aa r X i v : . [ qu a n t - ph ] J a n Path Integral Analysis of Arrival Times with a Complex Potential
J.J.Halliwell
Blackett LaboratoryImperial CollegeLondon SW7 2BZUK (Dated: October 31, 2018)
Abstract
A number of approaches to the arrival time problem employ a complex potential of a simplestep function type and the arrival time distribution may then be calculated using the stationaryscattering wave functions. Here, it is shown that in the Zeno limit (in which the potential becomesvery large), the arrival time distribution may be obtained in a clear and simple way using a pathintegral representation of the propagator together with the path decomposition expansion (in whichthe propagator is factored across a surface of constant time). This method also shows that thesame result is obtained for a wide class of complex potentials.
PACS numbers: . INTRODUCTION Some of the interesting outstanding problems in quantum theory concern situations inwhich time appears in a non-trivial way. One of the simplest such problems is the arrivaltime problem. In the one-dimensional version of this problem one considers an initial wavefunction concentrated in the region x > x = 0 between time τ and τ + dτ . The classical analysis of this problem is trivial, but the quantum analysis is not so,due in part to the fact that the usual machinery of quantum measurement refers to fixedmoments of time and not to measurements distributed over an interval of time.There are many different approaches to this problem [1]. One particular approach is toinclude a complex potential V ( x ) = − iV θ ( − x ) (1)in the Schr¨odinger equations, and to then compute the final state | ψ ( τ ) i = exp (cid:18) − i ~ H τ − V ~ θ ( − x ) τ (cid:19) | ψ i (2)where H is the free Hamiltonian. The intuitive idea behind this is that the part of the wavepacket that reaches the origin during the time interval [0 , τ ] is absorbed, so that N ( τ ) = h ψ ( τ ) | ψ ( τ ) i (3)is the probability of not crossing during the time interval. The probability of crossingbetween τ and τ + dτ is then Π( τ ) = − dNdτ (4)Such potentials were originally considered by Allcock in his seminal work [2] and havesubsequently appeared in detector models of arrival times [3, 4]. A recent interesting resultof Echanobe et al. is that under certain conditions a complex potential of the form Eq.(1)is essentially the same as pulsed measurements, in which the wave function is measured atdiscrete time intervals [5].The difficulty behind this approach, however, is that the wave function is not entirelyabsorbed by this complex potential and N ( τ ) includes parts of the initial wave function thathave reflected off the potential. The reflection is small for small V but then the resolutionof the measurement, which is proportional to ~ /V , is poor. On the other hand, there is a2ot of reflection for large V and indeed the wave function is entirely reflected in the limit V → ∞ . This is the quantum Zeno effect [6] and plagues many different approaches to thearrival time problem.Echanobe et al. have made an interesting proposal which embraces the Zeno effect inthe V → ∞ limit yet at the same time extracts the physics hidden within it by suitablenormalization [5]. They consider the limit V → ∞ of the expressionΠ N ( τ ) = Π( τ )1 − N ( ∞ ) (5)which is normalized since R ∞ dτ Π( τ ) = 1 − N ( ∞ ) . The point is that N ( ∞ ) represents thetotal amount of reflected wave function, so 1 − N ( ∞ ) represents the total probability ofcrossing during the time interval [0 , ∞ ) and this goes to zero as V → ∞ . The expressionΠ( τ ) also goes to zero as V → ∞ but the ratio Eq.