We present a simple method to deal with caustics in the semiclassical
approximation to the partition function of a one-dimensional quantum system.
The procedure, which makes use of complex trajectories, is applied to the
quartic double-well potential.Comment: 5 pages, 1 figure, Latex. Contribution to the Proceedings of the XXI
  Brazilian National Meeting on Particles and Fields (Sao Lourenco, October
  23-27, 2000

Cavalcanti, R. M.

de Carvalho, C. A. A.

Fraga, E. S.

Joras, S. E.

English

arXiv

We present a simple method to deal with caustics in the semiclassical approximation to the partition function of a one-dimensional quantum system. The procedure, which makes use of complex trajectories, is applied to the quartic double-well potential

De Carvalho, C.A.A.

Cavalcanit, R.M.

Fraga, E.S.

Joras, S.E.

UNT (University of North Texas) Digital Library

T.“’ 11 61 BNL-68026 Uniform semiclassical approximation in quantum statistical mechanics* ..\Y C. A. A. de Carva.lho,’ R. M. Cavalcanti,2 E. S. Fraga,3 and S. E. Jo&s” ’ Instituto de Fisica, Universidade Federal do Rio de Janeiro, Caixa Postal 68528, Rio de Janeiro, RJ 21945-970, Brazil ‘Institute de Fisica, Universidade de 5’60 Paulo, Caixa Postal 66318, Stio Paulo, SP 05315970, Brazil 3Nuclear Theory Group, Physics Departvnent, Brookhaven National Laboratory, Upton, NY 11973-5000, U.S.A. 4 High Energy Theory Group, Physics Department, Brown University, Providence, RI 02912, U.S.A. Abstract We present a simple method to deal with caustics in the semiclassical approximation to the part.ition function of a one-dimenkonal quantum system. The procedure, which makes use of complex trajectories, is ap- plied to the quartic double-well potential. 1 Introduction It is well known [l] that the pa.rtit,ion function for a particle of ma,ss m. interacting with a potential V(z) and a thermal reservoir at temperature T can be written as a pa.th integral (p = l/k~T): (zOlewpH IXO) = J ‘(/3h)zriJ [Dx(r)] e--s[+, c(fl)=ro prL S[x] = J [ dr $n:t” + V(x) . I 0 *\VOI+ supported by CNPq, F.\PESP, FU.JB/IJFR.I and the U.S. Department of Energy under Contract No. DE-ACOS-SXH10886. 1 -_ The semiclassical approximation to the density matrix element (2) is given by1 where X;(T) is a classical trajectory [i.e., it is a solution to the Euler-Lagrange equation, rnZ = V’(x)? subject to the boundary conditions ~(0) = z(/?fi) = ZO] that minimizes2 (globally or locally) the action S[Z], and f.!+ is the determinant of of the fluctuation operator I”[x;] q -ma: + V”[xE]. (A derivation of this result will be sketched in Section 2.) In Ref. [2] we have examined, for the sake of simplicity, potentials for which there is only one classical trajectory satisfying the above boundary conditions. In general, however, the number N of such solutions depends on ~0 and 8. A problem then occurs when we cross a caustic [the frontier between two regions --- of the (xo,~)-pl ane characterized by different values of N]: the r.h.s. of (4) diverges [3]. This divergence, however, is unphysical, being an artifact of the semiclassical approximation. The purpose of this work is’ to present a simple extension of the semiclassical approximation which circumvents this problem.3 (Due to limitations of space, here we shall only sketch the method. Details will be given elsewhere.) 