We derive here the Gutzwiller trace formula and the semiclassical zeta function,

Transcription

1 Chater 39 Semiclassical quantization (G. Vattay, G. Tanner and P. Cvitanović) We derive here the Gutzwiller trace formula and the semiclassical zeta function, the central results of the semiclassical quantization of classically chaotic systems. In chater 40 we will rederive these formulas for the case of scattering in oen systems. Quintessential wave mechanics effects such as creeing, diffraction and tunneling will be taken u in chater Trace formula Our next task is to evaluate the Green s function trace (36.15) in the semiclassical aroximation. The trace tr G sc (E) = d D q G sc (q, q, E) = tr G 0 (E) + d D q G j (q, q, E) receives contributions from long classical trajectories labeled by j which start and end in q after finite time, and the zero length trajectories whose lengths aroach zero as q q. First, we work out the contributions coming from the finite time returning classical orbits, i.e., trajectories that originate and end at a given configuration oint q. As we are identifying q with q, taking of a trace involves (still another!) stationary hase condition in the q q limit, S j (q, q, E) + S j(q, q, E) q i q =q q = 0, i q =q meaning that the initial and final momenta (38.40) of contributing trajectories should coincide i (q, q, E) i (q, q, E) = 0, q jth eriodic orbit, (39.1) 711 j

2 CHAPTER 39. SEMICLASSICAL QUANTIZATION 712 Figure 39.1: A returning trajectory in the configuration sace. The orbit is eriodic in the full hase sace only if the initial and the final momenta of a returning trajectory coincide as well. Figure 39.2: A romanticized sketch of S (E) = S (q, q, E) = (q, E)dq landscae orbit. Unstable eriodic orbits traverse isolated ridges and saddles of the mountainous landscae of the action S (q, q, E). Along a eriodic orbit S (E) is constant; in the transverse directions it generically changes quadratically. so the trace receives contributions only from those long classical trajectories which are eriodic in the full hase sace. For a eriodic orbit the natural coordinate system is the intrinsic one, with q axis ointing in the q direction along the orbit, and q, the rest of the coordinates transverse to q. The jth eriodic orbit contribution to the trace of the semiclassical Green s function in the intrinsic coordinates is tr G j (E) = 1 i (2π ) (d 1)/2 j dq q d d 1 q det D j 1/2 e i S j iπ 2 m j, j where the integration in q goes from 0 to L j, the geometric length of small tube around the orbit in the configuration sace. As always, in the stationary hase aroximation we worry only about the fast variations in the hase S j (q, q, E), and assume that the density varies smoothly and is well aroximated by its value D j (q, 0, E) on the classical trajectory, q = 0. The toological index m j (q, q, E) is an integer which does not deend on the initial oint q and not change in the infinitesimal neighborhood of an isolated eriodic orbit, so we set m j (E) = m j (q, q, E). The transverse integration is again carried out by the stationary hase method, with the hase stationary on the eriodic orbit, q = 0. The result of the transverse integration can deend only on the arallel coordinate tr G j (E) = 1 dq det D j (q, 0, E) i q det D j (q, 0, E) 1/2 e i S j iπ 2 m j, where the new determinant in the denominator, det D j = det 2 S (q, q, E) + 2 S (q, q, E) q i q j q i q + 2 S (q, q, E) j q i q j + 2 S (q, q, E) q, i q j is the determinant of the second derivative matrix coming from the stationary hase integral in transverse directions.