(5) is finite and independent of V , anddefines a reasonable normalized arrival time distribution function. Known results on thestationary scattering wave functions [7] yield the resultΠ( τ ) = 2 m / V / h ψ f ( τ ) | ˆ pδ (ˆ x )ˆ p | ψ f ( τ ) i (6)from which the normalized result isΠ N ( τ ) = ~ m h p i h ψ f ( τ ) | ˆ pδ (ˆ x )ˆ p | ψ f ( τ ) i (7)where | ψ f ( τ ) i is the freely evolved wave function and h p i is the average momentum in theinitial wave packet [5, 7].An interesting question in these expressions is the origin of the form of the ˆ pδ (ˆ x )ˆ p term,which is not obvious from the calculation of it in Ref.[7]. Compare this to the simplest guessfor the arrival time distribution function, the current density J ( t ) = ~ m h ψ f ( τ ) | (ˆ pδ (ˆ x ) + δ (ˆ x )ˆ p ) | ψ f ( τ ) i (8)which is sensible classically but is not always positive in the quantum case [8]. (Here δ (ˆ x ) = | ih | , where | i denotes the eigenstate | x i of the position operator at x = 0.) A simpleoperator re-ordering of J ( t ) gives the “ideal” arrival time distribution of KijowskiΠ K ( τ ) = ~ m h ψ f ( τ ) | ˆ p / δ (ˆ x )ˆ p / | ψ f ( τ ) i (9)which is clearly positive, but harder to relate to specific measurement schemes [9]. It wouldbe of value to find a derivation of Eq.(7) which gives some insight into the form of this3xpression. A further question concerns the role of the complex potential. Although Eq.(7)is independent of V , it is not clear to what extent the result depends on the specific choiceof complex potential, Eq.(1).In this paper, we show that both of these questions are answered very simply usingpath integral methods. We show that the general form (6) is derived very easily (up to aconstant, fixed by normalization). Secondly, the result (7) is seen to be true for a wide classof potentials of the form V ( x ) = − iV θ ( − x ) f ( x ) (10)where f ( x ) is any positive function.With the general complex potential Eq.(10), the arrival time distribution (4) is given byΠ( τ ) = 2 h ψ τ | V (ˆ x ) | ψ τ i = 2 V ~ Z −∞ dx f ( x ) | ψ ( x, τ ) | (11)where ψ ( x, τ ) is defined in Eq.(2). The key is therefore to evaluate the propagator g ( x , τ | x ,
0) = h x | exp (cid:18) − i ~ H τ − V ~ θ ( − ˆ x ) f (ˆ x ) τ (cid:19) | x i (12)for x < x >
0. This may be calculated using a sum over paths, g ( x , τ | x ,
0) = Z D x exp (cid:18) i ~ S (cid:19) (13)where S = Z τ dt (cid:18) m ˙ x + iV θ ( − x ) f ( x ) (cid:19) (14)and the sum is over all paths x ( t ) from x (0) = x to x ( τ ) = x .To deal with the step function form of the potential we need to split off the sections of thepaths lying entirely in x > x <
0. The way to do this is to use the path decompositionexpansion or PDX [10–12]. Each path from x > x < x = 0many times, but all paths have a first crossing, at time t , say. As a consequence of this, itis possible to derive the formula, g ( x , τ | x ,
0) = i ~ m Z τ dt g ( x , τ | , t ) ∂g r ∂x ( x, t | x , (cid:12)(cid:12) x =0 (15)Here, g r ( x, t | x ,
0) is the restricted propagator given by a sum over paths of the form (13)but with all paths restricted to x ( t ) >
