aa r X i v : . [ qu a n t - ph ] A p r Semiclassical Ehrenfest Paths
Rafael Liberalquino and Fernando Parisio
Departamento de F´ısica,Universidade Federal de Pernambuco,50670-901, Recife, Pernambuco, BrazilFone: 55-081-32227339email: [email protected]
Trajectories are a central concept in our understanding of classical phenomena andalso in rationalizing quantum mechanical effects. In this work we provide a way todetermine semiclassical paths, approximations to quantum averages in phase space,directly from classical trajectories. We avoid the need of intermediate steps, likeparticular solutions to the Schroedinger equation or numerical integration in phasespace by considering the system to be initially in a coherent state and by assumingthat its early dynamics is governed by the Heller semiclassical approximation. Ourresult is valid for short propagation times only, but gives non-trivial information onthe quantum-classical transition.
I. INTRODUCTION
Even though wave mechanics does not emcompass the concept of trajectory, this clas-sical notion often appears in the quantum formalism. On a fundamental level, there aretrajectory-based theories like path-integrals [1] and Bohmian mechanics [2] that aim to pro-vide interpretations and quantitative rules to deal with general quantum systems. Withinthe orthodox structure of wave mechanics we have the Ehrenfest theorem, which establishesthe phase-space paths followed by h ˆ q i ( t ) and h ˆ p i ( t ), providing an important, but deceptive,connection with classical mechanics. An illustration of the continuing interest in the concep-tual/experimental status of trajectories in quantum mechanics is a recent implementationof appropriate combinations of strong and weak measurements to determine average pathsof single photons in a two-slit interferometer [3].The whole of semiclassics is based on Newtonian paths, the goal being to express quantummechanical quantities in terms of real [4–6] or complex [7–14] classical trajectories and theirstability properties. These approximations are extensively used, e.g., in physical chemistryand also in linking the distinct structures of classical and quantum mechanics, where acritical point is the understanding of the emergence of classical chaos from a linear theory.Usually, in order to calculate semiclassical expressions h ˆ q i sc ( t ) and h ˆ p i sc ( t ), we first need anapproximate solution to the time dependent Schroedinger equation | ψ ( t ) i sc . The purpose ofthis manuscript is to provide a more direct connection between classical, semiclassical andquantum paths, and to discuss the fundamental differences between them. For example, howquantum non-locality shows up in the semiclassical trajectories (and vanishes as ¯ h → II. PRELIMINARY REMARKS
Given a particle of mass µ subjected to a potential V , the equations of motion for theexpectation values of position and momentum in quantum mechanics, provided by the Ehren-fest Theorem, lead to µ d d t h ˆ q i ψ = − * d ˆ V dˆ q + ψ , (1)where h ... i ψ = h ψ ( t ) | ... | ψ ( t ) i , | ψ ( t ) i being an arbitrary solution of the time-dependentSchroedinger equation. To simplify the notation we suppress the subscript ψ hereafter. Theformal similarity between the previous relation and Newton’s second law is quite misleading.This is partly due to the widely known fact thatd V d q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) h ˆ q i = * d ˆ V dˆ q + . (2)The two quantities in the above relation tend to coincide only in the limiting case of spatiallylocalized wave functions. Some authors refer to the fulfillment of this last condition alongwith Eq. (1) as the Ehrenfest Theorem [15]. There is, however, another important difference.While in classical mechanics, µ ¨ x = − d V / d x is a second order differential equation, relation(1) is an identity , not an equation. Once the quantum dynamics is resolved, | ψ ( t ) i can beused to calculate the quantities in (1) and to show that both sides of the equality invariablycoincide. As a consequence, if one resorts to any approximate method to get | ψ ( t ) i approx ≈| ψ ( t ) i , relation (1) ceases to hold exactly. In this case the question arises, which side of (1)better describes the exact quantum evolution of the mean values ? In principle we are freeto pick either d d t h ˆ q i approx or − µ * d ˆ V dˆ q + approx (3)as our trial approximate acceleration (or a linear combination of them). III. SEMICLASSICAL EHRENFEST DYNAMICSA. The ingredients