3 CHAPTER 39. SEMICLASSICAL QUANTIZATION 713 The ratio det D j /det D j is here to enforce the eriodic boundary condition for the semiclassical Green s function evaluated on a eriodic orbit. It can be given a meaning in terms of the monodromy matrix of the eriodic orbit by following observations det D = det D = q = (q, ) (q, q ) + q q = (, q q ), (q, q ). q q Defining the 2(D 1)-dimensional transverse vector x hase sace we can exress the ratio = (q, ) in the full det D = (, q q ) det D (q, ) = (x x ) x = det (M 1), (39.2) in terms of the monodromy matrix M for a surface of section transverse to the orbit within the constant energy E = H(q, ) shell. The classical eriodic orbit action S j (E) = (q, E)dq is an integral around a loo defined by the eriodic orbit, and does not deend on the starting oint q along the orbit, see figure The eigenvalues of the monodromy matrix are also indeendent of where M j is evaluated along the orbit, so det (1 M j ) can also be taken out of the q integral tr G j (E) = 1 1 i det (1 M j ) 1/2 er( i S j iπ 2 m j ) dq. q j Here we have assumed that M j has no marginal eigenvalues. The determinant of the monodromy matrix, the action S (E) = (q, E)dq and the toological index are all classical invariants of the eriodic orbit. The integral in the arallel direction we now do exactly. First, we take into account the fact that any reeat of a eriodic orbit is also a eriodic orbit. The action and the toological index are additive along the trajectory, so for rth reeat they simly get multilied by r. The monodromy matrix of the rth reeat of a rime cycle is (by the chain rule for derivatives) M r, where M is the rime cycle monodromy matrix. Let us denote the time eriod of the rime cycle, the single, shortest traversal of a eriodic orbit by T. The remaining integral can be carried out by change of variables dt = dq / q(t) L 0 dq T q(t) = dt = T. 0 Note that the satial integral corresonds to a single traversal. If you do not see why this is so, rethink the derivation of the classical trace formula (21.19) - that

4 CHAPTER 39. SEMICLASSICAL QUANTIZATION 714 derivation takes only three ages of text. Regrettably, in the quantum case we do not know of an honest derivation that takes less than 30 ages. The final result, the Gutzwiller trace formula tr G sc (E) = tr G 0 (E) + 1 i T r=1 1 det (1 M r ) 1/2 er( i S iπ 2 m ), (39.3) an exression for the trace of the semiclassical Green s function in terms of eriodic orbits, is beautiful in its simlicity and elegance. The toological index m (E) counts the number of changes of sign of the matrix of second derivatives evaluated along the rime eriodic orbit. By now we have gone through so many stationary hase aroximations that you have surely lost track of what the total m (E) actually is. The rule is this: The toological index of a closed curve in a 2D hase sace is the sum of the number of times the artial derivatives i q i for each dual air (q i, i ), i = 1, 2,..., D (no sum on i) change their signs as one goes once around the curve Average density of states We still have to evaluate tr G 0 (E), the contribution coming from the infinitesimal trajectories. The real art of tr G 0 (E) is infinite in the q q limit, so it makes no sense to write it down exlicitly here. However, the imaginary art is finite, and lays an imortant role in the density of states formula, which we derive next. The semiclassical contribution to the density of states (36.15) is given by the imaginary art of the Gutzwiller trace formula (39.3) multilied with 1/π. The contribution coming from the zero length trajectories is the imaginary art of (38.48) for q q integrated over the configuration sace d 0 (E) = 1 d D q Im G 0 (q, q, E), π The resulting formula has a retty interretation; it estimates the number of quantum states that can be accommodated u to the energy E by counting the available quantum cells in the hase sace. This number is given by the Weyl rule, as the ratio of the hase sace volume bounded by energy E divided by h D, the volume of a quantum cell, N sc (E) = 1 h D d D d D q Θ(E H(q, )). (39.4) where Θ(x) is the Heaviside function (36.20). N sc (E) is an estimate of the sectral staircase (36.19), so its derivative yields the average density of states d 0 (E) = d de N sc(e) = 1 h D d D d D q δ(e H(q, )), (39.5) recisely the semiclassical result (39.6). For Hamiltonians of tye 2 /2m + V(q), the energy shell volume in (39.5) is a shere of radius 2m(E V(q)). The