2 Improved semiclassical approximation In order to show how one can improve the semiclassical approximation so as to eliminate the unphysical divergences at the caustics, it is convenient to recall how (4) is derived. Briefly, one has to: (i) expand the action around a minimum Z,(T): S[X, + ~1 = S[XJ + 5’2 + SS, where SZ = is,“” &-r](r) P[z,(r)] V(T) and 6s = O(q31; (ii) th row away 65’; (iii) express V(T) in terms of the orthonormal modes of F, i.e., T(T) = Cj”=oajcpj(r), h w ere p’cpj(T) = Xjcpj(r), ~(0) = ‘P~(@=L) = 0; th en integral in (2) h S2 = $ ~~Zo Xju: and [Z&(T)] = J-J:=, duj/&%. The path as now become a product of Gaussian integrals. Performing the integrations one arrives at the “usual” semiclassical approximation to the density matrix element: (xole-prilxo) m e-S[z=llfi A-1/2, (5) where A = I$‘??, Xj = det j. If there are N minima, one has to add together their contributions, thus obtaining (4). When we cross a caustic, a classical trajectory Z,-(T) is created or annihilated. Precisely at this point, the lowest eigenvalue of F[xJ vanishes, thus making the integral s-“, duo exp(-Xoui/2h) diverge. This problem can be remedied by ‘Each t,erm in the sum on the r.h.s. of (4) is in fact the first term of a series. See Ref. [J]. “The Euclidean nature of the path integral allows one to discard saddle-points. “Ankerhold ef al. [4] have discussed the caustics problem near the top of a potential barrier. The present. work shows how to deal with the problem everywhere. 2 retaining fluctuations beyond quadratic in the subspace spanned by ~0 (the eigenmode of F-associated with X0). As a result of this procedure,4 we obtain an improved approximation to the density matrix element (2): where x gm(r) is the global minimum of S[x] and with 1 V(a0) = - XOU~ + 2 A couple of remarks are in order here: (i) we take for M [Eq. (S)] the smallest even integer such that the coefficient of af in V(Q) is positive for all values of x0 and ,I!?; this suffices to make the integral in (7) finite even when X0 vanishes; (ii) the factor Xi’2 in F cancels the factor X0 -II2 contained in AA112; combined with (i), this shows that the improved a.pproximation to (xole-Pfi 1x0), Eq. (6), is finite at the caustics; (iii) there is a one-to-one correspondence between the minima of S[x] and the minima of V(Q); therefore, it is not necessary to ex- plicitly add their contributions as in (4), for they are already contained in F. Although the procedure outlined above teaches us how to cross the caustics, it is not very convenient: in order to obtain the coefficients of V(Q) one has to find A0 and cpo(r). This, in general, is not an easy ta.sk, and makes the whole procedure very cumbersome. Instead, we shall present an alternative way of obtaining those coefficients, which is based on remark (iii) above. Let us assume that M = 4 in Eq. (8); this is the case for the quartic double- well potential, to be discussed in the next section. Then the “effective action” A(ao) q S[x,,]+V(ao) for the “critical” mode “po is a. fourth degree polynomial in a~. Let us assume it has three (real) extrema: a global minimum at a0 = 0, a loca.1 maximum at u > 0, and a local minimum at ‘u > U. This allows us to write such a polynomial as5 A(Q) = S[x:,,] + N [ $ uu ui - 3 (U + w) uz + a ui] . In order to relate the parameters in A(Q) to calculable quantities, we impose that d(v) = S[xlm] and d(u) = S[z,,,], w h ere zlrn (T) and x,~(T) are the local minimum and the.lowest saddle-point of S[z], respectively. This yields S[xlm] - S[xgm] _ d(2)) - d(0) = t3(2 - c) S[xsp] - S[xsm] - d(zi) - d(0) 2J-1 ’ (9) where 5 q ‘U/U. Another combination of parameters which becomes determined is p z cyu4: s[xs,] - S[:~rn] = d(n) - d(0) = & (a< - 1). (10) “The procedure adopted here resembles t,he treatment of caustics in optics [5] and in quant,um mechanics [6], suitably modified to take into account the Euclidean nature of the path integral in (2). 