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Semiclassical Mechanics with Molecular Applications$

M. S. Child

Print publication date: 2014

Print ISBN-13: 9780199672981

Published to Oxford Scholarship Online: October 2014

DOI: 10.1093/acprof:oso/9780199672981.001.0001

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(p.344) Appendix C Transformations in classical and quantum mechanics

(p.344) Appendix C Transformations in classical and quantum mechanics

Source:
Semiclassical Mechanics with Molecular Applications
Publisher:
Oxford University Press

This appendix concerns the connection between classical canonical transformations and the corresponding unitary transformations in quantum mechanics, with particular emphasis on the role of the classical generator. The ideas, which stem from Van Vleck (1928), were later reviewed by Fock (1959) and Van der Waerden (1967), and most recently expounded in an elegant review by Miller (1974) . The main ideas are introduced in Section C.1 and illustrated by reference to angle–action and energy–time systems in Section C.2, which includes certain results required elsewhere in the text. Section C.3 covers the theory of dynamical transformations along the lines of Miller’s path integral approach to classical S matrix theory. Section C.4 applies similar ideas to the semiclassical Green’s function. Finally, Section C.5 uses the theory to obtain uniform approximations to the Wigner 3j and 6j symbols, in a way that underlines their geometrical significance.

C.1 Classical and semiclassical transformations

A transformation (q,p)(q˜,p˜) is termed canonical in classica1 mechanics if the value of the Hamiltonian is preserved,

(C.1)
H(q,p)=H˜(q˜,p˜),

in such a way that Hamilton’s equations apply in both systems:

(C.2)
dq/dt=H/p,dp/dt=(H/q)dq˜/dt=H˜/p˜,dp˜/dt=(H˜/q˜),

conditions that are readily verified to require a unit Jacobian,

(C.3)
(q˜,p˜)/(q,p)=1.

The systematic theory of such transformations (Goldstein 1980; Percival and Richards 1982) is developed in terms of one or other of four possible generating functions F1(q,q˜), F2(q,p˜), F3(p,q˜) and F4(p,p˜), each dependent on one old and one new (p.345) variable, and with the remaining variables generated by the following partial derivative relations:

(C.4a)
p=(F1/q)q˜,p˜=(F1/q˜)q,
(C.4b)
p=(F2/q)p˜,q˜=(F2/p˜)q,
(C.4c)
q=(F3/p)q˜,p˜=(F3/q˜)p,
(C.4d)
q=(F4/p)p˜,p˜=(F4/p˜)p.

The equality between the mixed second derivatives of any of the Fi then automatically ensures the validity of (C.3), but the equations carry the awkward complication that, for example, (C.4a) yields mixed functions p(q,q˜) and p˜(q,q˜) that must be inverted in order to express (q˜,p˜) in terms of (q,p) or vice versa. The choice between the Fi in any context may be made for physical or mathematical convenience, because any one of the following interrelated set yields an equivalent transformation, as is readily confirmed by use of eqns (C.4a)–(C.4d):

(C.5a)
F2(q,p˜)=F1(q,q˜)+p˜q˜,
(C.5b)
F3(p,q˜)=F1(q,q˜)pq,
(C.5c)
F4(p,p˜)=F1(q,q˜)+p˜q˜pq.

For example, according to (C.4b) and (C.5a),

(C.7)
p=(F2/q)p˜=(F1/q)q˜+(F1/q˜)q(q˜/q)p+p˜(q˜/q)pq˜=(F2/p˜)q=(F1/q˜)q(q˜/p˜)q+q˜+p˜(q˜/p˜)q,

which reduce to the identities

(C.7)
p=p,q˜=q˜

after substitution from (C.4a) for the terms in F1.

Equations (C.4a)–(C.4d) and (C.5a )–(C.5c) are the ingredients of classical canonical transformation theory. The next step is to show that the classical generators Fi may also provide foundations for the corresponding quantum mechanical unitary transformation functions Ui, at least in a semiclassical sense. Consider for example the transformation

(C.8)
ψ(q)=U1(q,q˜)ψ˜(q˜)dq˜,

where U1(q,q˜) must be defined in such a way that (C.8) relates corresponding eigenfunctions of H(q,i/q) and H˜(q˜,i/q˜). (Note that if q˜ were an angle variable the operator i/q˜ would be modified by addition of a Maslov term, as in eqn (4.23)). Put in mathematical terms this requires that

(C.9)
H(q,i/q)ψ(q)=U1(q,q˜)H˜(q˜,i/q˜)ψ˜(q˜)dq˜,

(p.346) which is the quantum mechanical analogue of (C.1). In other words, in the light of (C.8), U1(q,q˜) must satisfy

(C.10)
H(q,i/q)U1(q,q˜)ψ˜(q˜)dq˜=U1(q,q˜)H˜(q˜,i/q˜)ψ˜(q˜)dq˜.

It is shown below that a solution to order 0 may be expressed in the form

(C.11)
U1(q,q˜)=A1(q,q˜)exp[iF1(q,q˜)/],

where F1(q,q˜) is the corresponding classical generator . To see this, note first, by virtue of (C.1), (C.4a) and (C.11), that, to order 0,

(C.12)
H(q,i/q)U1(q,q˜)=H(p,q)U1(q,q˜)=H˜(p˜,q˜)U1(q,q˜)=H˜(q˜,i/q˜)U1(q,q˜).

Secondly, provided the integrand takes the same value at both integration limits, it follows from (C.8), by repeated partial integration on the right-hand side of the equation below (again to order 0), that

(C.13)
H(q,i/q)ψ(q)=[H˜(q˜,i/q˜)U1(q,q˜)]ψ˜(q˜)dq˜=U1(q,q˜)H˜(q˜,i/q˜)ψ˜(q˜)dq˜,

which establishes the validity of (C.9).

Finally, the unitarity condition

(C.14)
U1(q,q˜)U1(q,q˜)dq˜=δ(qq)

may be used to express the pre-exponent in A1(q,q˜) in (C.11) in terms of partial derivatives of F1(q,q˜). The existence of the delta function is well understood in the normal semiclassical sense that rapid oscillations in the full integrand will lead to complete cancellation except when q=q; thus

(C.15)
A1(q,q˜)A1(q,q˜)exp(i[F1(q,q˜)F1(q,q˜)])dq˜=δ(qq).

All that remains is to fix the magnitude of A1(q,q˜) by approximating

(C.16)
F1(q,q˜)F1(q,q˜)(F1/q)q˜(qq)

and writing

(C.17)
p=(F1/q)q˜,

(p.347) as specified by (C.4a). The effect, provided that p varies monotonically with q˜, is that (C.15) may be expressed as

(C.18)
|A1(q,q˜)|2(q˜/p)q˜expip(qq)dp=δ(qq).

It follows, by comparison with the standard form (Dirac 1958)

(C.19)
exp(ixy)dx=2πδ(y),

that

(C.20)
|A1(q,q˜)|2=12πq˜pq1=12π2F1qq˜.

