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On the “standard” quantum mechanical approach to times of arrival
Physics Letters A 303 (2002) 154–165 www.elsevier.com/locate/pla
On the “standard” quantum mechanical approach to times of arrival C.R. Leavens Institute for Microstructural Sciences, National Research Council of Canada, Ottawa, K1A 0R6, Canada Received 20 August 2002; accepted 5 September 2002 Communicated by P.R. Holland
Abstract The “standard” approach to times of arrival advocated by Equsquiza, Muga and Baute associates arrivals from the left (right) with positive (negative) wavenumber components of the wave function at the instant of arrival. Elementary arguments and calculated results exhibiting paradoxical properties call into question this sign of k assignment of direction of arrival. 2002 Published by Elsevier Science B.V.
1. Introduction Recently there has been renewed interest [1–3] in the problem of deriving an expression for the distribution Π(T ; X) of intrinsic arrival times T at an arbitrary spatial point X for an ensemble of quantum particles each prepared in the same initial state ψ(x, t = 0). Several different approaches [4–8] all based on conventional nonrelativistic quantum mechanics lead to the same result: Π(T ; X) = Π+ (T ; X) + Π− (T ; X) with
∞ 2 h¯ dk Θ(±k)|k|1/2 exp(ikX)φ(k, T ) , Π± (T ; X) = 2πm
(1)
(2)
−∞
where Π+ (T ; X) and Π− (T ; X) denote the contributions to Π(T ; X) of arrivals at X from the left and from the right, respectively, Θ is the unit step function, and φ(k, t) is the Fourier transform of the time-evolved wave function ψ(x, t). It should be emphasized that according to (2) right-going (+) arrivals at X are associated only with positive wavenumbers (k > 0) and left-going (−) arrivals only with negative wavenumbers (k < 0). The derivation of Allcock [4], of Kijowski [5], of Grot, Rovelli and Tate [6], and of Delgado and Muga [7] are for the special case of free (f ) evolution, in which case φ(k, t) = φ (f ) (k, t) ≡ φ (f ) (k) exp(−i h¯ k 2 t/2m), where E-mail address: [email protected] (C.R. Leavens). 0375-9601/02/$ – see front matter 2002 Published by Elsevier Science B.V. PII: S 0 3 7 5 - 9 6 0 1 ( 0 2 ) 0 1 2 3 9 - 2
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φ (f ) (k) ≡ φ (f ) (k, 0) is the Fourier transform of the initial wave function ψ(x, 0). Logically, one is concerned only with arrivals subsequent to the time, here t = 0, at which the initial state is prepared. However, for reasons of mathematical convenience, these authors considered instead the artificial problem in which arrival times in the range [−∞, 0] are included by imagining that the quantum particle was prepared at t = −∞ in the state which, in the assumed absence of any interaction including that with the apparatus actually used to prepare the state at t = 0, would evolve to the desired initial state ψ(x, t = 0). The mathematical advantage—“resolution of the identity”—of treating the artificial problem is tied to the implicit assumption that for free evolution the particle is certain to reach x = X, for any finite X, once and only once, at some T (X) in the range [−∞, +∞], provided that φ(f ) (k = 0) is zero. In this case, Π(T ; X) given by (1) and (2) is an arrival-time probability distribution +∞ (i.e., −∞ dT Π(T ; X) = 1). The more recent derivation of Baute, Sala Mayato, Palao, Muga and Egusquiza (BSPME) [8] is for arbitrary potentials V (x, t) and for both stationary and non-stationary states. In general, in the presence of a scattering potential which can prevent some particles from reaching X and/or cause others to reach X more than once, the density Π(T ; X) given by (2) is no longer a probability distribution [8] and the above advantage of the artificial problem no longer exists. For the non-stationary case, it then seems sensible to consider only arrivals subsequent to t = 0, even—for the sake of generality—in the free evolution case. The “standard” arrival-time density includes all orders of arrival but as yet there is no prescription for extracting the individual contributions of the various orders (e.g., the contribution of first arrivals). Egusquiza, Muga and Baute [9] title their distillation of the various approaches leading to Eqs. (1) and (2) “standard” quantum mechanical approach to times of arrival, emphasizing their claim that (2) can be derived “without in any way distorting the standard framework of quantum mechanics”. However, even for the case of free evolution, the present author regards the above association of direction of arrival with sign of k as an unjustifed assumption that is not a part of conventional quantum mechanics [10]. Hence, despite the excellent agreement reported in Chapters 1, 8 and 10 of [3] with a numerical “quantum jump” time-of-flight simulation, the author has some reservations about referring to (2) as a result of standard quantum mechanics. In this Letter counterintuitive and/or paradoxical properties of the “standard” arrival-time distribution are pointed out, illustrated with simple concrete examples, and contrasted with the corresponding—and frequently counterclassical—properties of a non-standard arrival-time distribution, namely, that of Bohm’s alternative [11– 13] to conventional quantum mechanics. The case studies in the following three sections were selected to provide simple illustrations of the following formal argument. Consider the integral in (2) as a function of X, for a fixed value of T , and write it in the form ∞ (2πkc )
1/2 −∞
dk exp(ikX)φ˜ ± (k, T ) ≡ (2πkc )1/2 ψ˜ ± (X, T ), (2π)1/2
with kc an arbitrary positive wavenumber and 1/2 k ˜ φ± (k, T ) ≡ Θ(±k) φ(k, T ). kc
(3)
(4)
Since the integral in (2) is proportional to the Fourier transform ψ˜ ± (X, T ) of the function φ˜ ± (k, T ) which is zero for either k < 0 or k > 0, it follows that the support of Π± (T ; X) ∝ |ψ˜ ± (X, T )|2 is the entire X-axis. This is true even if the wave function ψ(x, t) is zero for all t over an extended spatial range due to the presence of an infinitely repulsive barrier. Hence, one has the paradoxical result that the “standard” arrival time distribution Π(T ; X) = Π+ (T ; X) + Π− (T ; X) obtained from (2) can be finite for X in the interior of a quantally forbidden region where the probability of finding the particle is exactly zero at any time t. It is important to know whether or not this predicted effect is of nonnegligible importance. This is investigated for the stationary-state case in Section 2 for a quantum particle in an energy eigenstate of a rectangular well with infinitely high walls. In Section 3 the freely-evolving time-dependent case is considered by collapsing the confining potential instantaneously to zero at t = 0 and calculating the resulting arrival-time distributions Π(T ; X) for T 0 at a point X a finite distance
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outside the (former) well region. In Section 4 arrival-time distributions are considered for an initial Gaussian wave packet incident on a perfectly reflecting infinite potential step at x = 0. The interesting properties of the “standard” arrival-time density illustrated in these case studies and their possible implications are discussed in Section 5. The following extremely brief sketch of Bohmian mechanics is intended to facilitate the above-mentionned comparisons. In Bohm’s theory [11–13] a quantum particle such as an electron is postulated to be an actually existing point-like particle and an accompanying wave which guides its motion. For a nonrelativistic quantum particle in the presence of a potential V (x, t), the time-evolution of the guiding wave ψ(x, t) is described by the Schrödinger equation and the trajectory of the point-like particle is determined by the equation-of-motion dx(t)/dt = [J (x, t)/|ψ(x, t)|2 ]|x=x(t ) where J (x, t) ≡ (h¯ /m)[ψ ∗ (x, t)∂ψ(x, t)/∂x] is the probability current density. (For the 1D system considered here there is no spin-dependent contribution [14].) It is further postulated that, for an ensemble of electrons all prepared in the same initial state ψ(x, 0), the probability of such a particle having initial position x (0) ≡ x(t = 0) is given by |ψ(x (0), 0)|2 . The expression for the intrinsic arrival-time probability distribution for those particles that do reach X subsequent to t = 0 is readily obtained when arrivals at X are possible [10]. It is given by Π(T ; X) = N|J (X, T )|; the right-going and left-going contributions are Π± (T ; X) = ±NJ± (X, T ) where the right- and left-going components of the probability current density are determined by the sign of J , i.e., J± (x, t) ≡ J (x, t)Θ[±J (x, t)]. The normalization factor N will usually be ignored (i.e., set equal to 1) in graphical comparisons with the BSPME arrival-time density.
