## 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

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

(C.2)
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conditions that are readily verified to require a unit Jacobian,

(C.3)
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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)
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(C.4b)
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(C.4c)
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(C.4d)
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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)
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(C.5b)
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(C.5c)
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For example, according to (C.4b) and (C.5a),

(C.7)
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which reduce to the identities

(C.7)
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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)
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where $U1(q,q˜)$ must be defined in such a way that (C.8) relates corresponding eigenfunctions of $H(q,−iℏ∂/∂q)$ and $H˜(q˜,−iℏ∂/∂q˜)$. (Note that if $q˜$ were an angle variable the operator $−iℏ∂/∂q˜$ would be modified by addition of a Maslov term, as in eqn (4.23)). Put in mathematical terms this requires that

(C.9)
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(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)
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It is shown below that a solution to order $ℏ0$ may be expressed in the form

(C.11)
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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 $ℏ0$,

(C.12)
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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 $ℏ0$), that

(C.13)
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which establishes the validity of (C.9).

Finally, the unitarity condition

(C.14)
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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)
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All that remains is to fix the magnitude of $A1(q,q˜)$ by approximating

(C.16)
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and writing

(C.17)
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(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)
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It follows, by comparison with the standard form (Dirac 1958)

(C.19)
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that

(C.20)
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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)
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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)
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so that on substitution in (C.8)

(C.23)
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where

(C.24)
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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)
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(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)
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conforms exactly to the classical expression in (C.5a). Finally, on applying the standard quadratic approximation about $q˜(q,p˜)$,

(C.27)
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where

(C.28)
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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)
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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)
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where $∥∂2Fi/∂x∂y˜∥$ 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)
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(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)
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The solution

(C.33)
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shows correctly that

(C.34)
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The corresponding $F1(R,t)$ generator is given according to (C.5a) by

(C.35)
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after some rearrangement in the light of (C.34). The required semiclassical unitary transform is therefore given by

(C.36)
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Consequently $ψ(R)$ transforms to

(C.37)
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which reduces at the stationary phase level to

(C.38)
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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)
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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)
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which describes a state with unit translational flux and a particular action I0C.2corresponding to the quantum number

(C.41)
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where δ‎ is the usual Maslov index. Since the action operator is given by $Iˆ=−iℏ(∂/∂α)+δℏ$ (see Sections 3.1 and 4.1), the Hamilton–Jacobi equation for the generator $F2(R,α;$ $E,I)$ takes the form

(C.42)
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and the solution is readily found to be

(C.43)
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where

(C.44)
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Notice, on applying (C.5a), that the variables conjugate to E and I in the transformed representation take the interesting forms

(C.45)
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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)
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The form of the associated unitary transform,

(C.47)
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(p.351) then implies, after some manipulation, that

(C.48)
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which is the result employed (as eqn (10.24)) in Chapter 10.

The scaled harmonic oscillator, with Hamiltonian

(C.49)
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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)
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has the form assumed in eqns (C.9)–(C.13) ($ℏ=1$ in the present units).

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

(C.51)
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Hence, following (C.4a),

(C.52)
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which rearrange to the canonical form (see (4.21))

(C.53)
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The analogue of eqns (C.8) and (C.10) therefore suggests a representation for the Cartesian wavefunction $ψn(q)$ of the form

(C.54)
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where

(C.55)
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and, according to (4.25),

(C.56)
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(p.352) The integration limits in (C.54) will be chosen later. Turning to the function $A1(q,α)$, the semiclassical form,

(C.57)
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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)
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can be solved exactly by substitution from (C.55) to yield

(C.59)
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Two fundamental solutions may be recognized:

(C.60)
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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)
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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)
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for n even, and

(C.62b)
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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)
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for even n may be shown to take the following value at $q=0$:

(C.64)
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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)
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It follows by use of the gamma function duplication formula (Abramowitz and Stegun 1965) that

(C.66a)
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with the numerical values $C0=1.0623,C2=1.0046,C4=1.0015$ and $Cn→1$ as $n→∞$. A similar comparison for $ψn′(0)$ leads to

(C.66b)
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!! 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)
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(p.354) yields the correct momenta:

(C.68)
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The corresponding unitary transform, denoted for consistency with what follows by

(C.69)
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may be confirmed to propagate the wavefunction from time t1 to t2 in the sense that (Feynman and Hibbs 1965)

(C.70)
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To see this, note that the exact initial free-motion wavefunction, at energy $E=p2/2m$, is

(C.71)
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so that on using the standard integral

(C.72)
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(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)
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where S is the action along a particular path,

(C.74)
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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)
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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)
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where

(C.77)
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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 $t2−t1,$ 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)
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where

(C.79)
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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)
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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)
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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 . 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)
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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)
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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)
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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)
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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)
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and in the second case