Taken together with what will prove a convenient phase convention (Miller 1974), this means that the semiclassical unitary transform is given in terms of the corresponding classical generator by the equation

(C.21)
U1(q,q˜)=12πi2F1qq˜1/2expiF1(q,q˜).

This assumes that p(q,q˜) is a monotonic function of q˜ at all values of q; otherwise one must sum over the various contributing branches.

Equation (C.21) applies to the transformation from one coordinate representation to another, but it is readily seen that the general structure is preserved even when one or both of the representations is a momentum one. To see this recall that ψ˜(q˜) is related to its momentum representative, ϕ˜(p˜), by the Fourier transformation (Dirac 1958)

(C.22)
ψ˜(q˜)=1(2πi)1/2exp(ip˜q˜/)ϕ˜(p˜)dp˜,

so that on substitution in (C.8)

(C.23)
ψ(q)=U2(q,p˜)ϕ˜(p˜)dp˜,

where

(C.24)
U2(q,p˜)=1(2πi)1/2U1(q,q˜)exp(ip˜q˜/)dq˜=1(2πi)1/2A1(q,q˜)expi[F1(q,q˜)+p˜q˜]dq˜.

Stationary phase reduction of the final integral now yields the analogue of (C.21), because by virtue of (C.4a) the stationarity condition

(C.25)
F1/q˜=p˜

(p.348) yields a point q˜(q,p˜) consistent with the transformation previously generated by F1(q,q˜). Moreover, the stationary value of the exponent

(C.26)
F2(q,p˜)=F1[q,q˜(q,p˜)]+p˜q˜(q,p˜)

conforms exactly to the classical expression in (C.5a). Finally, on applying the standard quadratic approximation about q˜(q,p˜),

(C.27)
U2(q,p˜)=A2(q,p˜)expiF2(q,p˜),

where

(C.28)
A2(q,p˜)=(2πi)1/2A1[q,q˜(q,p˜)][2πi/(2F1/q˜2)]1/2=12πi2F1/qq˜2F1/q˜21/2=12πiq˜qp˜1/2=12πi2F2qp˜1/2.

Equations (C.4a) and (C.4b) are required for this final reduction.

Similar analysis may be applied for any desired form of transformation, the general semiclassical result being that the unitary transform Ui(x,y˜), where x and y denote q or p, is related to the corresponding classical generator Fi(x,y˜) in the form

(C.29)
Ui(x,y˜)=±12πi2Fixy˜1/2expiFi(x,y˜),

where the sign in the pre-exponent is positive for i=2, 3 and negative for i=1,4 (Miller 1974). The generalization to f degrees of freedom takes the form

(C.30)
Ui(x,y˜)=12πif±2Fixy˜1/2expiFi(x,y˜),

where 2Fi/xy˜ denotes the Van Vleck or Hessian determinant of the second derivative matrix.

C.2 Energy–time and angle–action representations

An illuminating, yet simple, illustration of the theory is provided by the transformation between Cartesian (P,R) and energy–time (E,t) descriptions of free motion. The starting point is the exact coordinate representation of the flux normalized state,

(C.31)
ψ(R)=(P/m)1/2exp(iPR/),

(p.349) which is transformed to the energy–time picture by an F1(R,t) generator, which is itself obtained from an intermediate F2(R,E) generator determined by the Hamilton–Jacobi equation

(C.32)
HF2RE,R=12mF2RE2=E.

The solution

(C.33)
F2(R,E)=(2mE)1/2R

shows correctly that

(C.34)
P=(F2/R)E=(2mE)1/2,t=(F2/E)R=(m/2E)1/2R=mR/P.

The corresponding F1(R,t) generator is given according to (C.5a) by

(C.35)
F1(R,t)=F2[R,E(R,t)]E(R,t)t=mR2/2t,

after some rearrangement in the light of (C.34). The required semiclassical unitary transform is therefore given by

(C.36)
U1(R,t)=mR2πit21/2exp(imR2/2t).

Consequently ψ(R) transforms to

(C.37)
φ(t)=m2R2πiPt21/2expimR22t+iPRdR,

which reduces at the stationary phase level to

(C.38)
φ(t)=exp(iEt/),

where E=P2/2m.

The resultant form is exact in this simple case. It is also evident that the form for φ(t) in (C.38) must apply to any system with one degree of freedom; hence it is relatively uninformative as it stands. However, the reverse transformation

(C.39)
ψ(R)=U1(R,t)exp(iEt/)dt

offers a convenient classically based integral representation for ψ(R) in certain contexts. The harmonic oscillator representation given below is an angle–action (rather than energy–time ) version of this idea and similar energy–time forms for the harmonic oscillator and for quadratic barrier passage may be found in Feynman and Hibbs (1965), where they are derived by path integral methods.

(p.350) It is also useful to extend the above argument, for the free-motion case, to situations in which there is an additional internal degree of freedom, with Hamiltonian H0(I), because the resulting analogue of (C.38 ) is required in Chapter 10. One might, for example, start with the mixed representation

(C.40)
Ψ(R,n0)=(P/m)1/2exp(iPR/)δ(II0)

which describes a state with unit translational flux and a particular action I0C.2corresponding to the quantum number

(C.41)
n0=I0/δ,

where δ‎ is the usual Maslov index. Since the action operator is given by Iˆ=i(/α)+δ (see Sections 3.1 and 4.1), the Hamilton–Jacobi equation for the generator F2(R,α; E,I) takes the form

(C.42)
12mF2R2+H0F2α+δ=E,

and the solution is readily found to be

(C.43)
F2(R,α;E,I)=(Iδ)α+P(E,I)R,

where

(C.44)
P(E,I)={2m[EH0(I)]}1/2.

Notice, on applying (C.5a), that the variables conjugate to E and I in the transformed representation take the interesting forms

(C.45)
τ=F2/E=mR/P(E,I)αˉ=F2/I=α(H0/I)t=αωτ,

where ω is the frequency of the internal motion. The conjugate to E is of course the time (now denoted τ for later convenience), but the conjugate to I is no longer α‎ but the modified variable αˉ, which is a constant of the motion in this non-interacting picture because, by construction, α‎ varies with time as α=ωτ+ constant (see Section 4.1). The remaining step in the transformation of (C.40) is to define the mixed generator F(R,I;τ,αˉ) which may be reduced with the help of (C.45) to

(C.46)
F(R,I;τ,αˉ)=F2(R,α;E,I)(Iδ)(ααˉ)Eτ=mR22τ+(Iδ)αˉH0(I)τ.

The form of the associated unitary transform,

(C.47)
U(R,I;τ,αˉ)=12πi2FRt1/2exp[iF(R,I;τ,αˉ)/],

(p.351) then implies, after some manipulation, that

(C.48)
φ(τ,αˉ)=U(R,I;τ,αˉ)Ψ(R,n0)dRdI=1(2π)1/2exp(in0αˉiEτ/),

which is the result employed (as eqn (10.24)) in Chapter 10.