2. Stationary-state case Consider the special case of a bound-state energy eigenfunction ψ(x, t) = ψn (x, t) ≡ Rn (x) exp(−iEn t/h¯ )
(5)
with Rn (x) real so that J (x, t) = 0. For this case, it follows immediately from (2) that, as expected, Π+ (T ; X) and Π− (T ; X) are independent of T and it is the relative number of arrivals per unit time at various spatial points X that is of interest. It is also straightforward to show that Π+ and Π− are equal so that the classically motivated expression J Π (X, T ) ≡ Π+ (T ; X) − Π− (T ; X) for the probability current density is zero which is consistent with J (x, t) = 0 for all x and t. Now consider the special case of a quantum particle in a bound-state energy eigenfunction of a rectangular potential well with infinitely repulsive walls at x = 0 and x = a. The spatial part of the nth energy eigenfunction is Rn (x) =
1/2 2 sin(kn x)Θ(x)Θ(a − x), a
kn ≡
(2mEn )1/2 nπ ≡ , h¯ a
(6)
with n = 1, 2, . . . . If one deliberately repeats a frequently made mistake and calculates φn (k, T ) ≡ φn (k) × exp(−iEn T /h¯ ) by Fourier transforming just sin(kn x) rather than sin(kn x)Θ(x)Θ(a − x) then one obtains from (2) that Π+ (T ; X) = Π− (T ; X) = (1/2)(vn /a) with vn ≡ h¯ kn /m. This corresponds to the well-known, intuitively appealing, and (apart from the discrete nature of vn ) classical picture of a point-particle bouncing back and forth inside the well with a constant speed vn between its instantaneous elastic collisions with the infinitely hard walls. If one uses the correct Fourier transform kn 1/2 kn nπk nπk n n 1 − (−1) + i(−1) cos sin φn (k) = (7) kn kn nπ 2 kn2 − k 2 of Rn (x) in (2) then one obtains instead Π± (T ; X) =
1 n3 π 5
vn 2 Cn (X/a) + Sn2 (X/a) 2a
(8)
C.R. Leavens / Physics Letters A 303 (2002) 154–165
Fig. 1. Dependence of Π+ (T ; X) = Π− (T ; X) calculated with (2) on X for a quantum particle in the energy eigenstate ψn (x, t) of a rectangular potential well with infinitely strong confining potential: n = 1 (solid curve); n = 2 (dashed curve); n = 3 (long-short dashed curve). The probability of finding the quantum particle to the left of the dotted vertical line is zero.
157
Fig. 2. Dependence of Π+ (T ; X)/Π(T ; 0) calculated with (2) on X for a quantum particle in the energy eigenstate ψn=1 (x, t) of a rectangular well with confining potential of height V0 = 10E1 (long-short dashed curve), V0 = 102 E1 (short dashed curve), V0 = 103 E1 (dotted curve), V0 = 104 E1 (thin solid curve), and ∞ (thick solid curve). The ground state energy E1 is a function of V0 for finite V0 .
with ∞ Cn (X/a) = 0
∞ Sn (X/a) = 0
du u1/2 cos(Xu/a) − (−1)n cos (1 − X/a)u , 2 1 − (u/nπ)
du u1/2 sin(Xu/a) + (−1)n sin (1 − X/a)u . 2 1 − (u/nπ)
(9)
Π± (T ; X) is symmetric about X = a/2. Fig. 1 shows the dependence of Π± (T ; X)/(vn /2a) on X/a over the range −a/2 X a/2 for n = 1, 2 and 3. In each case, the predicted relative probability of a quantum particle arriving at a point X is strongly suppressed as X moves from the interior of the well into the quantally forbidden region X < 0, but it is far from negligible there for X just outside the well. Fig. 2 shows Π+ (T ; X) = Π− (T ; X) for the n = 1 energy eigenstate of the rectangular well with a series of finite confining potentials V0 Θ(−x)Θ(a − x) of increasing height V0 to show that there is convergence to the result presented in Fig. 1 for the limiting, and potentially problematic, infinite V0 case.
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Fig. 3. Effect of interference between k > 0 and k < 0 contributions. The “standard” result for Π(T ; X) is shown by the thin solid curve, the result for F (T ; X) by the thick solid curve.
It is informative to consider the function1 2 ∞ h¯ 1/2 F (T ; X) = dk |k| exp(ikX)φ(k, T ) 2πm
(10)
−∞
obtained by removing the theta function Θ(±k) from the integral ∞ in Eq. (2). For a given value of T , the integrals of Π(T ; X) and F (T ; X) over all X are both equal to (h¯ /m) −∞ dk |k||φ(k, T )|2 which provides a useful check on the accuracy of the numerical calculations. For the n = 1 energy eigenstate (6), Fig. 3 compares F (T ; X) with Π(T ; X) = Π+ (T ; X) + Π− (T ; X) as a function of X/a. Remarkably, F (T ; X) is also not zero in the quantally forbidden region |X − a/2| > a/2 even though it, unlike Π± (T ; X), does not involve the Fourier transform of a function of k that is exactly zero over an extended interval of k values. This suggests that the real culprit behind the predicted non-negligible value of Π(T ; X) for |X −a/2| > a/2 is the occurrence of the non-analytic function |k|1/2 in the integrand of (2). As usual, another possibility is human error in the numerical calculation. Now, (10) with the factor |k|1/2 replaced by 1 is proportional to Rn (X)2 and replaced by k is proportional to [(dRn (x)/dx)|x=X ]2 , both of which are exactly zero for |X − a/2| > a/2. Numerical evaluation of these two quantities after trivial changes to just one line of coding in the program for evaluating F (T ; X) did give zero to within numerical accuracy for |X − a/2| > a/2, strengthening the case for the factor |k|1/2 being the actual culprit. For the stationary-state wave function (5) with J (x, t) = 0 an arrival-time distribution is not defined within Bohmian mechanics because each particle in the ensemble is motionless—its energy En is completely in the form of quantum potential energy [11–13]—at some unknown position x (0) within the well, with the various x (0) s distributed according to Rn (x (0))2 . If x (0) = X then the particle is always at X; if x (0) = X then it never arrives at X. 1 This function is introduced only to illustrate a mathematical point and is not intended as a candidate arrival-time distribution.
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3. Time-dependent free-evolution case Assume that for t < 0 the quantum particle is in the ground-state ψn=1 (x, t) of the infinite rectangular well of the previous section and that at t = 0 the confining barriers are instantaneously removed. For t > 0 the Fourier transform of the resulting freely-evolving wave function ψ(x, t) is φ (f ) (k, t) = φ (f ) (k) exp(−i h¯ k 2 t/2m) with φ (f ) (k) = φn=1 (k) given by (7). The resulting “standard” result for the right- and left-going components of the arrival-time distribution for T 0 at the point X = 1.1a, obtained by numerical evaluation of (2), is shown in Fig. 4. Both components are nonzero at T = 0, with the left-going component decreasing monotonically with increasing T . Regarding the latter, it is as if there is a non-zero probability amplitude for left-going particles being to the right of X = 1.1a immediately prior to t = 0. The corresponding arrival-time distribution based on Bohmian mechanics is readily obtained from the wave function ψ(x, t) for t > 0 which is given by [15] ψ(x, t) =
im(x − a)2 1 2 1/2 w[u+ (x − a, t)] − w[u− (x − a, t)] exp(ik1a) exp 4i a 2h¯ t
2 imx − exp w[u+ (x, t)] − w[u− (x, t)] 2h¯ t
(11)
with u± (x, t) ≡ −
(1 + i)(±h¯ k1 − mx/t) , 2(mh¯ /t)1/2
(12)
Fig. 4. T -dependence of Π± (T ; X) for X = 1.1a. The “standard” results for Π+ and Π− are shown by the thick solid and dashed curves respectively; the result of Bohmian mechanics for Π+ /2 = J is shown by the thin solid curve (Π− is zero for the space–time region shown). The results are scaled by the characteristic time t1 ≡ a/v1 .
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2
w(u) ≡ exp −u2 erfc(−iu) ≡ exp −u2 1/2 π
∞
dz exp −z2 .
(13)
−iu
The corresponding expression for the probability current density for t > 0 is 2 2 2 2 v1 J (x, t) = w[u+ (x − a, t)] − w[u− (x − a, t)] + w[u+ (x, t)] − |w[u− (x, t)] 8a mx am − 2 cos k1 − + a w∗ [u+ (x, t)]w[u+ (x − a, t)] − w∗ [u− (x, t)]w[u− (x − a, t)] h¯ t 2h¯ t
∗ mx am ∗ a w [u+ (x, t)]w[u+ (x − a, t)] − w [u− (x, t)]w[u− (x − a, t)] . + + 2 sin k1 − 2h¯ t h¯ t (14) At least, for the range of T values investigated, J (X = 1.1a, T ) is non-negative so that within Bohmian mechanics Π− (T ; X = 1.1a) is zero and Π(T ; X = 1.1a) = Π+ (T ; X = 1.1a) = 2J (X = 1.1a, T ). To facilitate comparison with the “standard” result, which is not normalized to unity, the factor of 2 has been dropped in Fig. 4. A satisfactory feature of the Bohm arrival-time distribution is that it is zero at T = 0 as one would expect on physical grounds.
