Quantum trajectories and the Bohm time constant

Annals of Physics 331 (2013) 317–322

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Quantum trajectories and the Bohm time constant Antonio B. Nassar ∗ Physics Department, Harvard-Westlake School, 3700 Coldwater Canyon, Studio City, 91604, USA Department of Sciences, University of California, Los Angeles, Extension Program, 10995 Le Conte Avenue, Los Angeles, CA 90024, USA

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Article history: Received 9 November 2012 Accepted 12 January 2013 Available online 29 January 2013 Keywords: Bohmian trajectories Bohmian mechanics

abstract This work proposes a new logarithmic nonlinear Schrödinger equation to describe the dynamics of a wave packet under continuous measurement. Via the method of quantum trajectories formalism of the Bohmian model of quantum mechanics, it is shown that this continuous measurement alters the dynamical properties of the measured system. While the width of the wave packet may reach a stationary regime, its quantum trajectories converge asymptotically in time to classical trajectories. So, continuous measurements not only disturb the particle but compel it to eventually converge to a Newtonian regime. The rate of convergence depends on what is defined here as the Bohm time constant which characterizes the resolution time of the measurement. If the initial wave packet width is taken to be equal to 2.8 × 10−15 m (the approximate size of an electron) then the Bohm time constant is found to be about 6.8 × 10−26 s. © 2013 Elsevier Inc. All rights reserved.

The measurement problem is one of the most important conceptual difficulties in quantum mechanics. It implies that measurements typically fail to have outcomes of the sort the theory was created to explain. In particular, the theory of continuous quantum measurements has gained considerable interest in the last decades [1]. The mere presence of an observing apparatus should considerably affect the behavior of a system [2–5]. This work proposes a new logarithmic nonlinear Schrödinger equation to describe the dynamics of a wave packet under continuous measurement. This equation displays some fundamental differences from a previous model developed by Bialynicki-Birula and Mycielski [6] as an alternative to

∗ Correspondence to: Physics Department, Harvard-Westlake School, 3700 Coldwater Canyon, Studio City, 91604, USA. Tel.: +1 818 487 6663. E-mail addresses: [email protected], [email protected]. 0003-4916/$ – see front matter © 2013 Elsevier Inc. All rights reserved. doi:10.1016/j.aop.2013.01.009

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A.B. Nassar / Annals of Physics 331 (2013) 317–322

linear quantum mechanics. Via the method of quantum trajectories formalism of the Bohmian model of quantum mechanics, it is shown that the continuous measurement can dramatically alter the dynamical properties of the measured system. This trajectory approach reveals a greater understanding of various features of quantum mechanics. For example, the width of the wave packet may reach a stationary regime while its quantum trajectories converge asymptotically in time to classical trajectories. Continuous observations not only disturb the motion of particle but compel it to converge to a definite position. The rate of this convergence depends on what is defined here as the Bohm time constant which characterizes the resolution time of the quantum measurement. If the initial wave packet width is taken to be equal to 2.8 × 10−15 m (the approximate size of an electron) then the Bohm time constant is found to be about 6.8 × 10−26 s. Bohmian mechanics has recently attracted increasing attention from researchers [7–26]. Despite the uncertainty principle, the predictions of nonrelativistic quantum mechanics permit particles to have precise positions at all times. The simplest theory demonstrating that this is so is indeed Bohmian mechanics. It is shown that this example is not reliant on any assumption of collapse. One of the fundamental aspects of Bohmian mechanics is its ability to tackle more clearly the quantum measurement problem. The wave function plays a dual role in the Bohmian model; it determines the probability of the actual location of the particle and choreographs its motion as well. As pointed out by Bell [26], in physics the only observations we must consider are position observations, if only the positions of instrument pointers — a definite outcome in an individual measurement is determined by the relevant position variable associated with the apparatus. It is a great merit of the Bohmian picture to force us to consider this fact. The starting point here is to think of continuously monitoring a quantum free particle’s position and responding to the measured value along the pioneering work of Mensky [1]. To describe explicitly the monitoring a quantum particle position, Mensky defined quantum corridors including the effect of finite resolution of time and provided a weight functional in the Gaussian form:

 wx¯ [x] = exp −

1

τ

2 

x(t ′ ) − x¯ (t ′ )



t



dt

′

4δ 2

0

,

(1)

where the back influence of the selective measuring device onto the measured system is taken into account by a restricting path integral (RPI) and δ represents the measurement error which is achieved during the period τ of the measurement. The resulting evolution of the wave function of the quantum system ψ can be expressed in terms of the path-integral for the unnormalized wave function [1]:

ψ(x, t ) =



D[x(t ′ )] exp

 

t

i

h¯

dt 0

′1

2

mx˙ 2 (t ′ ) −

1

τ

t

 0

2 

x(t ′ ) − x¯ (t ′ ) dt ′ 4δ 2



ψ(x0 , 0),

(2)

where the path integral is over paths x(t ) satisfying x(0) = x0 and x(t ) = x. The last term in Eq. (2) is taken as a measure of the deviation of the observable x(t ). Along the restricted path integral (RPI) approach, Mensky also proposed a linear Schrödinger equation with a non-Hermitian effective Hamiltonian:

  ∂ψ(x, t ) ih¯ 2 [x(t ) − x¯ (t )] ψ(x, t ). ih¯ = H ( x, t ) − ∂t 4τ δ 2

(3)

An explicit connection to a wave packet approach can be established by approximating the last term of Eq. (2) around an average time t¯, i.e., ∼ exp −[(x(t¯) − x¯ (t¯))2 /4δ 2 ] exp(−t¯/τ ), where δ clearly stands for the position uncertainty (width of the wave packet) and τ characterizes the time constant of the measurement. Then, the square of the absolute value of Eq. (2) yields the probability density for different measurement outputs at different times by considering that continuous position measurements produce and maintain localization as a necessary result of the information it provides. This rationale entails a minimum-uncertainty wave packet solution around the measurement record x¯ (t ) as follows:

  −1/4 [x − x¯ (t )]2 |ψ(x, t )| = 2π δ (t ) exp − . 4δ 2 (t ) 

2

(4)

A.B. Nassar / Annals of Physics 331 (2013) 317–322

319

This minimum-uncertainty wave packet solution is further supported by recent, alternative stochastic approaches [27] which have demonstrated that individual quantum trajectories remain minimum-uncertainty localized wave packets for all times: the localization being stronger the smaller h¯ becomes. Now, since ln |ψ(x, t )|2 ∝ [x − x¯ (t )]2 , the minimum-uncertainty wave packet can be used to support a generalization of Eq. (3). Accordingly, we propose that the evolution of the wave function of the quantum system ψ(x, t ) under continuous measurement can be described in terms of the logarithmic nonlinear Schrödinger equation: ih¯

   ∂ψ(x, t )  = H (x, t ) − ih¯ κ ln |ψ(x, t )|2 − ln |ψ(x, t )|2 ψ(x, t ), ∂t

(5)

where κ characterizes the resolution of the measurement. The last term in Eq. (5) arises from the requirement that the integration of this equation with respect to the variable x under the condition that for a particle the expectation value of the energy ⟨E (t )⟩, defined as

⟨E (t )⟩ ≡



+∞

ψ ∗ (x, t )E (t )ψ(x, t )dx,

(6)

−∞

must be equal to the expectation values of the kinetic and potential energies [28,29]. Eq. (5) has several unique properties. First of all, it guarantees the separability of noninteracting subsystems. Other nonlinear modifications can introduce an interaction between two subsystems even when there no real forces acting between them. Besides, the stationary states can always be normalized. For other nonlinearities, stationary solutions have their norms fully determined and after multiplication by a constant they cease to satisfy the equation. In addition, the logarithmic nonlinear Schrödinger Eq. (5) possesses simple analytic solutions in a number of dimensions — especially non-spreading wave-packet solutions. It is fundamentally different from the equation proposed by Bialynicki-Birula and Mycielski [6] due to (1) the imaginary coefficient in front the logarithmic terms and (2) the last term ln |ψ(x, t )|2 . Eq. (5) can now solved via the method of quantum trajectories formalism of the Bohmian model of quantum mechanics [7–26]. To this end, the wave function is first expressed in the polar form [30]:

ψ(x, t ) = φ(x, t ) exp(iS (x, t )/h¯ ).