The scaled harmonic oscillator, with Hamiltonian

(C.49)
H=12(p2+q2)=n+12,

offers further insights into the scope of the transformation theory. Here the quantum number n rather than the action, I=n+12, is conveniently taken as the variable conjugate to the angle α‎, because according to eqn (4.23) its operator equivalent,

(C.50)
nˆ=Iˆ12=i/α,

has the form assumed in eqns (C.9)–(C.13) (=1 in the present units).

Following the above Hamilton–Jacobi route (eqns (C.33)– (C.35)) one readily finds that

(C.51)
F1(q,α)=F2[q,n(q,α)]n(q,α)α=12q2tanα+12α.

Hence, following (C.4a),

(C.52)
p=F1/q=qtanαn=(F1/α)=12q2sec2α12,

which rearrange to the canonical form (see (4.21))

(C.53)
q=(2n+1)1/2cosα,p=(2n+1)1/2sinα.

The analogue of eqns (C.8) and (C.10) therefore suggests a representation for the Cartesian wavefunction ψn(q) of the form

(C.54)
ψn(q)=U1(q,α)φn(α)dα,

where

(C.55)
U1(q,α)=A1(q,α)expi2q2tanα+i2α

and, according to (4.25),

(C.56)
φn(α)=(2π)1/2exp(inα).

(p.352) The integration limits in (C.54) will be chosen later. Turning to the function A1(q,α), the semiclassical form,

(C.57)
A1sc=12πi2F1qα1/2=qsec2α2πi1/2,

cannot be exact because ψn(q) must be either even or odd in q, for even or odd values of n. One can, however, obtain an exact representation in this special case because eqn (C.12), which takes the form

(C.58)
122q2+12q2U1(q,α)=iq+12U1(q,α),

can be solved exactly by substitution from (C.55) to yield

(C.59)
i2tanαA+iqtanαAq122Aq2=iAα.

Two fundamental solutions may be recognized:

(C.60)
A(q,α)=Bnsec1/2αfor n=2k=Bnqsec3/2αfor n=2k+1;

other forms involving higher powers of q may be reduced to these by integration by parts in (C.54). One interesting feature is that the coefficient choices

(C.61)
Bn=(2π)1/2(2n+1)1/4for n=2k=(2π)1/2(2n+1)1/4for n=2k+1

bring (C.60) into coincidence with (C.57) (apart from a phase factor) in the semiclassical sense that q and α‎ are related by (C.53).

The following integral representations are therefore suggested for ψn(q):

(C.62a)
ψn(q)=Cn(2n+1)1/42ππ/2π/2sec1/2αexpi2q2tanα+in+12αdα

for n even, and

(C.62b)
ψn(q)=Cnq2π(2n+1)1/4π/2π/2sec3/2αexpi2q2tanα+n+12αdα

for n odd. Here the integration limits have been chosen to ensure that ψn(q) is real (as may be verified by the substitution α=α) and such that sec1/2α is real over the integration range.

(p.353) Equations (C.62a) and (C.62b) have been expressed in this form to emphasize their semiclassical origin, and also because the as yet undetermined factors Cn are close to unity. To see the latter note that the standard form

(C.63)
ψn(q)=π1/4(2nn!)1/2Hn(q)exp12q2

for even n may be shown to take the following value at q=0:

(C.64)
ψn(0)=(1)n/2π1/42n/2[Γ(n+1)]1/2/Γn2+1,

where Γ(x) is the gamma function (Abramowitz and Stegun 1965), while the integral in (C.62a) at q=0 gives (Gradsteyn and Ryyzhik 1980)

(C.65)
20π/2sec1/2αcos(n+12)αdα=23/2πΓ(32)/Γ(n2+1)Γ(1n2).

It follows by use of the gamma function duplication formula (Abramowitz and Stegun 1965) that

(C.66a)
Cn=(2n+1)1/42Γn2+1/Γn+121/2for n=2k,

with the numerical values C0=1.0623,C2=1.0046,C4=1.0015 and Cn1 as n. A similar comparison for ψn(0) leads to

(C.66b)
Cn=(2n+1)1/4Γn+12/2Γn2+11/2for n=2k+1,

!! with the values C1=0.9885,C3=0.9976,C5=0.9980, etc.

This classically based integral representation has close analogies with forms given by Feynman and Hibbs (1965), Ovchinnikova (1974) and Boyer and Wolf (1975). It also provides a direct route to the harmonic uniform approximation discussed in Section B.4.

C.3 Dynamical transformations and the classical S matrix

Miller (1974) and coworkers have made elegant use of the foregoing theory in deriving semiclassical transition amplitudes, or S matrix elements, from the classical limit of the Feynman propagator (Feynman and Hibbs 1965). The underlying idea is that passage from a phase point (p1,q1) at time t1 to (p2,q2) at t2 constitutes a dynamical transformation whose classical generator determines the quantum mechanical propagator .

The simplest illustration applies to the case of free motion, for which it is readily verified that the generator

(C.67)
F1(q2t2;q1t1)=m(q2q1)22(t2t1)

(p.354) yields the correct momenta:

(C.68)
p2=F1/q2=p1=(F1/q1)=m(q2q1)(t2t1).

The corresponding unitary transform, denoted for consistency with what follows by

(C.69)
K(q2t2;q1t1)=12πi2F1q1q21/2exp[iF1(q2t2;q1t1)/],

may be confirmed to propagate the wavefunction from time t1 to t2 in the sense that (Feynman and Hibbs 1965)

(C.70)
ψ(q2t2)=K(q2t2;q1t1)ψ(q1t1)dq1.

To see this, note that the exact initial free-motion wavefunction, at energy E=p2/2m, is

(C.71)
ψ(q1t1)=exp[i(pq1Et1)/],

so that on using the standard integral

(C.72)
exp(iax2)dx=(2πi/a)1/2,

(C.70) yields an identical form to (C.71) except that the subscript 2 appears in place of 1. Equations (C.67)–(C.71) are exact for this simple case.

The corresponding exact propagator in more general situations may be expressed as a path integral (Feynman and Hibbs 1965),

(C.73)
K(q2t2;q1t1)=exp{iS[q(t)]/}dpath,

where S is the action along a particular path,

(C.74)
S[q(t)]={p(t)q˙(t)H[p(t),q(t)]}dt,

and the integral in (C.73) is taken over all possible phase space paths with end points q1 at t1 and q2 at t2. A convenient way to approximate S for a given path is to break it into short time segments of length Δti along which the momentum pi consistent with H at a given energy may be taken as constant; hence, following (C.67)–(C.68) the ith action increment becomes

(C.75)
ΔSi=F1(q2t2;q1t1)ViΔti=mΔqi2/2ΔtiViΔti,

where mΔqi=piΔti.