4. Time-dependent nonzero potential case We now approach a non-stationary state example of the paradoxical effect of Section 2 in an indirect way in order to illustrate further interesting properties of the “standard” and Bohm trajectory arrival-time distributions and also to make contact with the work of Yamada and Takagi [16,17]. We begin by first considering a simple example of the free propagation of a non-relativistic wave function that is antisymmetric about x = 0 at all times t [10]. The initial wave function (x − x0 )2 (x + x0 )2 (f ) + ik0 x − exp − − ik0 x ψ (x, 0) ≡ N exp − (15) 4('x)2 4('x)2 with
3/2 1/2 N = 2 π 'x 1 − exp −2('x)2k02 −
x02 2('x)2
−1/2 (16)
is a coherent linear superposition of two Gaussians with individual centroids ±x0 , mean wavenumbers ±k0 , respectively, and equal spatial widths 'x. It readily follows that −i h¯ k 2 t (f ) 1/2 φ (k, t) = N2π 'x exp 2m
2 × exp −(k − k0 ) ('x)2 − i(k − k0 )x0 − exp −(k + k0 )2 ('x)2 + i(k + k0 )x0 , (17) 1/2
i h¯ t ψ (f ) (x, t) = 2N'x ('x)2 − η1/2 exp −k02 ('x)2 + ik0 x0 exp β0 + β2 x 2 + i γ0 + γ2 x 2 2m × exp(δx + i,x) − exp(−δx − i,x) . (18) The corresponding probability current density is J (f ) (x, t) =
h¯ 2N 2 ('x)2 η1/2 exp −2k02 ('x)2 exp 2 β0 + β2 x 2 m × 4γ2 x[cosh(2δx) − cos(2,x)] + 2, sinh(2δx) − 2δ sin(2,x) .
(19)
C.R. Leavens / Physics Letters A 303 (2002) 154–165
In (18) and (19) −1 η ≡ 4('x)4 + (h¯ t/m)2 , ht ¯ , β2 ≡ −η('x)2 , β0 ≡ η 4('x)6k02 − ('x)2x02 − 2k0x0 ('x)2 m
h¯ t ηh¯ t , γ2 ≡ , γ0 ≡ −η 4k0 x0 ('x)4 + 4k02 ('x)4 − x02 2m 2m k0 h¯ t x0 ht ¯ δ ≡ 2η('x)2 x0 + , , ≡ 2η 2k0('x)4 − . m 2m
161
(20)
For x0 < 0 and k0 > 0, the quantity t0 ≡ −x0 /(h¯ k0 /m) provides a useful characteristic time, namely the time at which the centroids of the individual Gaussian components of (18) reach x = 0. At the instant t = t0 the wave function (18) has a string of nodes at the points xl = l/πk0 (l = 0, ±1, ±2, . . .). The probability density |ψ (f ) (x, t)|2 is zero at x = 0 for all t. Within (nonrelativistic) Bohmian mechanics a particle can reach a point x = X at a time t = T when |ψ(X, T )|2 = 0 only if v(X, T ) ≡ J (X, T )/|ψ(X, T )|2 = ±∞ [18]. From (18) and (19), respectively, it follows that as x → 0, for any t, |ψ (f ) (x, t)|2 → 0 as x 2 and J (f ) (x, t) → 0 as x 3 so that v(x, t) → 0 as x.2 Hence, according to Bohm’s theory a particle with wave function (18) for all time never reaches the point x = X = 0. This result disagrees with the finite probability of arrival at X = 0 predicted by the “standard” expression (see Fig. 5) and, when divorced from the ontology of Bohm’s theory, is counterintuitive to many because it means that a freely propagating point-like particle in the state (18), if initially approaching x = 0, must turn around before reaching that point. On the other hand, the conclusion that no particle reaches X = 0 is in complete agreement with that obtained by Yamada and Takagi [17] using the sum-over-Feynman-paths consistent histories approach.3 The closely related (via the method of images) case in which there is a perfectly reflecting (r) rectangular potential step of infinite height in the region x 0 provides a simple example of a non-stationary state with a quantally forbidden region. The initial wave function is ψ (r) (x, 0) ≡ 21/2Θ(−x)ψ (f ) (x, 0).
(21)
It follows from the symmetry of the problem that the time-evolved wave function ψ (r) (x, t) and associated probability current density J (r)(x, t) are ψ (r) (x, t) = 21/2Θ(−x)ψ (f ) (x, t);
J (r) (x, t) = 2Θ(−x)J (f ) (x, t).
The Fourier transform of ψ (r) (x, t) is g(t) δ + i(, − k) δ + i(, + k) φ (r) (k, t) = w − w − i(β2 + iγ2 )1/2 2(β2 + iγ2 )1/2 2(β2 + iγ2 )1/2
(22)
(23)
with 1/2 1/2 η exp −k02 ('x)2 + β0 + i(k0 x0 + γ0 ) . g(t) ≡ N'x ('x)2 − i h¯ t/2m
(24)
2 The coefficient multiplying x depends on t allowing v(x, t) to vanish at points (x, t) with x = 0 so that particle trajectories can turn around before reaching x = 0. It is easy to show that v(x, t = t0 ) 0 for all x < 0 and 0 for all x > 0. 3 Precisely because they are completely different concepts, there is no necessary contradiction in invoking Feynman paths and Bohm trajectories in the same argument. In particular, there is no inconsistency in the picture that the actual point-like particles of Bohm’s theory follow deterministic trajectories under the influence of a guiding wave the mathematical expression for which can be constructed using the virtual paths of Feynman’s theory. Inconsistencies can arise when one also interprets Feynman paths as actual paths followed by actual pointlike particles.
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Fig. 5. Comparison at T = t0 of the X dependence of the “standard” result for 2Π± (T ; X) calculated for the freely-evolving antisymmetric wave function ψ (f ) (x, t) (thin solid (+) and thin dashed (−) curves) with that for Π± (T ; X) calculated for the perfectly reflected wave function ψ (r) (x, t) (thick solid (+) and thick dashed (−) curves). Also shown is |J (r) (X, T = t0 )| which is equal to 2|J (f ) (X, T = t0 )| (not shown) for X 0 (long-short dashed curve). The parameters of the wave functions are x0 = −20 Å, k0 = 0.5 Å−1 and 'k = 0.1 Å−1 corresponding to 'x = 5 Å.
Since φ (r) (k, T ) is in general not equal to φ (f ) (k, T ), the “standard” expression (2) can exhibit the interesting property of what might be called single-particle spatial nonlocality: Π± (T ; X; ψ (r)) is not equal to 2Π± (T ; X; ψ (f ) ) in the half-space X 0 even though for all t the wave function ψ (r) (x, t) is identical to 21/2ψ (f ) (x, t) in that region. In contrast to this, the corresponding Bohm trajectory expression Π± (T ; X) = ±NJ (X, T )Θ[±J (X, T )] is local in the sense that, aside from the normalization factor, it depends only on the wave function and its spatial gradient at the space–time point (X, T ) of interest. Hence, Π± (T ; X; ψ (r)) is identical to Π± (T ; X; ψ (f ) ) for X 0 and, for the former case, no particles can arrive at a point X in the quantally forbidden region X 0. Another important difference between the two approaches is that J Π (X, T ) can be very different in shape from J (X, T ) in the “standard” approach while the two quantities are proportional in the Bohm trajectory approach. The above mentionned properties of the two approaches are illustrated in Figs. 5 and 6. For T = t0 ≡ |x0 |/(h¯ k0 /m), Fig. 5 compares the X-dependence of the “standard” results for Π± (T = t0 ; X; ψ (r) ) and 2Π± (T = t0 ; X; ψ (f ) ) with each other and with |J (r)(X, T = t0 )| = −J (r) (X, t0 ). For ψ (r) the finite probability of arrival in the quantally forbidden region X > 0, predicted by the “standard” theory, is far from negligible at t = t0 . The single-particle nonlocality is most evident near X = 0. For X 0, Fig. 6 compares J (r) (X, t0 ) with the “standard” calculations of J Π (X, t0 ) for ψ (r) and of 2J Π (X, t0 ) for ψ (f ) ; these three quantities have identical shapes within Bohmian mechanics. The nonlocal effect predicted by the “standard” theory is again evident.
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Fig. 6. Comparison of J (r) (X, T = t0 ) (long-short dashed curve) with the “standard” results for J Π (X, t0 ) for ψ (r) (thick solid curve) and 2J Π (X, t0 ) for ψ (f ) (thin solid curve).