50ne can easily check that A’(O) = A’(u) = A’(v) = 0. * , ‘, -. In terms of the calculable parameters [ and p, we can rewrite d(ao) as d(uo) G S[x,,] + Vs(uo/u), where Vs(z) = p [$tz2 - 6 (l+ <) .z3 + i z4]. There still remains one parameter to be determined, namely u. Fortunately, F does not depend on it. Indeed, identifying Vs(ao/u) with V(Q) yields X0 = pt/u2, which allows us to rewrite (7) solely in terms of E and ~1: duo e-V3(Qdu)/fi - - J-J f& -* dze-V3b)/” f (11) Co The discussion above can be easily adapted to the case in which d(uo) has only one extremum. In this case, d’(ao) has one real (UO = 0) and two complex conjugate roots (W and w*), the latter corresponding to the complex trajec- tories x,t(T) and xzt(~). Accordingly, one has d(u0) = S[x,,] + V~(ao/lwl), where VI(Z) = yy [ $- z2 - $ (cos 4) z3 + $ z4], with x G crlw14 and 4 5 arg(w). Identifying A (w ) with S[x:,t] then yields S[x,t] - S[xgm] =’ d(w) - d(0) = g (210” - w) = -$ (2e2”+ - e4i4) , (12) from which we can obtain x and 4. Finally, identifying V~(uo/iwl) with V(Q) leads to X0 = x/lw12, so that 3 = ,/& [I duo e-vl(aoflwl)fA = ,,/& LI d,-e-V’(“)l”. (13) 3 Application: the quartic double-well potential Let us consider the quartic double-well potential, V(x) = $ (x2 - u~)~, X > 0. In order to simplify notation, we replace x and r by q f x/u and 6 q WI-, respectively, where w s (Xu2/m)ll”. In the new variables, the equation of motion reads ;i = U’(q), where U(q) = a (q2 - 1)“. Closed trajectories have the form qc(o) = qt cd(u, k), (14) where cd is one of the Jacobian elliptic functions [7], u q dm (8 - O/2), 0 z /3hw, and k G dm. The turning point qt is fixed by the boundary condition q=(O) = qo. The classical action can be written as S[xc] = (h/g) I[qc], where g z liX/dw3 and &I = 0 CT(a) + 2sgn(qt - q0) s 4t d2 P-J(q) - U(e)1 drl. (15) QO Fklly, the determinant of the fluctuation operator is given by [2] fl = 4vw(qo - qt) d2 W(Yo) - U(e)] dqo fJ’(qt ) ( > dqt @,’ (16) Using these ingredients oue cau cornput,e both the usual and the improved semiclassical approximations to (r/oIe-fl’Iqo), Eqs. (4) and (6), respectively. The results are compa.red in Fig. 1. 4 Figure 1: p q (~ole-@Iqo) vs. qo for 0 = 5.0 and g = 0.3. The dashed line is obta.ined using the usua.1 semiclassical approximation [Eq. (4)]; the solid line is the result of the approximation discussed in Section 2. The equation q=(O) = qo (which determines the turning point qt) has three real solutions for qo < & = 0.33319.. .; as we cross the caustic, two of them (the ones associated with,a local minimum a,nd a saddle-point of the action) coalesce - thus causing the divergence in the usual semiclassical approximation - and reemerge at the other side as a pair of complex conjugate solutions. References [l] R.. P. Feynma.n, Statistical fifechanics (Addison-Wesley, New York, 1972); H. Kleinert, Path Integrals in Quantum Mechanic.s, Statistics and Polymer Ph.ysics (World Scientific, Singapore, 1995). [2] C. A. A. de Carvalho, R. M. Cava.lcanti, E. S. Fraga, a.nd S. E. Jor&s, Ann. Phys. (N.Y.) 273, 146 (1999); Phys. Rev. E 61. 6392 (2’000). [3] C. A. A. de C arvalho and R. M. Cavalca.nti, Braz. J. Phys. 27, 373 (1997). [4] J. Ankerhold and H. Grabert, Physica A 188, 568 (1992); F. J. Weiper, J. Ankerhold, a.nd H. Grabert, Physica A 223, 193 (1990). [5] M. Berry, in Physics of Defects, Les Houches Session XXXV (1980), edited by R. Balia.n, M. Kldman, a.nd J.-P. Poirier (North-Holland, Amsterdam, 1981); M. 1’. Berry a.nd C. Upstill, in Progress in Optics XV111, edited by E. Wolf (North-Holla.nd, Amsterdam, 1980). [G] G. Dangelmayr and W. Veit, Ann. Phys. (N.Y.) 118, 108 (1979); L. S. Schulman, Techn,iyues and Applications of Path Integration (3ohn Wiley, New York, 1981). [7] I. S. Gradshteyn a,ncl I. M. Ryzhik, Tuble of Integruls, Series and Products (Aca.demic Press, New York, 1965). 5 0.04 0.03 P 0.02 0.01 0 

UNIFORM SEMICLASSICAL APPROXIMATION IN QUANTUM STATISTICAL MECHANICS.

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Uniform semiclassical approximation in quantum statistical mechanics

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