(p.355) Such expedients are, however, unnecessary in the semiclassical treatment of small isolated species, because the fluctuations in S[q(t)] from one path to another may be assumed to be so large that constructive interference occurs only around the paths of minimum action, which are by Hamilton’s principle those traced out by the classical trajectories from q1 at t1 to q2 at t2 (Goldstein 1980; Percival and Richards 1982) . Hence in the semiclassical limit (Feynman and Hibbs 1965)

(C.76)
K(q2t2;q1t1)12πi2Sclq1q21/2exp[iScl(q2t2;q1t1)/],

where

(C.77)
Scl(q2t2;q1t1)=t1t2[pq˙H(p,q)]dt,

with the integral taken along a classical trajectory. Note that the pre-exponent in (C.76), which includes contributions from the immediately neighbouring paths, is fixed by unitarity (compare eqns (C.14)–(C.20)). The sum of terms of the Cartesian form in eqn (C.76), taken over all ‘root trajectories’ from q1 to q2 in time t2t1, is known is the Van Vleck (1928) propagator .

Henceforth it is assumed that the system is conservative (i.e. that H contains no explicit time dependence), in which case terms in H (or E) may be factored out of the multidimensional analogue of (C.76) in the form

(C.78)
κ(q2;q1)=K(q2t2;1t1)exp[iE(t2t1)/]=a(q2;q1)exp[iW(q2;q1)/],

where

(C.79)
W(q2;q1)=q1q2pdq,

with the integral taken along a trajectory from q1 to q2. This reduced propagator governs the evolution of the spatial part of the wavefunction in the form

(C.80)
ψ(q2)=κ(q2;q1)ψ(q1)dq1.

Since the trajectory used to determine κ(q1;q2) assumedly depends on q1 and q2, the initial and final momenta, generated by the equations

(C.81)
p1i(q2;q1)=(W/q1i),p2i(q2;q1)=W/q2i,

are also strictly dependent on q2 and q1. Note also that the double-ended nature of the boundary conditions may allow more than one classical path from q1 to q2, in which case (C.78) must be replaced by an appropriate sum.

(p.356) We turn now to the relation between the propagator and the scattering matrix in collisional applications of the theory. Miller (1974) argues that an observable transition from one quantum state n1 to another n2 involves not a coordinate change, but a change of action from I1 to 2. Hence the S matrix depends on a propagator κ(I2;I1) in an angle–action representation. Not surprisingly, if the transformations are followed through at the stationary phase level, the result is the primitive semiclassical approximation of eqn (9.37). Here, however, the aim is to obtain the integral representation of eqn (9.28), from which other results are deduced in Section 10.2.

It is assumed for simplicity that the system involves translational variables (P,R) and a single set of internal variables given in either angle–action (I,α) or Cartesian (p,x) form. It is also useful to remember, according to the discussion in Section C.2, that the semiclassical conjugate to α‎ is not I but

(C.82)
N=Iδ=n,

where δ‎ is the Maslov index and n the quantum number.

Before proceeding to the details, it is important to establish the dependence of these variables upon one another. For example, in considering a trajectory from asymptotic variables (P1, R1, N1, α1) to (P2, R2, N2, α2) the final quantities P2 and N2 are dependent on the choice of all four initial variables, R2 may be chosen arbitrarily, and α2 depends on R2 as well as the initial set, because (see eqn (4.5))

(C.83)
αi=ω(Ni)t+constant,

where ω(Ni)=H0/Ni, and a different value of R2 implies a different final time. Closer inspection shows that P2 and N2 are also interdependent because

(C.84)
E=12mP2+H0(I),

and the same is true of P1 and N1 if the energy E is taken as an independent variable. Finally, it is convenient to accommodate the interdependence of αi and Ri by defining the modified angle

(C.85)
αˉi=αi+ω(Ni)mRi/Pi=αi+ω(Ni)τi,

which is in effect the constant in eqn (C.83). The upshot is that the variables (N,αˉ,E,τ), which were introduced earlier in eqns (C.42)–(C.48), are the most convenient for conceptual purposes, because E is fixed and the outcome of the trajectory is independent of the τi, provided the motion starts and ends in an asymptotic region. The essential dependence is therefore either (N2,αˉ2) on (N1,αˉ1), or (N1,N2) on (αˉ1,αˉ2), or vice versa. The aim of what follows is to transform from the physically convenient practical variables, used to determine the trajectory, to the conceptual (E,τ,N,αˉ) system, and then to relate the S matrix to the propagator κ(αˉ2;αˉ1). The necessary transformation steps will be followed via the classical generators.

(p.357) Two possibilities are considered, according to whether the motion is followed in the (P, R, N, α‎) or (P, R, p, x) system. In the first case the dynamical transformation itself is induced by the F1 type generator

(C.86)
W(α2R2;α1R1)=α1α2Ndα+R1R2PdR,

and in the second case

(C.87)
W(x2R2;x1R1)=x1x2pdx+R1R2PdR,

with the integrals taken along classical trajectories.

The easiest route to the corresponding (τ1,αˉ1) to (τ2,αˉ2) generator involves an intermediate transformation to the (N,R) representation, using as the generator either

(C.88)
ρ(N2R2;N1R1)=W(α2R2;α1R1)N2α2+N1α1=N1N2αdN+R1R2PdR

or

(C.89)
ρ˜(N2R2;N1R1)=W(x2R2;x1R1)F2(x2,N2)+F2(x1,N1),

(p.358) where F2(x,N) is the type 2 generator from (p,x) to (N,α), given according to eqn (4.12) by

(C.91)
F2(x,N)=ax{2μ[H0(I)Vint(x)]}1/2dx,0<α<π=2πIax{2μ[H0(I)Vint(x)]}1/2dxπ<α<2π.

It is readily verified from (C.89) that

(C.91)
ρ/N2=α2,ρ/N1=α1,ρ/R2=P2,ρ/R1=P1,

and similarly for p˜ because by construction

(C.92)
F2/x=p,F2/N=α.

The next step is to transform to the (αˉ,τ) representation with the help of eqns (C.42)–(C.48). The resulting generator takes the form

(C.93)
λ(αˉ2,τ2;αˉ1τ1)=ρ(N2R2;N1R1)G(N2R2;αˉ2τ2)+G(N1R1;αˉ1τ1),

where

(C.94)
G(NR;αˉτ)=mR22τH0(I)τNαˉ;

alternatively, ρ˜(N2R2;N1R1) may be employed in place of ρ(N2R2;N1R1). Finally, (C.93) may be cast into a more appealing form by noting that the variables αi and Pi,

(C.95)
αi=(G/Ni)=αiˉ+(H0/Ni)τi=αiˉ+ω(Ni)τiPi=G/Ri=mRi/τi,

implied by (C.94) are necessarily consistent with those given by (C.91). Hence, on combining (C.83) with (C.88)–(C.95),

(C.96)
λ(αˉ2τ2;αˉ1τ1)=N1N2αdNP1P2RdP+N2αˉ2N1αˉ1+E(τ2τ1).

Note that although the Ni and Pi are dependent on the αˉi, the function λ(αˉ2τ2;αˉ1τ1) generates the proper conjugates to αˉi and τi in the form

(C.97)
λ/αˉ2=N2,λ/αˉ1=N1,λ/τ2=(λ/τ1)=E,

because, by virtue of (C.84) and (C.95), the terms in Ni/αˉj cancel exactly with those in Pi/αˉj.