(7)

Now, after substitution of Eq. (7) into Eq. (5), we obtain

       i ∂S h¯ 2 ∂ 2φ φ ∂S 2 ∂ S ∂φ ∂ 2S i ∂φ + φ =− − 2 2 + 2 + ih¯ ∂t h¯ ∂ t 2m ∂ x2 ∂x h¯ ∂x ∂x ∂x h¯   − ih¯ κ ln φ 2 − ⟨ln φ 2 ⟩ φ. 

(8)

Eq. (8) can be separated into real and imaginary parts. By defining the quantum hydrodynamical density ρ , velocity v and quantum potential Vqu respectively as

ρ(x, t ) = φ 2 (x, t ), 1 ∂S v= , m ∂x h¯ 2 ∂ 2 φ Vqu = − , 2mφ ∂ x2

(9) (10) (11)

we have

∂v ∂v 1 ∂ Vqu +v =− ∂t ∂x m ∂x

(12)

∂ρ ∂ + (ρv) + κ [ln ρ − ⟨ln ρ⟩] ρ = 0. ∂t ∂x

(13)

and

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A.B. Nassar / Annals of Physics 331 (2013) 317–322

Eq. (12) is an Euler-type equation describing trajectories of a fluid particle, with momentum p = mv , whereas Eq. (13) describes the evolution of the quantum fluid density ρ . This density is interpreted as the probability density of a particle being actually present within a specific region. Such an actual particle follows a definite space–time trajectory that is determined by its wave function through an equation of motion in accordance with the initial position, formulated in a way consistent with the Schrödinger time evolution. An essential and unique feature of the quantum potential is that the force arising from it is unlike a mechanical force of a wave pushing on a particle with a pressure proportional to the wave intensity. By assuming that initially the wave packet is centered at x = 0 and

 −1/2   ρ(x, 0) = 2π δ 2 (0) exp −x2 /2δ 2 (0) and ρ vanishes for |x| → ∞ at any time we may rewrite (see Eq. (4))

   −1/2 [x − x¯ (t )]2 . ρ(x, t ) = |ψ(x, t )|2 = 2π δ 2 (t ) exp − 2δ 2 (t )

(14)

Now, Eq. (14) can be readily used to demonstrate that +∞



[x − x¯ (t )]2 ρ(x, t )dx = δ 2 (t ).





(15)

−∞

Substitution of Eq. (14) into Eq. (13) yields:

  δ˙ (x − x¯ ) ˙ 1 ∂ρ 2˙ ¯ ¯ = − + x + ( x − x ) δ ρ, ∂t δ δ2 δ3

(16)

and

∂(ρv) = ∂x



     δ˙ δ˙ (x − x¯ ) −κ ρ+ − κ (x − x¯ ) + x˙¯ − ρ, δ δ δ2

(17)

which implies that

v(x, t ) =



 δ˙ − κ (x − x¯ ) + x˙¯ . δ

(18)

Substitution of Eq. (18) into Eq. (12) yields:



δ¨ (t ) − 2κ δ˙ (t ) + κ 2 δ(t ) −

h¯ 2 4m2 δ 3 (t )



(x − x¯ )1 + x¨¯ (t ) (x − x¯ )o = 0,

(19)

which implies that

¨ t ) − 2κ δ( ˙ t ) + κ 2 δ(t ) = δ(

h¯ 2 4m2 δ 3 (t )

(20)

and x¨¯ (t ) = 0.

(21)

Eqs. (21) and (20) show that a continuous measurement of a quantum wave packet gives specific features to its evolution: the appearance of distinct classical and quantum elements, respectively. This measurement consists of monitoring the position of the quantum system and the result is the measured classical path x¯ (t ) for t within an quantum uncertainty δ(t ). From Eq. (20), we note that for κ ̸= 0 a stationary regime can be reached and that the width of the wave packet can be related to the resolution of measurement as follows:

κ=

h¯ 2mδo2

,

(22)

which means that if an initially free wave packet is kept under a certain continuous measurement, its width may not spread in time.