In the present manuscript we will be concerned with semiclassical approximations a sc ( t ), p sc ( t ), and q sc ( t ) to the corresponding exact quantum expected values. Therefore, our basicingredients are the classical trajectory [ q c ( t ) and p c ( t )] and its stability properties given bythe tangent matrix m . Consider the initial conditions ( q , p ) leading, after a short time, tothe phase-space point ( q f , p f ). If we take the neighboring initial point ( q + δq , p + δp ),one ends up at ( q f + δq f , p f + δp f ). In the linear approximation m is defined by δq f bδp f c = m qq m qp m pq m pp δq bδp c , where b and c are constant parameters, that will be conveniently set, with dimensions ofposition and momentum, respectively.In the semiclassical domain it is natural to consider localized states, which uniformlytend to a classical point in phase space as ¯ h →
0. For this reason, a canonical coherent stateis taken as our initial ket, the corresponding wave function being given by the minimumuncertainty Gaussian packet ψ ( x ) = h x | z i = 1 π / b / e − ( x − q b + i ¯ h p ( x − q ) , (4)where | z i is a coherent state with z = ( q /b + ip /c ) / √
2. The position and momentumuncertainties are ∆ q = b/ √ p = c/ √
2, with b c = ¯ h . These relations also provideconvenient scales b and c used in the definition of the stability matrix. In order to satisfythe minimum uncertainty relation we must have b ∼ ¯ h − γ and c ∼ ¯ h γ for any γ . However,for γ = 1 /
2, either b/c or c/b diverges as ¯ h →
0, leading to a non-physical infinite squeezingin the classical limit. Thus, we assume from now on that b ∼ √ ¯ h and c ∼ √ ¯ h , (5)so that the classical limit corresponds to a rescaling, the geometrical nature of the stateremaining unchanged.Finally, given the classical path and initial state, we choose the simplest semiclassicalapproximation to describe the time evolution of | z i , namely, Heller’s “Thawed Gaussian”[16]: ψ sc ( x, t ) = π − / b − / q m qq + im qp e − ζ b ( x − q c ) + i ¯ h [ S + p c ( x − q c )+ q p ] , (6)where q c and p c are the classical position and momentum as functions of time, S is theclassical action, and ζ = ( m pp − im pq ) / ( m qq + im qp ). Due to its extreme simplicity, | ψ sc | is a Gaussian for all times and the Heller wave function is usually a poor approximation todescribe the quantum wave function for longer propagation times. Therefore, our resultswill be valid only on a short-time scale. This limitation, however, will not prevent us to getuseful information. B. Semiclassical trajectories
Since the above formula is a Gaussian for all times, its use together with the first optionin (3) leads simply to a sc = a c . In this case, no vestige of the quantum behavior is left. Onthe contrary, if one employs (6) and the second option in (3), the result is non-trivial. Thus,we define the semiclassical acceleration as a sc ≡ − µ * d ˆ V dˆ q + sc = − µ Z ∞−∞ d x | ψ sc ( x, t ) | d V d x , (7)where ψ sc ( x ) = h x | ψ ( t ) i sc . This definition provides corrections to the bare classical trajec-tory because the Gaussian extends over the space “sensing” the different values taken byd V / d x , i. e., the whole force field. It is important to note that the potential cannot besingular for any finite x , since the Gaussian is non-zero everywhere. We begin our analysisof (7) in a very general way, by assuming that the potential is an arbitrary polynomial: V ( x ) = N X n =0 α n x n . (8)In principle, N is supposed to be finite, but we may take N → ∞ in cases where convergenceof a sc can be guaranteed. Substituting the above relation in the general prescription (7) weget a sc = − µ X n nα n √ πσ ( t ) Z ∞−∞ d x x n − e − σ ( t )2 ( x − q c ) , (9)with σ ( t ) = b q m qq + m qp . This leads to a sc ( t ) = − µ N X n =1 nα n " σ ( t )2 i n − H n − " iq c ( t ) σ ( t ) , (10)where H k denotes the Hermite polynomial of degree k [notice that i − k H k ( iy ) is a real function]and the time dependence was made explicit. By integration we get p sc ( t ) and q sc ( t ). Werecall that in order to get these approximations to the quantum expected values one has onlyto solve the classical equations for q c ( t ) and the stability matrix m , not the Schroedingerequation. C. General properties
As any consistent semiclassical approximation, a sc must coincide with the classical andexact quantum values for Gaussian packets in linear and harmonic potentials ( V = const , V ∝ x , and V ∝ x ). This is indeed the case. For V = const , formula (7) gives immediately a sc = 0, while for V = α x , relation (10) gives the constant acceleration a sc = − α /µ .Finally, for V = α x , we get a sc = − α q c ( t ) /µ . Setting α = µω / a sc ( t ) = − ω q c ( t ), as expected. For higher order potentials the classical, semiclassical, and quantumresults no longer coincide.Before going into more quantitative examples, we establish the general lower order cor-rection to the classical acceleration in the general case of an analytic potential. To dothat we must realize that the Planck constant appears in (7) [or (10)] implicitly, through σ ∝ b ∝ √ ¯ h . Replacing H n ( y ) = [ n/ X k =0 (2 y ) n − k ( − k n ! k !