This function λ(αˉ2τ2;αˉ1τ1) plays the role of W(q2;q1) in (C.79); hence the semiclassical propagator takes the form

(C.98)
κ(αˉ2τ2;αˉ1τ1)=a(αˉ2τ2;αˉ1τ1)exp[iλ(αˉ2τ2;αˉ1τ1)/].

Moreover, it is clear from the energy dependences of iλ(αˉ2τ2;αˉ1τ1) and of the typical asymptotic wavefunction given by (C.48)

(C.99)
ψni(αˉiτi)=(2π)1/2exp[i(NiαˉiEτi)/],

that propagation with respect to τ is purely multiplicative, as one might expect by analogy with (C.74). The propagation equation is therefore

(C.100)
ψ(n1)(αˉ2τ2)=02πκ(αˉ2τ2;αˉ1τ1)ψn1(αˉ1τ1)dαˉ1,

where the superscript on the left-hand side implies propagation of ψn1. It follows, by analogy with (C.76), that

(C.101)
κ(αˉ2τ2;αˉ1τ1)=12πi2λαˉ1αˉ21/2exp[iλ(αˉ2τ2;αˉ1τ1)/].

(p.359) The relation between κ(αˉ2τ2;αˉ1τ1) and the S matrix is now readily established by noting that ψ(n1)(αˉ2τ2) must decompose as

(C.102)
ψ(n1)(αˉ2τ2)=n2Sn1n2ψ(n2)(αˉ2τ2),

so that on projecting out the n20th term from (C.100)

(C.103)
Sn10n20=02π02πψn20(αˉ2τ2)κ(αˉ2τ2;αˉ1τ1)ψn10(αˉ1τ1)dαˉ1dαˉ2=12π02π02πi2πn1αˉ2αˉ21/2exp[iΛn10n20(αˉ1αˉ2)]dαˉ1dαˉ2,

where

(C.104)
Λn10n20=n1n2αdnk1k2Rdk+(n2n20)αˉ2(n1n10)αˉ1.

Note that, for ease of comparison with (9.29), Ni and Pi have been replaced by

(C.105)
ni=Ni/andki=Pi/,

that the quantum numbers ni0 have been given the superscript to distinguish them from the ni which are functions of the αˉi (as are the ki), and that the mixed second derivative of λ in (C.101) has been evaluated with the help of (C.97).

Equation (C.104) is the most general integral form for the S matrix but it assumes knowledge of the trajectories from all initial to all final angles αˉ at the energy of interest. The following simpler form,

(C.106)
Sn10n20=12π02παˉ1αˉ2n11/2exp[iΔn10n20(αˉ2)]dαˉ2,

where

(C.107)
Δn10n20=n1n2αdnk1k2Rdk+(n2n20)αˉ2,

may be obtained by stationary phase integration with the help of the identity

(C.108)
n1αˉ2αˉ1n1αˉ1αˉ21=(n1,αˉ1)(αˉ2,αˉ1)(αˉ1,αˉ2)(n1,αˉ2)=αˉ1αˉ2n1,

where n2, k2 and αˉ1 are now to be taken as functions of n10 and αˉ2, while k1 is of course determined by E and n10. Reversal of the functional dependence of αˉ1 on αˉ2 transforms (C.106) to the form

(C.109)
Sn10n20=12π02παˉ2αˉ1n101/2exp[iΔn10n20(αˉ1)]dαˉ1,

(p.360) which is identical with the initial value representation of eqn (9.28). This closes the present analysis except to note that eqns (C.104) and (C.107) employ expressions for Λn10n20 and Δn10n20 derived from W(α2R2;α1R1) on the assumption that the trajectory is followed in the (N,α,P,R) system. It is readily verified by comparison between (C.88) and (C.89) that W(x2,R2;x1R1) would yield an equivalent expression, obtained by substituting

(C.110)
n1n2αdn=1x1x2pdxF2(x2,N2)+F1(x1,N1)

in eqn (C.107), which is applicable in situations where the internal motion is most conveniently followed in Cartesian variables.

C.4 The semiclassical Green’s function

The semiclassical Green’s function is given by Gutzwiller (1990) as a half Fourier transform of the Van Vleck (1928) propagator in eqn (C.76):

(C.111)
G(qq;E)=1i0dtK(qq;t)eiEt/=1i0dt12πifdet2Sqqt1/2×eiS[q,q,t]/+iEt/+phase,

where the subscript on the pre-exponent is a reminder that the derivatives are taken at fixed t. Moreover, S/t=E(q,q,t)—the negative of the energy of the trajectory from q to q in time t. The exponent is therefore stationary at times t such that E(q,q,t) coincides with the target energy E. Subsequent manipulations are simplified by the Legendre transformation

(C.112)
S[q,q,t]=qqp(q,E)dqEt=Wq,q,EEt,

where the reduced classical action integral

(C.113)
W(qq;E)=qqpdq

is the stationary value of the exponent in (C.111). Consequently the stationary phase approximation to G(qq;E) is given by

(C.114)
Gsc(qq;E)2π2πi(f+1)/2traj2St21det2Sqqt1/2×expiW(qq;E)/iμπ/2,

(p.361) where the sum is taken over all trajectories from q to q at energy E, and the Maslov index μ‎ counts the number of sign changes of the determinant at the so-called conjugate points along the trajectory (Gutzwiller 1990).

Significant simplifications of eqn (C.114) may be obtained by recognizing that W/E=t(E). In the first place it follows that 2S/t2=E/t=t/E1=2W/E21. Secondly, following Gutzwiller (1990), eqn (C.112) may be used to express the Van Vleck determinant in terms of det2W/qq. As a preliminary, note that

(C.115)
tqit=2WqiE+2WE2Eqit=0,

from which it follows that

(C.116)
qjSqit=qjWqiE+WEEqit=2Wqjqi+2WqjEEqi=2Wqjqi2WqjE2WqiE2WE21.

Terms like 2E/qjqi vanish because E=H(p,q) is independent of q. When expressed in terms of determinants, this means that

(C.117)
det2Sqjqit=2WE21det2W/qq2W/qE2W/Eq2W/E2,

where the coordinate derivatives of W are taken at constant E.

The determinant on the right must, however, be evaluated with care (Gutzwiller 1990), because det(2W/qq)=0. To see this note by the Hamilton–Jacobi equation that

(C.118)
H(p,q)=HW/q,q=E.

Hence, on differentiating with respect to qi,

(C.119)
pjHpjpjqi=qjq˙j2Wqjqi=0,

which implies that the matrix (2W/qq) is singular. Moreover, the nature of the singularity may be understood by adopting a coordinate system such that q1 runs along a particular classical trajectory, and that q2,q3,,qf are perpendicular to it. Consequently q˙=(q˙1,0,...,0), which means that 2W/qjqi=0 whenever (p.362) i=1 or j=1. It also follows, by differentiating (C.118) with respect to E, that q˙12W/q1E=1. Consequently

(C.120)
det2Sqqt=q˙1q˙12WE21det2Wq˜q˜E,

in which (2W/q˜q˜) denotes the reduced Hessian matrix with indices i or j restricted to (2,3,,f).