0. It vanishes when either end point is the origin but4ts derivative at x = 0 is non-zero (and in fact the derivative of g r corresponds to a sumover all paths in x > x = 0 [12]). It is also useful to record a PDX formulainvolving the last crossing time t , g ( x , τ | x ,
0) = − i ~ m Z τ dt ∂g r ∂x ( x , τ | x, t ) (cid:12)(cid:12) x =0 g (0 , t | x , t ) (16)These two formulae may be combined to give a first and last crossing version of the PDX, g ( x , τ | x ,
0) = ~ m Z τ dt Z t dt ∂g r ∂x ( x , τ | x, t ) (cid:12)(cid:12) x =0 g (0 , t | , t ) ∂g r ∂x ( x, t | x , (cid:12)(cid:12) x =0 (17)The restricted propagators in the two regions are easier to work with than Eq.(12) sincethe potential is zero throughout x > − iV f ( x ) throughout x < θ ( − x ) term. The problemof calculating the propagator Eq.(12) therefore essentially reduces to the easier problem ofcalculating it between two points lying on x = 0. This can sometimes be evaluated bya mode sum calculation [13], even though the full propagator Eq.(12) is not necessarilycalculable in this way.Returning to the first crossing PDX, Eq.(15), g r is the restricted propagator for the freeparticle , which is given by the method of images expression g r ( x , τ | x ,
0) = θ ( x ) θ ( x ) ( g f ( x , τ | x , − g f ( − x , τ | x , g f denotes the free particle propagator. It follows that ∂g r ∂x ( x, t | x , (cid:12)(cid:12) x =0 = 2 ∂g∂x (0 , t | x , θ ( x ) (19)Inserting this in Eq.(15), and rewriting it as an operator expression, we obtain the result h x | exp (cid:18) − i ~ H τ − V ~ θ ( − ˆ x ) f (ˆ x ) τ (cid:19) | x i = − m Z τ dt h x | exp (cid:18) − i ~ H ( τ − t ) − V ~ θ ( − ˆ x ) f (ˆ x )( τ − t ) (cid:19) × δ (ˆ x )ˆ p exp (cid:18) − i ~ H t (cid:19) | x i (20)We note the appearance of the combination δ (ˆ x )ˆ p which clearly corresponds to a “crossingoperator” (and in derivations of the PDX arises directly from a derivative of θ (ˆ x ) [11]). Nownote that the operator δ (ˆ x ) has the simple property that for any operator Aδ (ˆ x ) Aδ (ˆ x ) = δ (ˆ x ) h | A | i (21)5his property together with Eq.(20) inserted in Eq.(11) yieldsΠ( τ ) = 2 V ~ m Z τ dt ′ Z τ dt Z −∞ dx f ( x ) × h | exp (cid:18) i ~ H † ( τ − t ′ ) (cid:19) | x ih x | exp (cid:18) − i ~ H ( τ − t ) (cid:19) | i× h ψ | exp (cid:18) i ~ H t ′ (cid:19) ˆ p δ (ˆ x ) ˆ p exp (cid:18) − i ~ H t (cid:19) | ψ i (22)where H = H − iV θ ( − x ) f ( x ) is the total (non-hermitian) Hamiltonian. Now the key pointis that as V → ∞ , the integrals over t and t ′ are strongly concentrated around τ , so weobtain Π( τ ) ≈ C h ψ | exp (cid:18) i ~ H τ (cid:19) ˆ p δ (ˆ x ) ˆ p exp (cid:18) − i ~ H τ (cid:19) | ψ i (23)where C = 2 V ~ m Z τ dt ′ Z τ dt Z −∞ dx f ( x ) × h | exp (cid:18) i ~ H † ( τ − t ′ ) (cid:19) | x ih x | exp (cid:18) − i ~ H ( τ − t ) (cid:19) | i (24)Furthermore, by the change of variables s = τ − t , s ′ = τ − t ′ , it is easily seen that C is independent of τ for large V , so is just a constant. It is also easily seen that theapproximation (23) will hold for V ≫ k / m , where | k | is the largest momentum in theinitial state.Eq.(23) is the main result and is of precisely the form Eq.(6), up to an overall constant.Since Eq.(7) is obtained by normalization of Eq.(6), we therefore obtain Eq.(7). It holds for ageneral class of potentials, of the form Eq.(10), not just the simple case f ( x ) = 1 consideredin Refs.[5, 7]. The reason for this is that for large V the paths in the path integral are keptout of the region x <