(C.87)
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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)
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or

(C.89)
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(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)
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It is readily verified from (C.89) that

(C.91)
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and similarly for $p˜$ because by construction

(C.92)
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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)
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where

(C.94)
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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)
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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)
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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)
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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)
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Moreover, it is clear from the energy dependences of $iλ(αˉ2τ2;αˉ1τ1)$ and of the typical asymptotic wavefunction given by (C.48)

(C.99)
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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)
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where the superscript on the left-hand side implies propagation of $ψn1$. It follows, by analogy with (C.76), that

(C.101)
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(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)
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so that on projecting out the $n20$th term from (C.100)

(C.103)
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where

(C.104)
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Note that, for ease of comparison with (9.29), Ni and Pi have been replaced by

(C.105)
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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)
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where

(C.107)
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may be obtained by stationary phase integration with the help of the identity

(C.108)
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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)
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(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)
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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)
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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)
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where the reduced classical action integral

(C.113)
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is the stationary value of the exponent in (C.111). Consequently the stationary phase approximation to $G(q′′q′;E)$ is given by

(C.114)
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(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 $∂2S/∂t2=−∂E/∂t=−∂t/∂E−1=−∂2W/∂E2−1$. Secondly, following Gutzwiller (1990), eqn (C.112) may be used to express the Van Vleck determinant in terms of $det∂2W/∂q′′∂q′$. As a preliminary, note that

(C.115)
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from which it follows that

(C.116)
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Terms like $∂2E/∂qj′′∂qi′$ vanish because $E=H(p′,q′)$ is independent of $q′′$. When expressed in terms of determinants, this means that

(C.117)
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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 $det(∂2W/∂q′′∂q′)=0$. To see this note by the Hamilton–Jacobi equation that

(C.118)
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Hence, on differentiating with respect to $qi′$,

(C.119)
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which implies that the matrix $(∂2W/∂q′′∂q′)$ 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 $∂2W/∂qj′′∂qi′=0$ whenever (p.362) $i=1$ or $j=1$. It also follows, by differentiating (C.118) with respect to E, that $q˙1∂2W/∂q1∂E=1$. Consequently

(C.120)
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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)
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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)
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in which $pˉ$ is the momentum on the upturned potential energy surface, $Vˉq=−V(q)$ and $τ(E)=−∂Wˉ/∂E$. Thus

(C.123)
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or in the context of eqn (11.64)

(C.124)
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# 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 $|(j2j3)j1−m1⟩$ by the equations (Brink and Satchler 1968; Zare 1988)

(C.125)
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where the elements $Tj1m2$ define the unitary transformation

(C.126)
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subject to

(C.127)
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Note, for future reference, that the sum over m2 in (C.126 ) could be more symmetrically regarded as a sum over $m2−m3$ at fixed $m1$.

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 $ℏ$) with $J1$ fixed and $J2$ and $J3$ free to rotate around it, subject to the triangular constraint

(C.128)
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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 $m2−m3$ represented by the sum over m2 on the right, all other actions being fixed by $|J1|,m1$ and $m2−m3$. 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 $m2−m3$ 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)
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where

(C.130)
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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)
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(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)
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(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 $m2−m3$, 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)
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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)
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and

(C.135)
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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)
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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)
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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)
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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)
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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)
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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)
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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)
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from which it may be verified that

(C.143)
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Equation (C.140) was first given by Miller (1974). If two of the ji are much larger than the other, say $j2≃j3≫j1$, it is simpler to employ the 3j equivalent of the earlier result of Brussard and Tolhoek (1957), namely

(C.144)
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where $Dμνj(0,θ,0)$ is the rotation matrix (Brink and Satchler 1968) and

(C.145)
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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)
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Fig. C.2 Semiclassical (curves) and exact (points) 3j symbols, (a) $j10060−1060−50$ and (b) $1206070−10m10−m$. (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)
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with the positive and negative signs taken in the real and complex cases respectively. Finally,

(C.148)
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$αi0$ and $ηi0$ being the angles on the nearest boundary:

(C.149)
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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)
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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)
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(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;

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)
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where

(C.153)
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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)
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again acts as the proper F4 type generator, in the sense that

(C.155)
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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)
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(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)
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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)
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so that the semiclassical unitary transform analogous to (C.142) becomes

(C.159)
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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)
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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)
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where

(C.162)
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(p.372) (p.373) with $α˜i0$ and $η˜i0$ given by the analogues of (C.148), and

(C.163)
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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 symbols$a$ 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)

$aj1λ4.5λ3.5λλ−3.5λ2.5λ$

Table C.2 Quantum mechanical and semiclassical 6j symbols$b$ 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)

$bj1λ8λ7λ6.5λ7.5λ7.5λ$