Taken in conjunction with eqn (C.114) this means that

(C.121)
G(qq;E)2π12πi(f+1)/2trajq˙1q˙11det2Wq˜q˜E1/2×eiW(qq;E)/iμπ/2.

In conclusion it should be noted that eqns (C.115)–(C.117) may be extended to imaginary time propagation, with t=iτ, as required, for example in the instanton theory of chemical reactions in Section 11.4. The difference is that eqn (C.112) is replaced by

(C.122)
Sˉ[q,q,t]=qqpˉ(q,E)dqEt=Wˉq,q,E+Eτ,

in which pˉ is the momentum on the upturned potential energy surface, Vˉq=V(q) and τ(E)=Wˉ/E. Thus

(C.123)
det2Sˉqqt=q˙1q˙12WˉE21det2Wˉq˜q˜E,

or in the context of eqn (11.64)

(C.124)
det2Sˉqqt=q˙1q˙1τE1det2Wˉq˜q˜E.

C.5 Angular momentum coupling coefficients

The angular momentum vector coupling coefficients relate to transformations between one angular momentum coupling scheme and another. Semiclassical approximations for them have been obtained in a variety of ways (Racah 1951; Beidenharn 1953; Adler et al. 1956; Brussard and Tolhoek 1957; Ponzano and Regge 1968; Miller 1974; Schulten and Gordon 1975) . Following Miller (1974), they are here derived by use of classical generators, in a way that also borrows illuminating geometrical insights from Ponzano and Regge (1968) and Schulten and Gordon (1975).

(p.363) To start with the simplest case, the 3j symbol is a symmetrical version of the Clebsch–Gordan coefficient, which relates the uncoupled states |j2m2|j3m3 to the coupled states |(j2j3)j1m1 by the equations (Brink and Satchler 1968; Zare 1988)

(C.125)
j1j2j3m1m2m3=(1)j2j3+m1(2j1+1)1/2Tj1m2,

where the elements Tj1m2 define the unitary transformation

(C.126)
|(j2j3)j1m1=m2Tj1m2|j2m2|j3m3,

subject to

(C.127)
m1+m2+m3=0.

Note, for future reference, that the sum over m2 in (C.126 ) could be more symmetrically regarded as a sum over m2m3 at fixed m1.

Appendix C Transformations in classical and quantum mechanics

Fig. C.1 A graphical representation of eqn (C.125). The lower panel shows |Tj1m2|2 for j1=120, as given by (C.146) (curves) and by exact recursion relations (points). Wigner’s (1959) estimate is [(2j1+1)/4πA], where A is the area of the projected triangle. (Reprinted with permission from (Schulten and Gordon 1975). Copyright 1975, AIP Publishing LLP.)

The derivation that follows makes use of the classical analogue of eqn (C.126) depicted in Fig. C.1, in which the Ji have fixed magnitudes ji+12 (in units of ) with J1 fixed and J2 and J3 free to rotate around it, subject to the triangular constraint

(C.128)
J1+J2+J3=0,

consistent with (C.127). This choice of the senses of the Ji ensures the known symmetry of the 3j symbol (Brink and Satchler 1968; Zare 1988) under cyclic permutation of the indices. Seen in relation to the above equations, the triangle with a fixed edge J1 represents the left-hand side of (C.126), and the freedom of rotation about J1 corresponds to the uncertainty in m2m3 represented by the sum over m2 on the right, all other actions being fixed by |J1|,m1 and m2m3. The method of derivation involves constructing a semiclassical unitary transform equivalent to Tj1m2 in terms of an F4 type generator between the magnitude J1=j1+12, on one hand, and the difference m2m3 on the other, subject to the geometrical constraints in Fig. C.1. One vital observation, which may be confirmed by reference to Fig. 4.3, is that the angle α1, measured around the vector J1, constitutes the conjugate variable to the magnitude J1. Similarly, the azimuthal angles βi in the component representation

(C.129)
Ji=(Picosβi,Pisinβi,mi),

where

(C.130)
Pi=(Ji2mi2)1/2,

are the angles conjugate to the z components mi. Hence, bearing in mind the known symmetry of the 3j symbol under permutation of the indices, the natural choice for F4 (obtained by setting F3=0 in (C.5b)) may be written

(C.131)
F4=Ω(3)=i(Jiαi+miβi),

(p.364) where the angles αi and βi are geometrically determined functions of the Ji and mi. Moreover, the absolute orientation about the z axis is irrelevant, which makes it convenient to transform temporarily to new conjugate variables {Li,ηi} defined by the equations

(C.132)
m1=L2L3,η1=β3β2m2=L2L1,η2=β1β3m3=L1L2,η3=β2β1,4pt

(p.365) a construction that also later aids extension of the theory to 6j symbols; note also that Tj1m2, which determines the transformation between J1 and the difference m2m3, relates more symmetrically to a transformation from J1 to L1 at fixed L1+L2+L3.

In these new variables (C.131) transforms to

(C.133)
Ω(3)=i(Jiαi+Liηi),

which is the desired form for the F4 generator, because the angles αi and ηi are determined, in terms of the Ji and mi, by the following geometrical identities (Ponzano and Regge 1968):

(C.134)
cosα1=J12(m3m2)+(J32J22)m14P1F(J1,J2,J3),sinα1=J1A(P1,P2,P3)P1F(J1,J2,J3),

and

(C.135)
cosη1=P12P22P322P2P3=J12J22J322m2m32P2P3,sinη1=2A(P1,P2,P3)P2P3,

where F(J1,J2,J3) and A(P1,P2,P3) are respectively the areas of the triangle (J1,J2,J3) and its projection (P1,P2,P3) in the xy plane:

(C.136)
F(J1,J2,J3)=14[(J1+J2+J3)(J1+J2+J3)(J1J2+J3)(J1+J2J3)]1/2,A(P1,P2,P3)=14[(P1+P2+P3)(P1+P2+P3)(P1P2+P3)(P1+P2P3)]1/2.

Similar expressions for (α2,η2) and (α3,η3) are obtained from (C.134) and (C.135) by cyclic permutation of the indices. It is also evident by inspection of Fig. C.1 that any angular momentum set {Ji,mi} is equally consistent with a mirror-image geometry (obtained by reflection in the (J1,z) plane), in which the signs of the αi and ηi are reversed.

The validity of Ω(3) as an F4 generator, such that

(C.137)
Ω(3)/Ji=αi,Ω(3)/Li=ηi,

is most simply confirmed by direct but lengthy differentiation, while a more elegant but more abstract proof is outlined by Schulten and Gordon (1975). The important consequence is that the analogue of (C.29) yields the desired unitary transform

(p.366)

(C.138)
U4(3)(J1,L1)=12πi2Ω(3)J1L11/2exp(iΩ(3))=J12A1/2cos(Ω(3)+π/4),

where the sum that leads to the cosine is taken over the two equivalent orientations of the triangle, and the term π/4 arises because

(C.139)
2Ω(3)J1L1=α1L1=J12A

for the orientation specified by (C.134)–(C.135).