( n − k )! (11)into (10) and collecting the lower powers of σ we get a sc = a c − σ µ d V dq (cid:12)(cid:12)(cid:12) q = q c + O ( σ ) . (12)This shows that our result is consistent with classical mechanics plus a main correction oforder ¯ h and smaller terms (involving higher powers of ¯ h and higher derivatives of V ). Per-haps the most important point about the above relation is that these approximate quantumpaths cannot come from any effective or modified potential, because they are not Hamil-tonian. Differently from what happens in classical mechanics, the instantaneous dynamicsof a phase-space point depends upon higher order derivatives of the potential. This is atrace of quantum non-locality. While in the Newtonian mechanics the particle only needsto “know” its position [to get V ( x )] and the force to which it is subjected ( ∝ d V / d x ), bothlocal quantities, in the semiclassical case all derivatives of the potential are needed, whichamounts to knowing the function V for all values of x , no matter its distance to the particle.Any change in the potential in a far region instantly influences the semiclassical particle’spath. Of course, this non-locality vanishes as ¯ h → t = 0. This can be understood by realizing that our expression yields, by construction,the exact quantum result as t →
0. For sufficiently short times the evolution operator canbe written as ˆ I − i ¯ h ˆ Ht − h ˆ H t , so that the expectation value of momentum is, to secondorder in t h ˆ p i ( t ) = h z | (cid:18) ˆ I + i ¯ h ˆ Ht − h ˆ H t (cid:19) ˆ p (cid:18) ˆ I − i ¯ h ˆ Ht − h ˆ H t (cid:19) | z i = p + i ¯ h t h [ ˆ V (ˆ q ) , ˆ p ] i z − t µ *( ˆ p, ∂ ˆ V∂ ˆ q )+ z + O ( t )= p − t * ∂ ˆ V∂ ˆ q + z − t µ *( ˆ p, ∂ ˆ V∂ ˆ q )+ z + O ( t ) , (13)where { , } stands for the anticommutator. In the quantum mechanical case (where theEhrenfest Theorem holds) we define the exact acceleration as a quant = 1 µ dd t h ˆ p i ( t ) = − µ * ∂ ˆ V∂ ˆ q + z − t µ *( ˆ p, ∂ ˆ V∂ ˆ q )+ z + O ( t ) , (14)which is a sc for t = 0. This guarantees a consistent quantum-classical interplay in thesense that the semiclassical expression tends to become classical for ¯ h → t →
0. This closes our discussion for short-time propagation in1D analytic potentials. In the next section we illustrate the procedure by applying (10) inthe case of a cubic potential. In addition, we give an example involving a non-analyticalpotential, for which expression (10) is not valid and (7) must be used directly.
IV. APPLICATIONSA. Cubic potential
Here we consider in more detail the classical, semiclassical, and quantum accelerations inthe particular case of a cubic potential V = αx . The quantum result up to first order in t is a quant ≈ − αµ " h ˆ q i + tµ ( h ˆ q ˆ p i + h ˆ p ˆ q i ) = − αµ " q + b tµ q p . (15)Let us calculate a sc also up to first order in t . The full semiclassical expression reads a sc = − αµ " q c ( t ) + σ ( t ) (16)which coincides with (12) for a cubic potential. We have q c ( t ) = q + d q d t (cid:12)(cid:12)(cid:12) t + O ( t ), fromwhich it is easy to show that m qq ∼ m qp ∼ t , so that σ = b + O ( t ). Substituting theseexpansions into (16) we get a sc ≈ − αµ " q + b tµ q p . (17)Thus, in this case, the quantum and semiclassical results coincide not only to zeroth order,but also to first order in t . On the other hand we know from (12) that a sc goes to a c as¯ h →
0. We can immediately write a c ≈ − αµ " q + 2 tµ q p , (18)since b ∼ ¯ h . In particular, it is interesting to note that the classical equilibrium point q = 0and p = 0, leading to a c = 0, does not occur in the semiclassical and quantum cases, sincethere is always a “residual” acceleration a sc = a quant = 3 αb / µ . B. Wave packet impinging on a step potential
A crucial process in wave-packet dynamics is the collision with a potential wall. Here weexamine the evolution of h ˆ q i for the Gaussian packet (4) bouncing off a step potential, V ( x ) = , for x ≤ V , for x > , (19)for the cases where p < √ µV . From the classical trajectory, q c ( t ) = p tµ + q , for t ≤ t − p µ ( t − t ) , for t > t , (20)with t ≡ µq /p , we extract the two needed tangent matrix elements: m qq ( t ) = , for t ≤ t − , for t > t (21)and m qp ( t ) = ¯ htm qq ( t ) /µb . The semiclassical acceleration, as defined in (7), is simply a sc = − V µ | ψ sc (0 , t ) | (22)or, using (6), (20) and (21), a sc ( t ) = − V µ √ πσ ( t ) e − (cid:18) ptµ − x σ ( t ) (cid:19) , for t ≤ t e − (cid:16) p ( t − t µσ ( t ) (cid:17) , for t > t , (23)where σ ( t ) = b r(cid:16) ¯ htb m (cid:17) + 1. Numerical integration leads to q sc ( t ), which is depicted in figure1 along with q c ( t ) and the time evolution of the exact quantum average h ˆ q i ( t ). The initialposition and momentum are q = 0 and p = 1, and the barrier is located at q = 1 withheight V = 5, in a system of arbitrary units. The other parameters are b = 0 . µ = 1.In the top panel ¯ h = 0 .