5 CHAPTER 39. SEMICLASSICAL QUANTIZATION 715 surface of a d-dimensional shere of radius r is π d/2 r d 1 /Γ(d/2), so the average exercise 39.2 density of states is given by 2m d 0 (E) = D 2 d π D2 d D q [2m(E V(q))] D/2 1, (39.6) Γ(D/2) and V(q)<E N sc (E) = 1 π D/2 h D d D q [2m(E V(q))] D/2. (39.7) Γ(1 + D/2) V(q)<E Physically this means that at a fixed energy the hase sace can suort N sc (E) distinct eigenfunctions; anything finer than the quantum cell h D cannot be resolved, so the quantum hase sace is effectively finite dimensional. The average density of states is of a articularly simle form in one satial dimension d 0 (E) = T(E) 2π, (39.8) where T(E) is the eriod of the eriodic orbit of fixed energy E. In two satial dimensions the average density of states is d 0 (E) = ma(e) 2π 2, (39.9) where A(E) is the classically allowed area of configuration sace for which V(q) < E. The semiclassical density of states is a sum of the average density of states and the oscillation of the density of states around the average, d sc (E) = d 0 (E)+d osc (E), where d osc (E) = 1 π T r=1 follows from the trace formula (39.3). cos(rs (E)/ rm π/2) det (1 M r ) 1/2 (39.10) exercise 39.3 exercise Regularization of the trace The real art of the q q zero length Green s function (38.48) is ultraviolet divergent in dimensions d > 1, and so is its formal trace (36.15). The short distance behavior of the real art of the Green s function can be extracted from the real art of (38.48) by using the Bessel function exansion for small z Y ν (z) 1 π Γ(ν) ( z ν 2) for ν 0 2 π (ln(z/2) + γ) for ν = 0, where γ = is the Euler constant. The real art of the Green s function for short distance is dominated by the singular art G sing ( q q m 1 Γ((d 2)/2) for d 2 2, E) = 2 π d q q 2 d 2. m (ln(2m(e V) q q /2 ) + γ) for d = 2 2π 2

6 CHAPTER 39. SEMICLASSICAL QUANTIZATION 716 The regularized Green s function G reg (q, q, E) = G(q, q, E) G sing ( q q, E) is obtained by subtracting the q q ultraviolet divergence. For the regularized Green s function the Gutzwiller trace formula is tr G reg (E) = iπd 0 (E) + 1 e r( i S (E) iπ 2 m (E)) T i det (1 M r. (39.11) 1/2 ) Now you stand where Gutzwiller stood in You hold the trace formula in your hands. You have no clue how good is the 0 aroximation, how to take care of the sum over an infinity of eriodic orbits, and whether the formula converges at all. r= Semiclassical sectral determinant The roblem with trace formulas is that they diverge where we need them, at the individual energy eigenvalues. What to do? Much of the quantum chaos literature resonds to the challenge of wrestling the trace formulas by relacing the delta functions in the density of states (36.16) by Gaussians. But there is no need to do this - we can comute the eigenenergies without any further ado by remembering that the smart way to determine the eigenvalues of linear oerators is by determining zeros of their sectral determinants. A sensible way to comute energy levels is to construct the sectral determinant whose zeroes yield the eigenenergies, det (Ĥ E) sc = 0. A first guess might be that the sectral determinant is the Hadamard roduct of form det (Ĥ E) = (E E n ), n but this roduct is not well defined, since for fixed E we multily larger and larger numbers (E E n ). This roblem is dealt with by regularization, discussed below in aendix Here we offer an imressionistic sketch of regularization. The logarithmic derivative of det (Ĥ E) is the (formal) trace of the Green s function d 1 ln det (Ĥ E) = = tr G(E). de E E n n This quantity, not surrisingly, is divergent again. The relation, however, oens a way to derive a convergent version of det (Ĥ E) sc, by relacing the trace with the regularized trace d de ln det (Ĥ E) sc = tr G reg (E). The regularized trace still has 1/(E E n ) oles at the semiclassical eigenenergies, oles which can be generated only if det (Ĥ E) sc has a zero at E = E n, see figure By integrating and exonentiating we obtain

7 CHAPTER 39. SEMICLASSICAL QUANTIZATION 717 Figure 39.3: A sketch of how sectral determinants convert oles into zeros: The trace shows 1/(E E n ) tye singularities at the eigenenergies while the sectral determinant goes smoothly through zeroes. ( E ) det (Ĥ E) sc = ex de tr G reg (E ) Now we can use (39.11) and integrate the terms coming from eriodic orbits, using the relation (38.17) between the action and the eriod of a eriodic orbit, ds (E) = T (E)dE, and the relation (36.19) between the density of states and the sectral staircase, dn sc (E) = d 0 (E)dE. We obtain the semiclassical zeta function det (Ĥ E) sc = e iπnsc(e) ex 1 e ir(s / m π/2) r det (1 M r ) 1/2. (39.12) r=1 We already know from the study of classical evolution oerator sectra of cha- ter 22 that this can be evaluated by means of cycle exansions. The beauty of this formula is that everything on the right side the cycle action S, the toological index m and monodromy matrix M determinant is intrinsic, coordinate-choice indeendent roerty of the cycle. chater One-dof systems It has been a long trek, a stationary hase uon stationary hase. Let us check whether the result makes sense even in the simlest case, for quantum mechanics in one satial dimension. In one dimension the average density of states follows from the 1-dof form of the oscillating density (39.10) and of the average density (39.8) d(e) = T (E) 2π + r T (E) π cos(rs (E)/ rm (E)π/2). (39.13) The classical article oscillates in a single otential well with eriod T (E). There is no monodromy matrix to evaluate, as in one dimension there is only the arallel coordinate, and no transverse directions. The r reetition sum in (39.13) can be rewritten by using the Fourier series exansion of a delta sike train n= We obtain δ(x n) = d(e) = T (E) 2π k= e i2πkx = cos(2πkx). k=1 δ(s (E)/2π m (E)/4 n). (39.14) n