The final result for the semiclassical 3j symbol, obtained by substituting U4(3) for Tj1m2 in (C.125), is therefore

(C.140)
tj1j2j3m1m2m3=(1)σ1(2πA)1/2cos(Ω(3)+π/4),

where Ω(3) is most conveniently expressed for computational purposes in the following form derived from (C.127), (C.131) and (C.132):

(C.141)
Ω(3)=J1α1+J2α2+J3α3m2η1+m1η2.

Here Ji=ji+12, the angles αi and ηi are determined by (C.134) and (C.135) plus their cyclic permutations, and the area A(P1,P2,P3) is given by (C.130) and (C.136). The phase term σ‎ may be determined by comparison between (C.142) and the identity (Brink and Satchler 1968)

(C.142)
j1j2j3000=(1)(j1+j2+j3)2(2πA)1/2[1+(1)j1+j2+j3],

from which it may be verified that

(C.143)
σ=j1+j2j3+1.

Equation (C.140) was first given by Miller (1974). If two of the ji are much larger than the other, say j2j3j1, it is simpler to employ the 3j equivalent of the earlier result of Brussard and Tolhoek (1957), namely

(C.144)
j1j2j3m1m2m3(1)m3j3(2j3+1)1/2Dm1,j3j2j1(0,θ,0),

where Dμνj(0,θ,0) is the rotation matrix (Brink and Satchler 1968) and

(C.145)
cosθ=m3j3+1/2.

Schulten and Gordon (1975) obtained eqn (C.140) (albeit with an apparent sign error in the expression for cosα1) by the interesting device of converting the 3j symbol recurrence relations to JWKB-type differential equations. This approach has (p.367) the added benefit that it also covers the non-classical range of {ji,mi} combinations for which the area A(P1,P2,P3) is imaginary and for which the angles given by eqns (C.134) and (C.135) are complex. The following uniform approximations cover both real and complex situations and, as usual, smooth out the spurious singularities at the boundaries (Schulten and Gordon 1975):

(C.146)
(j1j2j3m1m2m3)=(1)σz1/4(2A)1/2×{cosΩ0Ai(z)sinΩ0Bi(z)Ω<Ω0cosΩ0Bi(z)sinΩ0Ai(z)Ω>Ω0,
Appendix C Transformations in classical and quantum mechanics

Fig. C.2 Semiclassical (curves) and exact (points) 3j symbols, (a) j10060106050 and (b) 120607010m10m. (Reprinted with permission from (Schulten and Gordon 1975). Copyright 1975, AIP Publishing LLP.)

where the Airy functions Ai(z) and Bi(z) (Abramowitz and Stegun 1965) have argument

(C.147)
z=±32|ΩΩ0|2/3,

with the positive and negative signs taken in the real and complex cases respectively. Finally,

(C.148)
Ω0(3)=J1α10+J2α20+J3α30m2η10+m1η20,

αi0 and ηi0 being the angles on the nearest boundary:

(C.149)
αi0={0if 0Reαiπ/2πif π/2Reαiπηi0={0if 0Reηiπ/2πif π/2Reηiπ.

It is evident from Fig. C.2 that this uniform result provides an excellent approximation to the exact 3j symbol, regarded as a function of j1 at fixed m2 or vice versa. One also clearly sees the familiar oscillatory patterns in the bright classical ranges flanked by exponential decay into the shadow regions.

The theory of the 6j symbol follows similar lines. In this case

(C.150)
j1j2j3l1l2l3=[(2j1+1)(2l1+1)]1/2Tj1l1,

where the transformation elements Tj1l1 connect states with common internal labels, l2,l3 and j3, and a common resultant j2 but with different intermediate coupling schemes. In the j1 set, labelled |(l2,l3)j1,j3;j2,l2 and l3 have a resultant j1, which couples to j3 to form j2, while in the l1 set, labelled |(l2,j3)l1,l3;j2, the initial coupling is between l2 and j3 to form l1, which then combines with l3 to reach the same resultant j2. Thus Tj1l1 defined by the equation

(C.151)
|(l2,l3)j1,j3;j2=l1Tj1l1|(l2,j3)l1,l3;j2.

(p.368) (p.369) The corresponding classical situation is illustrated in Fig. C.3, in which Ji=ji+12 and Li=li+12. Thus, in geometrical terms, the left-hand side of (C.151) corresponds to the fixed triangles (J1,L2,L3) and (J1,J2,J3) with a common edge J1, while variation of the arbitrary dihedral angle α˜1 about J1 gives rise to a range of L1 values, consistent with the triangles (L1,L2,J3) and (L1,J2,L3), which is the classical analogue of the uncertainty in l1 implied by the sum on the right-hand side of (C.151).

This geometrical picture also shows that the choice of the symbol Li in eqns (C.128)–(C.143) was far from accidental because, as underlined by Ponzano and Regge (1968), the 3j symbol may be regarded as a large li approximation to its 6j counterpart;

Appendix C Transformations in classical and quantum mechanics

Fig. C.3 A graphical representation of eqn (C.151). The lower panel shows |Tj1l1|2 for l1=190, as given by (C.161) (curves) and by exact recursion relations (points). Wigner’s (1959) estimate is [(2l1+1)/4πV], where V is the volume of the tetrahedron. (Reprinted with permission from (Schulten and Gordon 1975). Copyright 1975, AIP Publishing LLP.)

(C.152)
j1j2j3l1l2l3(1)j1+j2+j3+2(l1+l2+l3)(2R)1/2j1j2j3m1m2m3,

where

(C.153)
m1=l2l3,m2=l3l1,m3=l1l2R=13(l1+l2+l3).

The geometrical analogue is that the edges Li of the 6j tetrahedron in Fig. C.3 go over, as Li, to those of a vertical prism, with an upper triangular face (J1,J2,J3), as in Fig. C.3, and a mean height R. Not surprisingly, therefore, the dihedral angles α˜i and η˜i in Fig. C.3 play precisely the same role in the 6j context as that played by αi and ηi in the theory of the 3j symbol. In particular the function

(C.154)
Ω(6)=i(Jiα˜i+Liη˜i)

again acts as the proper F4 type generator, in the sense that

(C.155)
Ω(6)/Ji=α˜iand Ω(6)/Li=η˜i,

but the expressions for α˜i and η˜i in terms of Ji and Li are much more complicated, namely (Ponzano and Regge 1968; Schulten and Gordon 1975):

(C.156)
cosα˜1=2J12L12J12(J12+L22+L32)J22(J12+L22L32)J32(J12L22+L32)16F(J1J2J3)F(J1L2L3)sinα˜1=3J1V2F(J1J2J3)F(J1L2L3)cosη˜1=2J12L12L12(L12+L22+L32)J22(L12+L22J32)L32(L12L22+J32)16F(L1J2L3)F(L1L2J3)sinη˜1=3L1V2F(L1J2L3)F(L1L2J3).