05 and in the bottom panel ¯ h = 0 . h (see the bottom plot). V. CONCLUSION AND PERSPECTIVES
We have developed a direct way to determine semiclassical trajectories for short propa-gation times. Despite this limitation we were able to reproduce some important quantumfeatures, like the tendency of turning points to recede in the quantum regime. In addition,we identified how quantum non-locality manifests itself in the semiclassical phase-space, de-stroying the original Hamiltonian structure. In this context it is not possible to resort toany extended Hamiltonian formalism [17] (see also[18]) .Perhaps, a promising perspective to be drawn from this approach is related to its gen-eralization to higher dimensional problems in connection with chaos. On the one hand wehave classical mechanics which presents sensitivity to initial conditions in a variety of sys-tems, and, on the other hand is quantum mechanics, whose linearity prevents exponentialseparation of h r i and h r + d r i . In principle, our semiclassical paths should be in the middle-way, which raises the question on the behavior of the Liapunov exponents associated tosemiclassical paths and on their dependence on ¯ h . VI. AKNOWLEDGEMENTS
Funding from CNPq and FACEPE (APQ-1415-1.05/10) is gratefully acknowledged. [1] R. P. Feynman and A. R. Hibbs,
Quantum mechanics and path integrals (Dover, New York,1965).[2] D. Bohm, Phys. Rev. , 166 (1952).[3] S. Kocsis et al., Science, , 1170 (2011).[4] M. Novaes, J. Math. Phys. , 102102 (2005).[5] L. D. C. Jaubert and M. A. M. de Aguiar, Phys. Scripta , 363 (2007).[6] R. N. P. Maia, F. Nicacio, and R. O. Vallejos, Phys. Rev. Lett. , 184102 (2008).[7] A. L. Xavier and M. A. M. de Aguiar, Ann. Phys. , 458 (1996).[8] M. Baranger, M. A. M. de Aguiar, F. Keck, H. J. Korsch, and B. Schellhaas, J. Phys. A ,7227 (2001).[9] A. D. Ribeiro, M. A. M. de Aguiar, and M. Baranger, Phys. Rev. E ,066204 (2004).[10] F. Parisio and M. A. M. de Aguiar, J. Phys. A , 9317 (2005).[11] M. Novaes and M. A. M. de Aguiar, Phys. Rev. A , 032105 (2005).[12] D. G. Levkov, A. G. Panin, and S. M. Sibiryakov, Phys. Rev. E , 046209 (2007). t < x > t < x > FIG. 1: Semiclassical trajectories x sc ( t ) (red), quantum averages h x i ( t ) (blue) and classical paths(green dashed lines). The initial position and momentum are q = 0 and p = 1, the otherparameters being b = 0 . µ = 1 in a system of arbitrary units. In the left figure ¯ h = 0 .
05 andin the right figure ¯ h = 0 .
1. The time interval in the horizontal axis ends soon after the classicalturning time.[13] Y. Goldfarb, J. Schiff, and D. J. Tannor, J. Chem. Phys. , 164114 (2008).[14] A. D. Ribeiro, Phys. Lett. A , 812 (2011).[15] L. E. Ballentine, Y. Yang, and J. P. Zibin, Phys. Rev. A , 2854 (1994).[16] E. J. Heller, J. Chem. Phys. , 1544 (1975).[17] A. K. Pattanayak and W. C. Schrieve, Phys. Rev. E , 3601 (1994).[18] F. Cooper, S-Y Pi, and P. N. Stancioff. Phys. Rev. D34