8 CHAPTER 39. SEMICLASSICAL QUANTIZATION 718 This exression can be simlified by using the relation (38.17) between T and S, and the identity (19.7) δ(x x ) = f (x) δ( f (x)), where x is the only zero of the function f (x ) = 0 in the interval under consideration. We obtain d(e) = δ(e E n ), n where the energies E n are the zeroes of the arguments of delta functions in (39.14) S (E n )/2π = n m (E)/4, where m (E) = m = 2 for smooth otential at both turning oints, and m (E) = m = 4 for two billiard (infinite otential) walls. These are recisely the Bohr- Sommerfeld quantized energies E n, defined by the condition ( (q, E n )dq = h n m 4 ). (39.15) In this way the trace formula recovers the well known 1-dof quantization rule. In one dimension, the average of states can be exressed from the quantization condition. At E = E n the exact number of states is n, while the average number of states is n 1/2 since the staircase function N(E) has a unit jum in this oint N sc (E) = n 1/2 = S (E)/2π m (E)/4 1/2. (39.16) The 1-dof sectral determinant follows from (39.12) by droing the monodromy matrix art and using (39.16) ( det (Ĥ E) sc = ex i 2 S + iπ ) 2 m ex 1 r e i rs iπ 2 rm. (39.17) Summation yields a logarithm by r t r /r = ln(1 t) and we get r det (Ĥ E) sc = e i 2 S + im 4 + iπ 2 (1 e i S i m 2 ) = 2 sin ( S (E)/ m (E)/4 ). (39.18) So in one dimension, where there is only one eriodic orbit for a given energy E, nothing is gained by going from the trace formula to the sectral determinant. The sectral determinant is a real function for real energies, and its zeros are again the Bohr-Sommerfeld quantized eigenenergies (39.15) Two-dof systems For flows in two configuration dimensions the monodromy matrix M has two eigenvalues Λ and 1/Λ, as exlained in sect Isolated eriodic orbits can be ellitic or hyerbolic. Here we discuss only the hyerbolic case, when the eigenvalues are real and their absolute value is not equal to one. The determinant

9 CHAPTER 39. SEMICLASSICAL QUANTIZATION 719 aearing in the trace formulas can be written in terms of the exanding eigenvalue as det (1 M r ) 1/2 = Λ r 1/2 ( 1 1/Λ r ), and its inverse can be exanded as a geometric series 1 det (1 M r ) 1/2 = 1 Λ r 1/2 Λ kr. k=0 With the 2-dof exression for the average density of states (39.9) the sectral determinant becomes det (Ĥ E) sc = e i mae 2 2 ex e ir(s / m π/2) r Λ r=1 r k=0 1/2 Λ kr = e i mae e i S iπ 2 m Λ 1/2 Λ k. (39.19) k=0 Résumé Sectral determinants and dynamical zeta functions arise both in classical and quantum mechanics because in both the dynamical evolution can be described by the action of linear evolution oerators on infinite-dimensional vector saces. In quantum mechanics the eriodic orbit theory arose from studies of semi-conductors, and the unstable eriodic orbits have been measured in exeriments [2] on the very aradigm of Bohr s atom, the hydrogen atom, this time in strong external fields. In ractice, most quantum chaos calculations take the stationary hase aroximation to quantum mechanics (the Gutzwiller trace formula, ossibly imroved by including tunneling eriodic trajectories, diffraction corrections, etc.) as the oint of dearture. Once the stationary hase aroximation is made, what follows is classical in the sense that all quantities used in eriodic orbit calculations - actions, stabilities, geometrical hases - are classical quantities. The roblem is then to understand and control the convergence of classical eriodic orbit formulas. While various eriodic orbit formulas are formally equivalent, ractice shows that some are referable to others. Three classes of eriodic orbit formulas are frequently used: Trace formulas. The trace of the semiclassical Green s function tr G sc (E) = dq G sc (q, q, E) is given by a sum over the eriodic orbits (39.11). While easiest to derive, in calculations the trace formulas are inconvenient for anything other than the leading