5. Discussion Thinking classically, to say that a point-particle arrived at position x = X at time t = T from the left (right) means that it was at x = X at t = T and was in the region x < X (x > X) for all times t in the interval T −'t t < T for some 't > 0. This statement is a meaningful one for a particle which moves along a continuous trajectory x(t) with a well-defined velocity (i.e., |dx(t)/dt| < ∞). Returning to the quantum world, the concept of directed arrival time for a quantum particle still seems to imply the existence of some kind of trajectory, at least for a finite interval before the instant of arrival. In Bohmian mechanics, particle trajectories are explicitly postulated and, when arrivals at X can occur in the temporal range of interest, derivation of the arrival time probability distribution Π(T ; X) and its right-going and left-going components Π+ (T ; X) and Π− (T ; X), respectively, is relatively trivial even in the presence of a scattering potential. Moreover, it is not difficult to decompose a given distribution into contributions from first, second, etc., arrivals. In conventional quantum mechanics the idea of an individual quantum particle following such an actual trajectory is taboo. A possible way around this difficulty is to time the arrivals of particles moving along the virtual trajectories of Feynman’s path integral theory, assuming that the resulting arrival-time probability distribution is identical to the desired one. Yamada and Takagi [16,17] have shown that this approach does not work for free Schrödinger particles, at least in part because the velocity associated with a typical Feynman path is singular almost everywhere along it. The approach fares somewhat better for free Dirac particles moving always at the speed of light along Feynman “checkerboard” paths in 1 + 1 dimensions. The derivation of Π(T ; X) and its right- and left-going components is straightforward [19] but the results are different from those obtained using Bohmian mechanics (e.g., Π is proportional to the probability density rather than to the absolute value of the probability current density). Moreover, an intrinsic first-arrival-time distribution is not well-defined because Π in general contains a nonzero contribution arising from interference between first and later (i.e., second, third, . . . ) arrivals [20]. Now, it is true that none of the derivations [5–9] of the “standard” arrival-time density make explicit use of particle trajectories, actual or virtual. However, they all associate arrivals from the left (right) with positive (negative) wavenumbers k and, for the artificial problem, make use of the assumption that every freely evolving quantum particle in the ensemble arrives at any fixed finite point X once and only once at some time T between
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−∞ and +∞, provided that φ (f ) (k = 0, t) = φ (f ) (k = 0, 0) is zero. Are these self-evident facts or do they betray an underlying intuitive picture in which each free particle in the ensemble has a momentum p = h¯ k that is constant in time? It is a mathematical fact that |Φ (f ) (p = h¯ k, t)|2 ≡ h¯ −1 |φ (f ) (k, t)|2 = h¯ −1 |φ (f ) (k)|2 is independent of time. Within conventional quantum mechanics this implies only that the probability of finding the momentum of a free particle to be between p and p + dp at time t is independent of t, not that the momentum of each such particle is independent of time prior to the momentum measurement, unless the particle is prepared in a momentum eigenstate. The suggestion that the “standard” arrival-time distribution might be based on the implicit assumption of a time-independent momentum for an individual free particle may not be so far-fetched because in “General Principles of Quantum Mechanics” Pauli explicitly states that the probability density in momentum (p = h¯ k) space is constant in time for free particles “since the momentum of the particle itself is constant” [21]. If this were the case then a free particle would have a trajectory x(t) = x(0) + p(0)t/m with two “hidden variables” x(0) and p(0) = p, a concept completely foreign to standard quantum mechanics. In any case, the assumption of a timeindependent momentum for each free particle does provide a scenario in which the concept of directed arrival times is meaningful and every free particle with φ (f ) (k = 0) = 0 must arrive at any finite X once and only once at some finite time T when the range of T is artificially extended from [0, +∞] to [−∞, +∞], as has been claimed [6] in the “standard” theory. However, not only is the implied trajectory foreign to conventional quantum mechanics, it is in conflict with the “quantum backflow effect”, i.e., with the possibility for the probability current density J (x, t) associated with a freely evolving wave function ψ(x, t) with no negative wavenumber components to be negative for some positions x for finite time intervals [t1 (x), t2 (x)] [4,5,22]. As has been discussed elsewhere [2], this effect seems impossible to reconcile with the “standard” arrival time distribution: how can Π− (T ; X) be zero for all T , as it must according to (2) when φ (f ) (k < 0, t) = 0, if there exists a finite time interval [t1 (X), t2 (X)] during which—according to the continuity equation—the probability of finding the quantum particle to the left of x = X is increasing with time? The backflow regime presents no difficulty in Bohmian mechanics: because of the guiding field, the “free” particle is not actually free and reverses its direction of motion in such a way that Π+ (T ; X) = 0 and Π− (T ; X) = −NJ (X, T ) when J (X, T ) < 0. In this Letter it has been argued that the assignment of direction of arrival underpinning the “standard” approach can lead to the further paradoxical property of a finite probability of arrival in a quantally forbidden region. This has been confirmed by explicit calculation. The interesting possibility of single-particle nonlocality was also shown to be a property of the “standard” theory. In the process it was realized that truncation of the k-space integrals in (2) was a sufficient but not necessary condition for the above-mentionned paradoxical property and that the nonanalytic factor of |k|1/2 could also be a major, if not the major, cause. The various derivations [5–9] of (2) reveal in detail how the factor of |k|1/2 arises. Basically, it is the result of demanding that the expression for the density of arrival times T at the position X chosen by an experimenter have the same formal structure as that for the probability density P [O(t)] associated with the measurement of the observable O at the instant of time t chosen by the experimenter—a very different situation as emphasized by Mielnik [23]. Now, the derivation of the arrival-time distribution N|J (X, T )| of Bohmian mechanics was carried out in x-space but the result can, of course, be expressed in k-space by Fourier transforming ψ(x, t). The result involves the product of two integrals over (all) k, one containing a factor of k 0 φ(k, T ) and the other a factor of k 1 φ(k, T ). Hence, it is not surprising on dimensional grounds alone that demanding that the arrival-time density have the form of the modulus squared of a single integral over k leads to an integrand containing the (nonanalytic) factor |k|1/2φ(k, T ). This suggests that adherence to the standard form should be relaxed in attempting to construct the arrival-time distribution for a quantum particle, assuming that the concepts of direction and order of arrival are meaningful within conventional quantum mechanics. Words such as “weird” and “bizarre” are so commonplace in discussions of quantum mechanics that it has led in some to a complacency in which any paradoxical (technically correct) theoretical result based on the conventional formalism is met with the comment “Oh, that’s just quantum mechanics” and the belief (invariably a safe one, it seems) that ultimately the paradox will be resolved and the predicted behaviour will be confirmed by experiment. However, some areas of quantum physics—such as timing quantum events—involve new aspects that might require
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some extension or modification of standard quantum theory. It is probably safe to say that most physicists would encounter some extension of their version of standard quantum mechanics in the “standard” approach to times of arrival advocated by Egusquiza, Muga and Baute [9] and be surprised by the amount of effort and degree of sophistication required in this approach to solve such an apparently simple problem as the time of arrival of a nonrelativistic quantum particle. This unexpected difficulty and the paradoxical results reported in this paper could very well be symptoms that the “standard” approach has somehow gotten off on the wrong track. Two possibilities have been suggested: (1) the concept of direction of arrival is not meaningful without some concept of trajectory and, in particular, the association of arrivals from the left (right) with positive (negative) wavenumbers is not justified; (2) probability distributions or densities for event times or time durations should not necessarily have the same form as probability distributions for observables at an instant of time.