A.B. Nassar / Annals of Physics 331 (2013) 317–322

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Now, the associated Bohmian trajectories [9,31–33] of an evolving ith particle of the ensemble with an initial position xoi can be calculated by first substituting x˙ i (t ) = vi (x, t )

(23)

into Eq. (18). Then, with the help of Eq. (22), we obtain: xi (t ) = vo t + xoi e−h¯ t /2mδo , 2

(24)

where vo , the initial group velocity of the wave packet, is the pre-measurement velocity of any individual particle of the ensemble and xoi is the initial position of the ith individual particle corresponding to the wave function given by Eq. (7). In turn, the velocity and acceleration of the ith-particle of the initial Gaussian ensemble at any instant t can be calculated from Eq. (24), namely:

vi (t ) = vo −

h¯ 2mδo2

xoi e−h¯ t /2mδo

2

(25)

and ai ( t ) =

h¯ 2 4m2 δo4

xoi e−h¯ t /2mδo . 2

(26)

The resulting picture is worth noticing: (1) The classical regime is found by setting h¯ → 0 in Eqs. (26)–(25). (2) The quantum trajectories converge asymptotically in time to Newtonian trajectories. If xoi = 0, then the particle follows the Newtonian trajectory at any time. If, however, xoi is positive, then the particles distributed in the right half of the initial ensemble are accelerated whereas the particles distributed in the left half of the initial ensemble are decelerated. Nevertheless, there is only a temporary asymmetry in the Bohmian velocities between any two symmetric particles since the rate of the asymmetry diminishes with time. After a short time, the distance in position space shifted by the particles initially lying at positive and negative xoi ′ s converges to a constant value. So, continuous measurements not only disturb the particle but compel it to eventually converge to a classical position. The rate of this convergence depends on what we define here as the Bohm time constant which determines the time resolution of the quantum measurement:

τB ≡

2mδo2 h¯

.

(27)

Finally, from Eq. (11) we have that quantum force is given by Fqu

  ∂ Vqu ∂ h¯ 2 h¯ 2 h¯ 2 2 ¯ ( x − x ) + = (x − x¯ ), =− =− − ∂x ∂x 8mδo4 4mδo2 4mδo4

(28)

which with the help of Eq. (24) yields Fqu =

h¯ 2 4mδo4

xoi e−t /τB .

(29)

So, the convergence of the quantum particle trajectories to classical trajectories is due to the influence of the measuring apparatus through the quantum force [34]. The quantum force is directly proportional to the initial position of the ith-particle and decays exponentially in time (it drops 63% of its initial value after a time constant τB ). Likewise, the position, velocity and acceleration of the wave packet approach their classical values. So, continuous observation of a wave packet may lead to a gradual freezing of the quantum features of the particle. Finally, if the initial wave packet width for an electron is taken to be equal to 2.8 × 10−15 m (the approximate size of an electron [35]) then the Bohm time constant of the measurement assumes the value (see Eq. (27)):

τB =

(9.1 × 10−31 )(2.8x10−15 )2 = 6.8 × 10−26 s. (6.63 × 10−34 )/2π

(30)

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A.B. Nassar / Annals of Physics 331 (2013) 317–322