10 CHAPTER 39. SEMICLASSICAL QUANTIZATION 720 eigenvalue estimates, as they tend to be divergent in the region of hysical interest. In classical dynamics trace formulas hide under a variety of aellations such as the f α or multifractal formalism; in quantum mechanics they are known as the Gutzwiller trace formulas. Zeros of Ruelle or dynamical zeta functions 1/ζ(s) = (1 t ), t = 1 Λ 1/2 e i S iπm /2 yield, in combination with cycle exansions, the semiclassical estimates of quantum resonances. For hyerbolic systems the dynamical zeta functions have good convergence and are a useful tool for determination of classical and quantum mechanical averages. Sectral determinants, Selberg-tye zeta functions, Fredholm determinants, functional determinants are the natural objects for sectral calculations, with convergence better than for dynamical zeta functions, but with less transarent cycle exansions. The 2-dof semiclassical sectral determinant (39.19) det (Ĥ E) sc = e iπn sc(e) 1 eis / iπm /2 Λ 1/2 Λ k k=0 is a tyical examle. Most eriodic orbit calculations are based on cycle exansions of such determinants. As we have assumed reeatedly during the derivation of the trace formula that the eriodic orbits are isolated, and do not form families (as is the case for integrable systems or in KAM tori of systems with mixed hase sace), the formulas discussed so far are valid only for hyerbolic and ellitic eriodic orbits. For deterministic dynamical flows and number theory, sectral determinants and zeta functions are exact. The quantum-mechanical ones, derived by the Gutzwiller aroach, are at best only the stationary hase aroximations to the exact quantum sectral determinants, and for quantum mechanics an imortant concetual roblem arises already at the level of derivation of the semiclassical formulas; how accurate are they, and can the eriodic orbit theory be systematically imroved? Commentary Remark Gutzwiller quantization of classically chaotic systems. The derivation given here and in sects and 39.1 follows closely the excellent exosition [1] by Martin Gutzwiller, the inventor of the trace formula. The derivation resented here is self contained, but refs. [3, 4] might also be of hel to the student. Remark Zeta functions. age 412. For zeta function nomenclature, see remark 22.4 on

11 CHAPTER 39. SEMICLASSICAL QUANTIZATION 721 References [1] M. C. Gutzwiller, Chaos in Classical and Quantum Mechanics (Sringer, New York, 1990). [2] A. Holle, J. Main, G. Wiebusch, H. Rottke, and K. H. Welge, Quasi-Landau sectrum of the chaotic diamagnetic hydrogen atom, Phys. Rev. Lett. 61, (1988). [3] R. G. Littlejohn, The Van Vleck formula, Maslov theory, and hase sace geometry, J. Stat. Phys. 68, 7 50 (1992). [4] L. Reichl, The Transition to Chaos: Conservative Classical Systems and Quantum Manifestations, 2nd ed. (Sringer, New York, 2004).

12 EXERCISES 722 Exercises Monodromy matrix from second variations of the action. Show that D j /D j = (1 M) (39.20) Volume of d-dimensional shere. Show that the volume of a d-dimensional shere of radius r equals π d/2 r d /Γ(1 + d/2). Show that Γ(1 + d/2) = Γ(d/2)d/ Average density of states in 1 dimension. Show that in one dimension the average density of states is given by (39.8) d(e) = T(E) 2π, where T(E) is the time eriod of the 1-dimensional motion and show that N(E) = S (E) 2π, (39.21) where S (E) = (q, E) dq is the action of the orbit Average density of states in 2 dimensions. Show that in 2 dimensions the average density of states is given by (39.9) d(e) = ma(e) 2π 2, where A(E) is the classically allowed area of configuration sace for which U(q) < E. exertracescl - 11feb2002 ChaosBook.org edition16.0, Feb