References [1] [2] [3] [4]
[5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23]
J.G. Muga, R. Sala, J.P. Palao, Superlatt. Microstruct. 23 (1998) 833. J.G. Muga, C.R. Leavens, Phys. Rep. 338 (2000) 353. J.G. Muga, R. Sala Mayato, I.L. Egusquiza (Eds.), Time in Quantum Mechanics, Springer, Berlin, 2002. G.R. Allcock, Ann. Phys. (N.Y.) 53 (1969) 253; G.R. Allcock, Ann. Phys. (N.Y.) 53 (1969) 286; G.R. Allcock, Ann. Phys. (N.Y.) 53 (1969) 311. J. Kijowski, Rep. Math. Phys. 6 (1974) 362; J. Kijowski, Phys. Rev. A 59 (1999) 897. N. Grot, C. Rovelli, R.S. Tate, Phys. Rev. A 54 (1996) 4676. V. Delgado, J.G. Muga, Phys. Rev. A 56 (1997) 3425. A.D. Baute, R. Sala Mayato, J.P. Palao, J.G. Muga, I.L. Egusquiza, Phys. Rev. A 61 (2000) 022118. I.L. Egusquiza, J.G. Muga, A.D. Baute, in: J.G. Muga, R. Sala Mayato, I.L. Egusquiza (Eds.), Time in Quantum Mechanics, Springer, Berlin, 2002. C.R. Leavens, Phys. Rev. A 58 (1998) 840. D. Bohm, B. Hiley, The Undivided Universe: An Ontological Interpretation of Quantum Mechanics, Routledge, London, 1993. P.R. Holland, The Quantum Theory of Motion, Cambridge Univ. Press, Cambridge, 1993. J.T. Cushing, A. Fine, S. Goldstein (Eds.), Bohmian Mechanics and Quantum Theory: An Appraisal, Kluwer, Dordrecht, 1996. P.R. Holland, Phys. Rev. A 60 (1999) 4326. S. Brouard, J.G. Muga, Phys. Rev. A 54 (1996) 3055. N. Yamada, S. Takagi, Prog. Theor. Phys. 85 (1991) 985. N. Yamada, S. Takagi, Prog. Theor. Phys. 86 (1991) 599. K. Berndl, in: J.T. Cushing, A. Fine, S. Goldstein (Eds.), Bohmian Mechanics and Quantum Theory: An Appraisal, Kluwer, Dordrecht, 1996. C.R. Leavens, Phys. Lett. A 272 (2000) 160. D. Alonso, R. Sala Mayato, C.R. Leavens, Phys. Rev. A, in press. W. Pauli, General Principles of Quantum Mechanics, Springer, Berlin, 1980. A.J. Bracken, G.F. Melloy, J. Phys. A 27 (1994) 2197. B. Mielnik, Found. Phys. 24 (1994) 1113.
On the “standard” quantum mechanical approach to times of arrival C.R. Leavens Institute for Microstructural Sciences, National Research Council of Canada, Ottawa, K1A 0R6, Canada Received 20 August 2002; accepted 5 September 2002 Communicated by P.R. Holland
Abstract The “standard” approach to times of arrival advocated by Equsquiza, Muga and Baute associates arrivals from the left (right) with positive (negative) wavenumber components of the wave function at the instant of arrival. Elementary arguments and calculated results exhibiting paradoxical properties call into question this sign of k assignment of direction of arrival. 2002 Published by Elsevier Science B.V.
1. Introduction Recently there has been renewed interest [1–3] in the problem of deriving an expression for the distribution Π(T ; X) of intrinsic arrival times T at an arbitrary spatial point X for an ensemble of quantum particles each prepared in the same initial state ψ(x, t = 0). Several different approaches [4–8] all based on conventional nonrelativistic quantum mechanics lead to the same result: Π(T ; X) = Π+ (T ; X) + Π− (T ; X) with
∞ 2 h¯ dk Θ(±k)|k|1/2 exp(ikX)φ(k, T ) , Π± (T ; X) = 2πm
(1)
(2)
−∞
where Π+ (T ; X) and Π− (T ; X) denote the contributions to Π(T ; X) of arrivals at X from the left and from the right, respectively, Θ is the unit step function, and φ(k, t) is the Fourier transform of the time-evolved wave function ψ(x, t). It should be emphasized that according to (2) right-going (+) arrivals at X are associated only with positive wavenumbers (k > 0) and left-going (−) arrivals only with negative wavenumbers (k < 0). The derivation of Allcock [4], of Kijowski [5], of Grot, Rovelli and Tate [6], and of Delgado and Muga [7] are for the special case of free (f ) evolution, in which case φ(k, t) = φ (f ) (k, t) ≡ φ (f ) (k) exp(−i h¯ k 2 t/2m), where E-mail address: [email protected] (C.R. Leavens). 0375-9601/02/$ – see front matter 2002 Published by Elsevier Science B.V. PII: S 0 3 7 5 - 9 6 0 1 ( 0 2 ) 0 1 2 3 9 - 2
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φ (f ) (k) ≡ φ (f ) (k, 0) is the Fourier transform of the initial wave function ψ(x, 0). Logically, one is concerned only with arrivals subsequent to the time, here t = 0, at which the initial state is prepared. However, for reasons of mathematical convenience, these authors considered instead the artificial problem in which arrival times in the range [−∞, 0] are included by imagining that the quantum particle was prepared at t = −∞ in the state which, in the assumed absence of any interaction including that with the apparatus actually used to prepare the state at t = 0, would evolve to the desired initial state ψ(x, t = 0). The mathematical advantage—“resolution of the identity”—of treating the artificial problem is tied to the implicit assumption that for free evolution the particle is certain to reach x = X, for any finite X, once and only once, at some T (X) in the range [−∞, +∞], provided that φ(f ) (k = 0) is zero. In this case, Π(T ; X) given by (1) and (2) is an arrival-time probability distribution +∞ (i.e., −∞ dT Π(T ; X) = 1). The more recent derivation of Baute, Sala Mayato, Palao, Muga and Egusquiza (BSPME) [8] is for arbitrary potentials V (x, t) and for both stationary and non-stationary states. In general, in the presence of a scattering potential which can prevent some particles from reaching X and/or cause others to reach X more than once, the density Π(T ; X) given by (2) is no longer a probability distribution [8] and the above advantage of the artificial problem no longer exists. For the non-stationary case, it then seems sensible to consider only arrivals subsequent to t = 0, even—for the sake of generality—in the free evolution case. The “standard” arrival-time density includes all orders of arrival but as yet there is no prescription for extracting the individual contributions of the various orders (e.g., the contribution of first arrivals). Egusquiza, Muga and Baute [9] title their distillation of the various approaches leading to Eqs. (1) and (2) “standard” quantum mechanical approach to times of arrival, emphasizing their claim that (2) can be derived “without in any way distorting the standard framework of quantum mechanics”. However, even for the case of free evolution, the present author regards the above association of direction of arrival with sign of k as an unjustifed assumption that is not a part of conventional quantum mechanics [10]. Hence, despite the excellent agreement reported in Chapters 1, 8 and 10 of [3] with a numerical “quantum jump” time-of-flight simulation, the author has some reservations about referring to (2) as a result of standard quantum mechanics. In this Letter counterintuitive and/or paradoxical properties of the “standard” arrival-time distribution are pointed out, illustrated with simple concrete examples, and contrasted with the corresponding—and frequently counterclassical—properties of a non-standard arrival-time distribution, namely, that of Bohm’s alternative [11– 13] to conventional quantum mechanics. The case studies in the following three sections were selected to provide simple illustrations of the following formal argument. Consider the integral in (2) as a function of X, for a fixed value of T , and write it in the form ∞ (2πkc )
1/2 −∞
dk exp(ikX)φ˜ ± (k, T ) ≡ (2πkc )1/2 ψ˜ ± (X, T ), (2π)1/2
with kc an arbitrary positive wavenumber and 1/2 k ˜ φ± (k, T ) ≡ Θ(±k) φ(k, T ). kc
(3)
(4)
Since the integral in (2) is proportional to the Fourier transform ψ˜ ± (X, T ) of the function φ˜ ± (k, T ) which is zero for either k < 0 or k > 0, it follows that the support of Π± (T ; X) ∝ |ψ˜ ± (X, T )|2 is the entire X-axis. This is true even if the wave function ψ(x, t) is zero for all t over an extended spatial range due to the presence of an infinitely repulsive barrier. Hence, one has the paradoxical result that the “standard” arrival time distribution Π(T ; X) = Π+ (T ; X) + Π− (T ; X) obtained from (2) can be finite for X in the interior of a quantally forbidden region where the probability of finding the particle is exactly zero at any time t. It is important to know whether or not this predicted effect is of nonnegligible importance. This is investigated for the stationary-state case in Section 2 for a quantum particle in an energy eigenstate of a rectangular well with infinitely high walls. In Section 3 the freely-evolving time-dependent case is considered by collapsing the confining potential instantaneously to zero at t = 0 and calculating the resulting arrival-time distributions Π(T ; X) for T 0 at a point X a finite distance
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outside the (former) well region. In Section 4 arrival-time distributions are considered for an initial Gaussian wave packet incident on a perfectly reflecting infinite potential step at x = 0. The interesting properties of the “standard” arrival-time density illustrated in these case studies and their possible implications are discussed in Section 5. The following extremely brief sketch of Bohmian mechanics is intended to facilitate the above-mentionned comparisons. In Bohm’s theory [11–13] a quantum particle such as an electron is postulated to be an actually existing point-like particle and an accompanying wave which guides its motion. For a nonrelativistic quantum particle in the presence of a potential V (x, t), the time-evolution of the guiding wave ψ(x, t) is described by the Schrödinger equation and the trajectory of the point-like particle is determined by the equation-of-motion dx(t)/dt = [J (x, t)/|ψ(x, t)|2 ]|x=x(t ) where J (x, t) ≡ (h¯ /m)[ψ ∗ (x, t)∂ψ(x, t)/∂x] is the probability current density. (For the 1D system considered here there is no spin-dependent contribution [14].) It is further postulated that, for an ensemble of electrons all prepared in the same initial state ψ(x, 0), the probability of such a particle having initial position x (0) ≡ x(t = 0) is given by |ψ(x (0), 0)|2 . The expression for the intrinsic arrival-time probability distribution for those particles that do reach X subsequent to t = 0 is readily obtained when arrivals at X are possible [10]. It is given by Π(T ; X) = N|J (X, T )|; the right-going and left-going contributions are Π± (T ; X) = ±NJ± (X, T ) where the right- and left-going components of the probability current density are determined by the sign of J , i.e., J± (x, t) ≡ J (x, t)Θ[±J (x, t)]. The normalization factor N will usually be ignored (i.e., set equal to 1) in graphical comparisons with the BSPME arrival-time density.