(p.370) (p.371) Here the various area terms F(J1J2J3) etc. are given by (C.136) and V is the volume of the tetrahedron, which is conveniently calculated from the Caley determinant

(C.157)
V2=12880L12L22L321L120J32J221L22J320J121L32J22J120111110.

Similar expressions for the remaining α˜i and η˜i are again obtained by cyclic permutation. Furthermore each angular momentum set {Ji,Li} is again realizable in two ways, one given by (C.156) and the other in which the signs of all the angles are reversed. The validity of (C.155) may be verified by direct differentiation, or by the more abstract argument outlined by Ponzano and Regge (1968).

The final step in the primitive semiclassical argument is to note, by virtue of (C.155) and (C.156), that

(C.158)
2Ω(6)J1L1=α˜1L1J1=J1L16V,

so that the semiclassical unitary transform analogous to (C.142) becomes

(C.159)
U4(J1,L1)=(J1L1/3πV)1/2cos(Ω(6)+π/4).
Appendix C Transformations in classical and quantum mechanics

Fig. C.4 Semiclassical (curves) and exact (points) 6j symbols {j80150190230120}. (Reprinted with permission from (Schulten and Gordon 1975). Copyright 1975, AIP Publishing LLP.)

Alternatively, on replacing Tj1l1 in (C.150) by U4(J1,L1),

(C.160)
j1j2j3l1l2l3=112πV1/2cos(Ω(6)+π/4),

where Ω(6) and V are defined by (C.154)–(C.157) with Ji=ji+12 and Li=li+12 .

Equation (C.160) was first suggested on heuristic grounds by Ponzano and Regge (1968)

and later derived from the recurrence relations by Schulten and Gordon (1975). The latter also give the following uniform Airy approximations which extend the approximation to angular momentum combinations for which the volume V is negative and the angles α˜i and η˜i complex:

(C.161)
{j1j2j3l1l2l3}=(Z212|V|)1/2{cosΩ0(6)Ai(Z)sinΩ0(6)Bi(Z),Ω(6)<Ω0(6)cosΩ0(6)Bi(Z)sinΩ0(6)Ai(Z),Ω(6)>Ω0(6)

where

(C.162)
Ω0(6)=i(Jiα˜i0+Liη˜i0),

(p.372) (p.373) with α˜i0 and η˜i0 given by the analogues of (C.148), and

(C.163)
Z=±32|ΩΩ0|1/2.

As usual the upper and lower signs in (C.163) apply in the classical and non-classical cases respectively.

The comparison in Fig. C.4 shows that eqn (C.161 ) gives an excellent approximation, at least for moderately large quantum numbers, and the numerical accuracy is found to be equally good for integer and half-integer quantum numbers (Schulten and Gordon 1975). Further information on the accuracy of the approximation in relation to the magnitudes of the angular momenta is contained in Tables C.1 and C.2, which are taken from scaling tests by Schulten and Gordon (1975) . It is evident that the errors are typically of order 10 per cent or less even for quite small quantum numbers.

Table C.1 Quantum mechanical and semiclassical 6j symbolsa taken from (Schulten and Gordon 1975)!!.

j1

λ=1

λ=4

λ=6

λ=32

1

QM

0.2789(00)

0.9692(−01)

0.0977(−01)

0.0806(−02)

SC

0.3034(00)

0.9520(−01)

0.0926(−01)

0.0758(−02)

2

QM

− 0.9535(−01)

0.0457(−01)

0.1288(−01)

0.0297(−02)

SC

− 0.9303(−01)

0.0523(−01)

0.1285(−01)

0.0285(−02)

3

QM

− 0.6742(−01)

− 0.0141(−01)

0.1378(−01)

0.6773(−02)

SC

− 0.6975(−01)

− 0.0143(−01)

0.1378(−01)

0.6773(−02)

4

QM

0.1533(00)

− 0.2083(−01)

− 0.0825(−01)

− 0.0275(−02)

SC

0.1558(00)

− 0.2158(−01)

− 0.0823(−01)

− 0.0281(−02)

5

QM

− 0.1564(00)

0.5313(−01)

0.1021(−01)

− 0.4379(−02)

SC

− 0.1566(00)

0.5315(−01)

0.1023(−01)

− 0.4376(−02)

6

QM

0.1099(00)

0.1711(−01)

0.4149(−03)

0.8876(−05)

SC

0.1090(00)

0.1704(−01)

0.4145(−03)

0.8870(−05)

7

QM

− 0.5536(−01)

0.9524(−03)

0.3496(−08)

0.6071(−15)

SC

− 0.5441(−01)

0.9452(−03)

0.3488(−08)

0.6064(−15)

8

QM

0.1800(−01)

0.4072(−05)

0.1804(−18)

0.6568(−36)

SC

0.1727(−01)

0.3915(−05)

0.1738(−18)

0.6332(−36)

aj1λ4.5λ3.5λλ3.5λ2.5λ

Table C.2 Quantum mechanical and semiclassical 6j symbolsb taken from (Schulten and Gordon 1975)!!.

j1

λ=1

λ=4

λ=8

λ=16

1

QM

0.3491(−01)

0.3226(−02)

0.0513(−02)

0.0302(−03)

SC

0.3482(−01)

0.3155(−02)

0.0499(−02)

0.0292(−03)

3

QM

0.1891(−01)

0.0185(−02)

− 0.1458(−02)

0.2156(−03)

SC

0.1905(−01)

0.0180(−02)

− 0.1458(−02)

0.2157(−03)

5

QM

− 0.2359(−01)

0.2973(−02)

0.0887(−02)

0.1358(−03)

SC

− 0.2382(−01)

0.2982(−02)

0.0889(−02)

0.1362(−03)

7

QM

0.0129(−01)

0.1603(−02)

− 0.0698(−02)

0.0315(−03)

SC

0.0152(−01)

0.1569(−02)

− 0.0699(−02)

0.0315(−03)

9

QM

0.1677(−01)

− 0.2800(−02)

0.0854(−02)

0.0697(−03)

SC

0.1671(−01)

− 0.2800(−02)

0.0854(−02)

0.0696(−03)

11

QM

− 0.2135(−01)

0.2264(−02)

− 0.0020(−02)

− 0.3562(−03)

SC

− 0.2147(−01)

0.2259(−02)

0.0022(−02)

− 0.3561(−03)

13

QM

− 0.2521(−01)

0.1724(−02)

0.1184(−02)

0.4040(−03)

SC

0.2527(−01)

0.1731(−02)

− 0.1183(−02)

0.4039(−03)

15

QM

0.0271(−01)

0.1171(−06)

0.5415(−12)

0.2293(−22)

SC

0.0257(−01)

0.1095(−06)

0.5051(−12)

0.2136(−22)

bj1λ8λ7λ6.5λ7.5λ7.5λ