1,

₀

> is large since [v(j) - 1] < 1 in that case. The term"' [v(j) + 1]-

²

corresponding to the contribution from the unphysical zeroth-order state

11'.PVn ,-m> to the normalization of the 11'i ^,o> eigenstate will be much smaller.)

In the alternative derivation of the overlap formulas given in this

Appendix, any unphysical zeroth-order states with negative quantum

numbers for n

₁

or n

₂

are omitted by fiat.

³⁹

-56-