2. Stationary-state case Consider the special case of a bound-state energy eigenfunction ψ(x, t) = ψn (x, t) ≡ Rn (x) exp(−iEn t/h¯ )
(5)
with Rn (x) real so that J (x, t) = 0. For this case, it follows immediately from (2) that, as expected, Π+ (T ; X) and Π− (T ; X) are independent of T and it is the relative number of arrivals per unit time at various spatial points X that is of interest. It is also straightforward to show that Π+ and Π− are equal so that the classically motivated expression J Π (X, T ) ≡ Π+ (T ; X) − Π− (T ; X) for the probability current density is zero which is consistent with J (x, t) = 0 for all x and t. Now consider the special case of a quantum particle in a bound-state energy eigenfunction of a rectangular potential well with infinitely repulsive walls at x = 0 and x = a. The spatial part of the nth energy eigenfunction is Rn (x) =
1/2 2 sin(kn x)Θ(x)Θ(a − x), a
kn ≡
(2mEn )1/2 nπ ≡ , h¯ a
(6)
with n = 1, 2, . . . . If one deliberately repeats a frequently made mistake and calculates φn (k, T ) ≡ φn (k) × exp(−iEn T /h¯ ) by Fourier transforming just sin(kn x) rather than sin(kn x)Θ(x)Θ(a − x) then one obtains from (2) that Π+ (T ; X) = Π− (T ; X) = (1/2)(vn /a) with vn ≡ h¯ kn /m. This corresponds to the well-known, intuitively appealing, and (apart from the discrete nature of vn ) classical picture of a point-particle bouncing back and forth inside the well with a constant speed vn between its instantaneous elastic collisions with the infinitely hard walls. If one uses the correct Fourier transform kn 1/2 kn nπk nπk n n 1 − (−1) + i(−1) cos sin φn (k) = (7) kn kn nπ 2 kn2 − k 2 of Rn (x) in (2) then one obtains instead Π± (T ; X) =
1 n3 π 5
vn 2 Cn (X/a) + Sn2 (X/a) 2a
(8)
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Fig. 1. Dependence of Π+ (T ; X) = Π− (T ; X) calculated with (2) on X for a quantum particle in the energy eigenstate ψn (x, t) of a rectangular potential well with infinitely strong confining potential: n = 1 (solid curve); n = 2 (dashed curve); n = 3 (long-short dashed curve). The probability of finding the quantum particle to the left of the dotted vertical line is zero.
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Fig. 2. Dependence of Π+ (T ; X)/Π(T ; 0) calculated with (2) on X for a quantum particle in the energy eigenstate ψn=1 (x, t) of a rectangular well with confining potential of height V0 = 10E1 (long-short dashed curve), V0 = 102 E1 (short dashed curve), V0 = 103 E1 (dotted curve), V0 = 104 E1 (thin solid curve), and ∞ (thick solid curve). The ground state energy E1 is a function of V0 for finite V0 .
with ∞ Cn (X/a) = 0
∞ Sn (X/a) = 0
du u1/2 cos(Xu/a) − (−1)n cos (1 − X/a)u , 2 1 − (u/nπ)
du u1/2 sin(Xu/a) + (−1)n sin (1 − X/a)u . 2 1 − (u/nπ)
(9)
Π± (T ; X) is symmetric about X = a/2. Fig. 1 shows the dependence of Π± (T ; X)/(vn /2a) on X/a over the range −a/2 X a/2 for n = 1, 2 and 3. In each case, the predicted relative probability of a quantum particle arriving at a point X is strongly suppressed as X moves from the interior of the well into the quantally forbidden region X < 0, but it is far from negligible there for X just outside the well. Fig. 2 shows Π+ (T ; X) = Π− (T ; X) for the n = 1 energy eigenstate of the rectangular well with a series of finite confining potentials V0 Θ(−x)Θ(a − x) of increasing height V0 to show that there is convergence to the result presented in Fig. 1 for the limiting, and potentially problematic, infinite V0 case.
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Fig. 3. Effect of interference between k > 0 and k < 0 contributions. The “standard” result for Π(T ; X) is shown by the thin solid curve, the result for F (T ; X) by the thick solid curve.
It is informative to consider the function1 2 ∞ h¯ 1/2 F (T ; X) = dk |k| exp(ikX)φ(k, T ) 2πm
(10)
−∞
obtained by removing the theta function Θ(±k) from the integral ∞ in Eq. (2). For a given value of T , the integrals of Π(T ; X) and F (T ; X) over all X are both equal to (h¯ /m) −∞ dk |k||φ(k, T )|2 which provides a useful check on the accuracy of the numerical calculations. For the n = 1 energy eigenstate (6), Fig. 3 compares F (T ; X) with Π(T ; X) = Π+ (T ; X) + Π− (T ; X) as a function of X/a. Remarkably, F (T ; X) is also not zero in the quantally forbidden region |X − a/2| > a/2 even though it, unlike Π± (T ; X), does not involve the Fourier transform of a function of k that is exactly zero over an extended interval of k values. This suggests that the real culprit behind the predicted non-negligible value of Π(T ; X) for |X −a/2| > a/2 is the occurrence of the non-analytic function |k|1/2 in the integrand of (2). As usual, another possibility is human error in the numerical calculation. Now, (10) with the factor |k|1/2 replaced by 1 is proportional to Rn (X)2 and replaced by k is proportional to [(dRn (x)/dx)|x=X ]2 , both of which are exactly zero for |X − a/2| > a/2. Numerical evaluation of these two quantities after trivial changes to just one line of coding in the program for evaluating F (T ; X) did give zero to within numerical accuracy for |X − a/2| > a/2, strengthening the case for the factor |k|1/2 being the actual culprit. For the stationary-state wave function (5) with J (x, t) = 0 an arrival-time distribution is not defined within Bohmian mechanics because each particle in the ensemble is motionless—its energy En is completely in the form of quantum potential energy [11–13]—at some unknown position x (0) within the well, with the various x (0) s distributed according to Rn (x (0))2 . If x (0) = X then the particle is always at X; if x (0) = X then it never arrives at X. 1 This function is introduced only to illustrate a mathematical point and is not intended as a candidate arrival-time distribution.
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3. Time-dependent free-evolution case Assume that for t < 0 the quantum particle is in the ground-state ψn=1 (x, t) of the infinite rectangular well of the previous section and that at t = 0 the confining barriers are instantaneously removed. For t > 0 the Fourier transform of the resulting freely-evolving wave function ψ(x, t) is φ (f ) (k, t) = φ (f ) (k) exp(−i h¯ k 2 t/2m) with φ (f ) (k) = φn=1 (k) given by (7). The resulting “standard” result for the right- and left-going components of the arrival-time distribution for T 0 at the point X = 1.1a, obtained by numerical evaluation of (2), is shown in Fig. 4. Both components are nonzero at T = 0, with the left-going component decreasing monotonically with increasing T . Regarding the latter, it is as if there is a non-zero probability amplitude for left-going particles being to the right of X = 1.1a immediately prior to t = 0. The corresponding arrival-time distribution based on Bohmian mechanics is readily obtained from the wave function ψ(x, t) for t > 0 which is given by [15] ψ(x, t) =
im(x − a)2 1 2 1/2 w[u+ (x − a, t)] − w[u− (x − a, t)] exp(ik1a) exp 4i a 2h¯ t
2 imx − exp w[u+ (x, t)] − w[u− (x, t)] 2h¯ t
(11)
with u± (x, t) ≡ −
(1 + i)(±h¯ k1 − mx/t) , 2(mh¯ /t)1/2
(12)
Fig. 4. T -dependence of Π± (T ; X) for X = 1.1a. The “standard” results for Π+ and Π− are shown by the thick solid and dashed curves respectively; the result of Bohmian mechanics for Π+ /2 = J is shown by the thin solid curve (Π− is zero for the space–time region shown). The results are scaled by the characteristic time t1 ≡ a/v1 .