Acknowledgment Part of this work was done during the Summer of 2012 at the UCLA Physics and Astronomy Department. Correspondence with L. Marchildon, I. Bialynicki-Birula, R. Tumulka and S. Goldstein is gratefully acknowledged. References [1] M.B. Mensky, Phys. Lett. A231 (1997) 1; Phys. Lett. A307 (2003) 85; Internat. J. Theoret. Phys. 37 (1998) 273; Continuous Quantum Measurement and Path Integrals, IOP Publisher, Bristol, 1993. See references therein. [2] D. Home, M.A.B. Whitaker, J. Phys. A: Math. Gen. 25 (1992) 657, and references therein. [3] B. Misra, E.C.G. Sudarshan, J. Math. Phys. 18 (1977) 756; C.B. Chin, E.C.G. Sudarshan, B. Misra, Phys. Rev. D 16 (1977) 520. [4] A. Peres, Amer. J. Phys. 48 (1980) 931; A. Peres, A. Ron, Phys. Rev. A 42 (1990) 5720. [5] W.M. Itano, D.J. Heinzen, J.J. Bollinger, D.J. Wineland, Phys. Rev. A 41 (1990) 2295. [6] I. Bialynicki-Birula, Brazilian J. Phys. 35 (2005) 211. [7] D. Dürr, S. Goldstein, R. Tumulka, N. Zanghi, Phys. Rev. Lett. 93 (2004) 090402. See references therein. [8] E. Guay, L. Marchildon, J. Phys. A: Math. Gen. 36 (2003) 5617. [9] A.S. Sanz, F. Borondo, S. Miret-Artés, Phys. Rev. B61 (2000) 7743. [10] D. Bohm, Phys. Rev. 85 (1952) 166; Phys. Rev. 85 (1952) 180. [11] R. Tumulka, Amer. J. Phys. 72 (2004) 1220, See references therein. [12] Oleg V. Prezhdo, Craig Brooksby, Phys. Rev. Lett. 86 (2001) 3215. [13] Y. Nogamia, F.M. Toyamab, W. van Dijk, Phys. Lett. A270 (2000) 279. [14] A.R. Plastino, M. Casas, A. Plastino, Phys. Lett. A281 (2001) 297. [15] N.C. Dias, J.N. Prata, Phys. Lett. A302 (2002) 261. [16] A.J. Makowski, P. Pepowski, S.T. Dembiski, Phys. Lett. A266 (2000) 241. [17] L. Shifren, R. Akis, D.K. Ferry, Phys. Lett. A274 (2000) 75. [18] P. Falsaperla, G. Fonte, Phys. Lett. A316 (2003) 382. [19] A. Datta, P. Ghose, M.K. Samal, Phys. Lett. A322 (2004) 277. [20] C. Colijn, E.R. Vrscay, Phys. Lett. A300 (2002) 334. [21] G. Potel, M. Muoz-Alear, F. Barranco, E. Vigezzi, Phys. Lett. A299 (2002) 125. [22] R.G. Stomphorst, Phys. Lett. A292 (2002) 213. [23] Md.M. Ali, A.S. Majumdar, D. Home, Phys. Lett. A304 (2002) 61. [24] G.E. Bowman, Phys. Lett. A298 (2002) 7. [25] S. Goldstein, in: E. N. Zalta (Ed.), Bohmian mechanics in Stanford Encyclopedia of Philosophy. http://plato.stanford.edu/ archives/win2002/entries/qm-bohm/. [26] J.S. Bell, Speakable and Unspeakable in Quantum Mechanics, Cambridge University Press, Cambridge, 1987. [27] W.T. Strunz, L. Diósi, N. Gisin, T. Yu, Phys. Rev. Lett. 83 (1999) 4909, and references therein. [28] M.D. Kostin, J. Chem. Phys. 57 (1972) 3589. [29] For a free particle, the expectation value of the total energy must be equal to the expectation value of the kinetic

 +∞

[30] [31] [32] [33]

∂ψ(x,t )

2

 +∞

∂ 2 ψ(x,t )

P 2 (t )

 +∞

h¯ 2 ¯ energy, i.e., ih¯ −∞ ψ ∗ (x, t ) ∂ t dx = − 2m −∞ ψ ∗ (x, t ) ∂ x2 dx or ⟨E (t )⟩ = ⟨ 2m ⟩ since −∞ ([x − x(t )] − δ 2 (t )) |ψ(x, t )|2 dx = 0 by using Eq. (14). See also Eq. (15). A.B. Nassar, J. Math. Phys. 27 (1986) 2949. A.K. Pan, Pram. J. of Phys. 74 (2010) 867. P. Holland, Ann. Phys. 315 (2005) 505. R.E. Wyatt, Quantum Dynamics with Trajectories, Springer, New York, 2005. See pages 61, 150 and 151.  



[34] From Eq. (14), we note that the expectation value of the quantum force vanishes at all times: Fqu h2

¯

2mδ 2



= − m1

∂ Vqu ∂x

=

⟨x − x¯ (t )⟩ = 0.

[35] Experiments to measure the size of the electron consist on colliding two beams of electrons against each other and counting how many are scattered and altered their trajectories. By counting the collisions, and knowing how many particles we have thrown, we can estimate the average size of each particle in the beam.