0, only entering it at just before the final time, so the result has verylimited dependence on the detailed form of the potential in x <
0. The result Eq.(23), andin particular the ˆ pδ (ˆ x )ˆ p form, follow as an almost immediate consequence of the PDX forlarge V , together with the simple property Eq.(21).To fully verify the validity of this path integral approach, we now calculate the constant C in the case f ( x ) = 1 and look for detailed agreement with the unnormalized result Eq.(6).Eq.(24) may be written, C = 2 V ~ m Z −∞ dx | φ ( x ) | (25)6here, after a change of variables s = τ − t , φ ( x ) = Z τ ds h x | exp (cid:18) − i ~ H s − V ~ θ ( − ˆ x ) s (cid:19) | i (26)The integrand may be represented as a sum over paths from x = 0 at time s = 0 to thefinal point x < τ . We may use the last crossing PDX, Eq.(16), to split this into asum over paths from x = 0 at s = 0, to the last crossing at x = 0, s = s , and from therepropagating entirely in x < x at time s = τ . This reads φ ( x ) = 1 m Z τ ds Z s du h x | exp (cid:18) − i ~ H ( s − u ) − V ~ ( s − u ) (cid:19) ˆ p | ih | exp (cid:18) − i ~ Hu (cid:19) | i (27)In the second bracket expression, H is the total Hamiltonian so still includes the θ ( − x )potential. This is the propagator along the edge of an imaginary step potential which,fortunately was calculated in Ref.[13] using a mode sum method (for the case of a real steppotential, but this is readily continued to imaginary values), and we have h | exp (cid:18) − i ~ Hu (cid:19) | i = (cid:16) m πi ~ (cid:17) / (1 − e − V u/ ~ )( V / ~ ) u / (28)Noting that Z τ ds Z s du = Z τ du Z τu ds (29)and changing variables to v = s − u , we see that for large V the dominant contributioncomes from close to u = 0 and v = 0. We may therefore let τ → ∞ , and the integrals thenfactor into a product φ ( x ) = 1 m Z ∞ dv h x | exp (cid:18) − i ~ H v − V ~ v (cid:19) ˆ p | i Z ∞ du (cid:16) m πi ~ (cid:17) / (1 − e − V u/ ~ )( V / ~ ) u / (30)The first integral may be evaluated using the familiar formula [14] Z ∞ dt (cid:16) m πi ~ t (cid:17) / exp (cid:18) i ~ (cid:20) Et + mx t (cid:21)(cid:19) = (cid:16) m E (cid:17) exp (cid:18) i ~ | x |√ mE (cid:19) (31)with E = iV , and then applying − i ~ ∂/∂x . The second is evaluated using the formula, Z ∞ dx (1 − e − x ) x / = 2 √ π (32)We thus obtain φ ( x ) = (cid:18) mV (cid:19) / exp (cid:18) − (1 − i ) ~ p mV | x | (cid:19) (33)7nserting in Eq.(25), this gives the final result C = 2 m / V / (34)The path integral method used here therefore gives precise agreement with the earlier resultEq.(6) obtained by scattering methods. [1] J.G.Muga, R.Sala Mayato and I.L.Egusquiza (eds), Time in Quantum Mechanics (Springer,Berlin, 2002); J.G.Muga and C.R.Leavens, Phys.Rep.338, 353 (2000).[2] G.R.Allcock, Ann.Phys 53, 253 (1969); 53, 286 (1969); 53, 311 (1969).[3] J.J.Halliwell, Prog.Th.Phys.102, 707 (1999).[4] J.A.Damborenea, I.L.Egusquiza, G.C.Hegerfeld and J.G.Muga, Phys. Rev. A66, 052104(2002).[5] J.Echanobe, A. del Campo and J.G.Muga, quant-ph 0712.0670 (2007).[6] B.Misra and E.C.G.Sudarshan, J.Math.Phys. 18, 756 (1977).[7] J.G.Muga, D.Seidel and G.C.Hegerfeldt, J.Chem.Phys. 122, 154106 (2005).[8] See, for example, J. G. Muga, J. P. Palao and C. R. Leavens Phys.Lett. A253, 21 (1999), andreferences therein.[9] J.Kijowski, Rep.Math.Phys. 6, 361 (1974).[10] A.Auerbach and S.Kivelson, Nucl.Phys. B257, 799 (1985); P. van Baal, in Lectures on PathIntegration