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2
w(u) ≡ exp −u2 erfc(−iu) ≡ exp −u2 1/2 π
∞
dz exp −z2 .
(13)
−iu
The corresponding expression for the probability current density for t > 0 is 2 2 2 2 v1 J (x, t) = w[u+ (x − a, t)] − w[u− (x − a, t)] + w[u+ (x, t)] − |w[u− (x, t)] 8a mx am − 2 cos k1 − + a w∗ [u+ (x, t)]w[u+ (x − a, t)] − w∗ [u− (x, t)]w[u− (x − a, t)] h¯ t 2h¯ t
∗ mx am ∗ a w [u+ (x, t)]w[u+ (x − a, t)] − w [u− (x, t)]w[u− (x − a, t)] . + + 2 sin k1 − 2h¯ t h¯ t (14) At least, for the range of T values investigated, J (X = 1.1a, T ) is non-negative so that within Bohmian mechanics Π− (T ; X = 1.1a) is zero and Π(T ; X = 1.1a) = Π+ (T ; X = 1.1a) = 2J (X = 1.1a, T ). To facilitate comparison with the “standard” result, which is not normalized to unity, the factor of 2 has been dropped in Fig. 4. A satisfactory feature of the Bohm arrival-time distribution is that it is zero at T = 0 as one would expect on physical grounds.
4. Time-dependent nonzero potential case We now approach a non-stationary state example of the paradoxical effect of Section 2 in an indirect way in order to illustrate further interesting properties of the “standard” and Bohm trajectory arrival-time distributions and also to make contact with the work of Yamada and Takagi [16,17]. We begin by first considering a simple example of the free propagation of a non-relativistic wave function that is antisymmetric about x = 0 at all times t [10]. The initial wave function (x − x0 )2 (x + x0 )2 (f ) + ik0 x − exp − − ik0 x ψ (x, 0) ≡ N exp − (15) 4('x)2 4('x)2 with
3/2 1/2 N = 2 π 'x 1 − exp −2('x)2k02 −
x02 2('x)2
−1/2 (16)
is a coherent linear superposition of two Gaussians with individual centroids ±x0 , mean wavenumbers ±k0 , respectively, and equal spatial widths 'x. It readily follows that −i h¯ k 2 t (f ) 1/2 φ (k, t) = N2π 'x exp 2m
2 × exp −(k − k0 ) ('x)2 − i(k − k0 )x0 − exp −(k + k0 )2 ('x)2 + i(k + k0 )x0 , (17) 1/2
i h¯ t ψ (f ) (x, t) = 2N'x ('x)2 − η1/2 exp −k02 ('x)2 + ik0 x0 exp β0 + β2 x 2 + i γ0 + γ2 x 2 2m × exp(δx + i,x) − exp(−δx − i,x) . (18) The corresponding probability current density is J (f ) (x, t) =
h¯ 2N 2 ('x)2 η1/2 exp −2k02 ('x)2 exp 2 β0 + β2 x 2 m × 4γ2 x[cosh(2δx) − cos(2,x)] + 2, sinh(2δx) − 2δ sin(2,x) .
(19)
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In (18) and (19) −1 η ≡ 4('x)4 + (h¯ t/m)2 , ht ¯ , β2 ≡ −η('x)2 , β0 ≡ η 4('x)6k02 − ('x)2x02 − 2k0x0 ('x)2 m
h¯ t ηh¯ t , γ2 ≡ , γ0 ≡ −η 4k0 x0 ('x)4 + 4k02 ('x)4 − x02 2m 2m k0 h¯ t x0 ht ¯ δ ≡ 2η('x)2 x0 + , , ≡ 2η 2k0('x)4 − . m 2m
161
(20)
For x0 < 0 and k0 > 0, the quantity t0 ≡ −x0 /(h¯ k0 /m) provides a useful characteristic time, namely the time at which the centroids of the individual Gaussian components of (18) reach x = 0. At the instant t = t0 the wave function (18) has a string of nodes at the points xl = l/πk0 (l = 0, ±1, ±2, . . .). The probability density |ψ (f ) (x, t)|2 is zero at x = 0 for all t. Within (nonrelativistic) Bohmian mechanics a particle can reach a point x = X at a time t = T when |ψ(X, T )|2 = 0 only if v(X, T ) ≡ J (X, T )/|ψ(X, T )|2 = ±∞ [18]. From (18) and (19), respectively, it follows that as x → 0, for any t, |ψ (f ) (x, t)|2 → 0 as x 2 and J (f ) (x, t) → 0 as x 3 so that v(x, t) → 0 as x.2 Hence, according to Bohm’s theory a particle with wave function (18) for all time never reaches the point x = X = 0. This result disagrees with the finite probability of arrival at X = 0 predicted by the “standard” expression (see Fig. 5) and, when divorced from the ontology of Bohm’s theory, is counterintuitive to many because it means that a freely propagating point-like particle in the state (18), if initially approaching x = 0, must turn around before reaching that point. On the other hand, the conclusion that no particle reaches X = 0 is in complete agreement with that obtained by Yamada and Takagi [17] using the sum-over-Feynman-paths consistent histories approach.3 The closely related (via the method of images) case in which there is a perfectly reflecting (r) rectangular potential step of infinite height in the region x 0 provides a simple example of a non-stationary state with a quantally forbidden region. The initial wave function is ψ (r) (x, 0) ≡ 21/2Θ(−x)ψ (f ) (x, 0).
(21)
It follows from the symmetry of the problem that the time-evolved wave function ψ (r) (x, t) and associated probability current density J (r)(x, t) are ψ (r) (x, t) = 21/2Θ(−x)ψ (f ) (x, t);
J (r) (x, t) = 2Θ(−x)J (f ) (x, t).
The Fourier transform of ψ (r) (x, t) is g(t) δ + i(, − k) δ + i(, + k) φ (r) (k, t) = w − w − i(β2 + iγ2 )1/2 2(β2 + iγ2 )1/2 2(β2 + iγ2 )1/2
(22)
(23)
with 1/2 1/2 η exp −k02 ('x)2 + β0 + i(k0 x0 + γ0 ) . g(t) ≡ N'x ('x)2 − i h¯ t/2m
(24)
2 The coefficient multiplying x depends on t allowing v(x, t) to vanish at points (x, t) with x = 0 so that particle trajectories can turn around before reaching x = 0. It is easy to show that v(x, t = t0 ) 0 for all x < 0 and 0 for all x > 0. 3 Precisely because they are completely different concepts, there is no necessary contradiction in invoking Feynman paths and Bohm trajectories in the same argument. In particular, there is no inconsistency in the picture that the actual point-like particles of Bohm’s theory follow deterministic trajectories under the influence of a guiding wave the mathematical expression for which can be constructed using the virtual paths of Feynman’s theory. Inconsistencies can arise when one also interprets Feynman paths as actual paths followed by actual pointlike particles.
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Fig. 5. Comparison at T = t0 of the X dependence of the “standard” result for 2Π± (T ; X) calculated for the freely-evolving antisymmetric wave function ψ (f ) (x, t) (thin solid (+) and thin dashed (−) curves) with that for Π± (T ; X) calculated for the perfectly reflected wave function ψ (r) (x, t) (thick solid (+) and thick dashed (−) curves). Also shown is |J (r) (X, T = t0 )| which is equal to 2|J (f ) (X, T = t0 )| (not shown) for X 0 (long-short dashed curve). The parameters of the wave functions are x0 = −20 Å, k0 = 0.5 Å−1 and 'k = 0.1 Å−1 corresponding to 'x = 5 Å.
Since φ (r) (k, T ) is in general not equal to φ (f ) (k, T ), the “standard” expression (2) can exhibit the interesting property of what might be called single-particle spatial nonlocality: Π± (T ; X; ψ (r)) is not equal to 2Π± (T ; X; ψ (f ) ) in the half-space X 0 even though for all t the wave function ψ (r) (x, t) is identical to 21/2ψ (f ) (x, t) in that region. In contrast to this, the corresponding Bohm trajectory expression Π± (T ; X) = ±NJ (X, T )Θ[±J (X, T )] is local in the sense that, aside from the normalization factor, it depends only on the wave function and its spatial gradient at the space–time point (X, T ) of interest. Hence, Π± (T ; X; ψ (r)) is identical to Π± (T ; X; ψ (f ) ) for X 0 and, for the former case, no particles can arrive at a point X in the quantally forbidden region X 0. Another important difference between the two approaches is that J Π (X, T ) can be very different in shape from J (X, T ) in the “standard” approach while the two quantities are proportional in the Bohm trajectory approach. The above mentionned properties of the two approaches are illustrated in Figs. 5 and 6. For T = t0 ≡ |x0 |/(h¯ k0 /m), Fig. 5 compares the X-dependence of the “standard” results for Π± (T = t0 ; X; ψ (r) ) and 2Π± (T = t0 ; X; ψ (f ) ) with each other and with |J (r)(X, T = t0 )| = −J (r) (X, t0 ). For ψ (r) the finite probability of arrival in the quantally forbidden region X > 0, predicted by the “standard” theory, is far from negligible at t = t0 . The single-particle nonlocality is most evident near X = 0. For X 0, Fig. 6 compares J (r) (X, t0 ) with the “standard” calculations of J Π (X, t0 ) for ψ (r) and of 2J Π (X, t0 ) for ψ (f ) ; these three quantities have identical shapes within Bohmian mechanics. The nonlocal effect predicted by the “standard” theory is again evident.
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Fig. 6. Comparison of J (r) (X, T = t0 ) (long-short dashed curve) with the “standard” results for J Π (X, t0 ) for ψ (r) (thick solid curve) and 2J Π (X, t0 ) for ψ (f ) (thin solid curve).
5. Discussion Thinking classically, to say that a point-particle arrived at position x = X at time t = T from the left (right) means that it was at x = X at t = T and was in the region x < X (x > X) for all times t in the interval T −'t t < T for some 't > 0. This statement is a meaningful one for a particle which moves along a continuous trajectory x(t) with a well-defined velocity (i.e., |dx(t)/dt| < ∞). Returning to the quantum world, the concept of directed arrival time for a quantum particle still seems to imply the existence of some kind of trajectory, at least for a finite interval before the instant of arrival. In Bohmian mechanics, particle trajectories are explicitly postulated and, when arrivals at X can occur in the temporal range of interest, derivation of the arrival time probability distribution Π(T ; X) and its right-going and left-going components Π+ (T ; X) and Π− (T ; X), respectively, is relatively trivial even in the presence of a scattering potential. Moreover, it is not difficult to decompose a given distribution into contributions from first, second, etc., arrivals. In conventional quantum mechanics the idea of an individual quantum particle following such an actual trajectory is taboo. A possible way around this difficulty is to time the arrivals of particles moving along the virtual trajectories of Feynman’s path integral theory, assuming that the resulting arrival-time probability distribution is identical to the desired one. Yamada and Takagi [16,17] have shown that this approach does not work for free Schrödinger particles, at least in part because the velocity associated with a typical Feynman path is singular almost everywhere along it. The approach fares somewhat better for free Dirac particles moving always at the speed of light along Feynman “checkerboard” paths in 1 + 1 dimensions. The derivation of Π(T ; X) and its right- and left-going components is straightforward [19] but the results are different from those obtained using Bohmian mechanics (e.g., Π is proportional to the probability density rather than to the absolute value of the probability current density). Moreover, an intrinsic first-arrival-time distribution is not well-defined because Π in general contains a nonzero contribution arising from interference between first and later (i.e., second, third, . . . ) arrivals [20]. Now, it is true that none of the derivations [5–9] of the “standard” arrival-time density make explicit use of particle trajectories, actual or virtual. However, they all associate arrivals from the left (right) with positive (negative) wavenumbers k and, for the artificial problem, make use of the assumption that every freely evolving quantum particle in the ensemble arrives at any fixed finite point X once and only once at some time T between
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−∞ and +∞, provided that φ (f ) (k = 0, t) = φ (f ) (k = 0, 0) is zero. Are these self-evident facts or do they betray an underlying intuitive picture in which each free particle in the ensemble has a momentum p = h¯ k that is constant in time? It is a mathematical fact that |Φ (f ) (p = h¯ k, t)|2 ≡ h¯ −1 |φ (f ) (k, t)|2 = h¯ −1 |φ (f ) (k)|2 is independent of time. Within conventional quantum mechanics this implies only that the probability of finding the momentum of a free particle to be between p and p + dp at time t is independent of t, not that the momentum of each such particle is independent of time prior to the momentum measurement, unless the particle is prepared in a momentum eigenstate. The suggestion that the “standard” arrival-time distribution might be based on the implicit assumption of a time-independent momentum for an individual free particle may not be so far-fetched because in “General Principles of Quantum Mechanics” Pauli explicitly states that the probability density in momentum (p = h¯ k) space is constant in time for free particles “since the momentum of the particle itself is constant” [21]. If this were the case then a free particle would have a trajectory x(t) = x(0) + p(0)t/m with two “hidden variables” x(0) and p(0) = p, a concept completely foreign to standard quantum mechanics. In any case, the assumption of a timeindependent momentum for each free particle does provide a scenario in which the concept of directed arrival times is meaningful and every free particle with φ (f ) (k = 0) = 0 must arrive at any finite X once and only once at some finite time T when the range of T is artificially extended from [0, +∞] to [−∞, +∞], as has been claimed [6] in the “standard” theory. However, not only is the implied trajectory foreign to conventional quantum mechanics, it is in conflict with the “quantum backflow effect”, i.e., with the possibility for the probability current density J (x, t) associated with a freely evolving wave function ψ(x, t) with no negative wavenumber components to be negative for some positions x for finite time intervals [t1 (x), t2 (x)] [4,5,22]. As has been discussed elsewhere [2], this effect seems impossible to reconcile with the “standard” arrival time distribution: how can Π− (T ; X) be zero for all T , as it must according to (2) when φ (f ) (k < 0, t) = 0, if there exists a finite time interval [t1 (X), t2 (X)] during which—according to the continuity equation—the probability of finding the quantum particle to the left of x = X is increasing with time? The backflow regime presents no difficulty in Bohmian mechanics: because of the guiding field, the “free” particle is not actually free and reverses its direction of motion in such a way that Π+ (T ; X) = 0 and Π− (T ; X) = −NJ (X, T ) when J (X, T ) < 0. In this Letter it has been argued that the assignment of direction of arrival underpinning the “standard” approach can lead to the further paradoxical property of a finite probability of arrival in a quantally forbidden region. This has been confirmed by explicit calculation. The interesting possibility of single-particle nonlocality was also shown to be a property of the “standard” theory. In the process it was realized that truncation of the k-space integrals in (2) was a sufficient but not necessary condition for the above-mentionned paradoxical property and that the nonanalytic factor of |k|1/2 could also be a major, if not the major, cause. The various derivations [5–9] of (2) reveal in detail how the factor of |k|1/2 arises. Basically, it is the result of demanding that the expression for the density of arrival times T at the position X chosen by an experimenter have the same formal structure as that for the probability density P [O(t)] associated with the measurement of the observable O at the instant of time t chosen by the experimenter—a very different situation as emphasized by Mielnik [23]. Now, the derivation of the arrival-time distribution N|J (X, T )| of Bohmian mechanics was carried out in x-space but the result can, of course, be expressed in k-space by Fourier transforming ψ(x, t). The result involves the product of two integrals over (all) k, one containing a factor of k 0 φ(k, T ) and the other a factor of k 1 φ(k, T ). Hence, it is not surprising on dimensional grounds alone that demanding that the arrival-time density have the form of the modulus squared of a single integral over k leads to an integrand containing the (nonanalytic) factor |k|1/2φ(k, T ). This suggests that adherence to the standard form should be relaxed in attempting to construct the arrival-time distribution for a quantum particle, assuming that the concepts of direction and order of arrival are meaningful within conventional quantum mechanics. Words such as “weird” and “bizarre” are so commonplace in discussions of quantum mechanics that it has led in some to a complacency in which any paradoxical (technically correct) theoretical result based on the conventional formalism is met with the comment “Oh, that’s just quantum mechanics” and the belief (invariably a safe one, it seems) that ultimately the paradox will be resolved and the predicted behaviour will be confirmed by experiment. However, some areas of quantum physics—such as timing quantum events—involve new aspects that might require
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some extension or modification of standard quantum theory. It is probably safe to say that most physicists would encounter some extension of their version of standard quantum mechanics in the “standard” approach to times of arrival advocated by Egusquiza, Muga and Baute [9] and be surprised by the amount of effort and degree of sophistication required in this approach to solve such an apparently simple problem as the time of arrival of a nonrelativistic quantum particle. This unexpected difficulty and the paradoxical results reported in this paper could very well be symptoms that the “standard” approach has somehow gotten off on the wrong track. Two possibilities have been suggested: (1) the concept of direction of arrival is not meaningful without some concept of trajectory and, in particular, the association of arrivals from the left (right) with positive (negative) wavenumbers is not justified; (2) probability distributions or densities for event times or time durations should not necessarily have the same form as probability distributions for observables at an instant of time.
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