Chattaraj P.K. (Ed.) Quantum Trajectories

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294 Quantum Trajectories

–2

–10

–5

510

FIGURE 18.5 Quantum caves for the head-on collision of two Gaussian wave packets with the

initial conditions: x

=−10 =−x

, v

= 2 =−v

and σ

√

2, and maximal interfer-

ence occurs at t = 5 in real space. These caves are formed with the isosurfaces |Ψ(z, t)|=0.053

(lighter gray sheets) and |∂Ψ(z, t)/∂z|=0.106 (darker gray sheets). The complex quantum

trajectories launched from two branches of the isochrone reach the real axis at t = 5. All

quantities are given in atomic units. (Reproduced with permission from Chou, C. C., Sanz,

A. S., Miret-Artés, S., and Wyatt, R. E., Phys. Rev. Lett., 102, 250401, 2009. Copyright APS

2009.)

function describing the head-on collision of two Gaussian wave packets was analyt-

ically extended to the complex plane.

As systematically analyzed for local structures of the QMF and its PVF around

stagnation points and poles [42, 43], complex quantum trajectories display helical

wrapping around stagnation tubes and hyperbolic deﬂection near vortical tubes, as

shown in Figure 18.5. These tubes are prominent features of quantum caves in space-

time Argand plots. Stagnation and vortical tubes alternate with each other, and the

centers of the tubes are stagnation and vortical curves. In addition, the equation for

approximate quantum trajectories around stagnation points was also derived. In con-

trast, Pólya trajectories display hyperbolic deﬂection around stagnation tubes and heli-

cal wrapping around vortical tubes. The intriguing topological structure consisting of

these tubes develops quantum caves in space-timeArgandplots, and quantum interfer-

ence leads to the formation of quantum caves and produces this topological structure.

Furthermore, the divergence and vorticity of the QMF characterizes the turbulent ﬂow

of trajectories; however, the PVF describes an incompressible and irrotational ﬂow

because of its vanishing divergence and vorticity except at poles in the complex plane.

For the case where the relative propagation velocity is larger than the spreading

rate of the wave packets, trajectories keep circulating around stagnation tubes for ﬁnite

times during interference and then escape as time progresses. This counterclockwise

Analytical Studies of Complex Quantum Trajectories 295

circulation of trajectories launched from different positions around the same stagna-

tion tubes can be viewed as a resonance process. The whole process shows a type

of long-range correlation among trajectories arising from different starting points. In

addition, the wrapping time for an individual trajectory can be determined accord-

ing to the divergence and vorticity of the QMF. The variation in the wrapping times

between trajectories is connected to the distribution of nodes in the wave function

and stagnation points of the QMF, and the average wrapping time over trajectories

from the isochrone arriving at the real axis at maximal interference was calculated as

one of the deﬁnitions for the lifetime of interference.

The rotational dynamics of the nodal line arising from the interference of these two

wave packets was thoroughly analyzed in the complex plane. Since the interference

features are transiently observed on the real axis only when the nodal line is near

the real axis, we can deﬁne the lifetime for the interference process corresponding

to the time interval for the nodal line to rotate from θ =−10

◦

to θ =+10

◦

.As

shown in Figure 18.6, the nodal line rotates counterclockwise from the initial angle

to a limiting value with an angular displacement ∆θ = π/2, and its intersections with

nodal trajectories determine the positions of nodes. In addition, the distance between

nodes increases with time. For the case shown in Figure 18.6, interference features

are observed on the real axis only when the nodal line is near the real axis, and this

occurs between about θ(3.52) =−10

◦

and θ(7.32) = 10

◦

. Thus, the lifetime for the

interference features is about ∆t = 3.8.

–4

–2

0 2 4

–3

–2

–1

t = 5

t = 10

t = ∞

t = 0

FIGURE 18.6 For the total wave function with the initial conditions: x

=−10 =−x

= 2 =−v

and σ

√

2, the evolution of the nodal line at t = 0, 2.5, 5, 7.5, 10 (black solid

lines) and t =∞(black dashed line) is shown, and the arrows indicate the rotation direction.

Nodes and stagnation points are denoted by dots; nodal trajectories are shown as dotted lines

passing through the nodal points. All quantities are given in atomic units. (Reproduced with

permission from Chou, C. C., Sanz, A. S., Miret-Artés, S., and Wyatt, R. E., Phys. Rev. Lett.,

102, 250401, 2009. Copyright APS 2009.)

296 Quantum Trajectories

For the case where the relative propagation velocity is approximately equal to

or smaller than the spreading rate of the wave packets, the rapid spreading of the

wave packets leads to the distortion of quantum caves. Because of the rapid spreading

rate of the wave packets, interference features develop very rapidly and remain visible

asymptotically in time.Therefore, the wrapping time becomes inﬁnity, and this implies

the inﬁnite survival of such interference features. However, the rotational dynamics

of the nodal line clearly explains the persistent interference features observed on the

real axis.

In contrast with conventional quantum mechanics, two counter-propagating wave

packets always interfere with each other in the complex plane, and this leads to a

persistent pattern of nodes and stagnation points. Thus, the rotational dynamics of the

nodal line in the complex plane clearly explains the transient or persistent appearance

of interference features observed on the real axis, and it offers a uniﬁed and elegant

method to deﬁne the lifetime for interference features. Therefore, these results show

that the complex trajectory method provides a novel and insightful understanding of

quantum interference at a fundamental level.

18.3.4 C

OMPLEX-EXTENDED BORN PROBABILITY DENSITY

Several studies have been devoted to issues concerning the probability density and

ﬂux continuity in the complex plane and probability conservation along complex

quantum trajectories [47–50]. First, Poirier proposed a complexiﬁed analytic proba-

bility density ρ

(z) in the complex plane deﬁned by the product of the wave function

and its generalized complex conjugate, ρ

(z, t) = Ψ(z, t)Ψ

∗

, t). This probabil-

ity density is consistent with the analytic continuation of the traditional probability

density ρ(x, t) deﬁned along the real axis. Although this probability density is com-

plex valued off the real axis, it satisﬁes a complexiﬁed ﬂux continuity equation in

the complex plane. However, it has been pointed out that this probability density

produces additional nodes because it satisﬁes the Schwarz reﬂection principle [48].

These additional nodes are located in the mirror images with respect to the real axis

of the original nodes of the wave function, and they are not a feature of the original

complex-extended wave function Ψ(z, t). Thus, the probability density proposed by

Poirier does not match the node structure of the wave function. On the other hand,

John proposed an approach to determine a conserved probability density in the com-

plex plane by solving an appropriate equation. However, it was found that conserved

probability densities may not exist in some regions in the complex plane [49].

The complex-extended Born probability density deﬁned by replacing x with z in

the Born probability density of quantum mechanics, ρ

(z, t) = Ψ(z, t)Ψ

∗

(z, t) where

time remains real valued, was recently studied in the complex plane [48, 50]. This

probability density is real valued and nonnegative in complex space. It is important to

note that this probability density is continuous but not analytic in the complex plane.

Although we cannot obtain a complex-version continuity equation for the Born prob-

ability density because of its nonanalyticity, we can derive an arbitrary Lagrangian–

Eulerian (ALE) rate equation for this probability density in the complex plane.

This equation gives the rate of change in the density along an arbitrary path.If

the grid point velocity is equal to zero, the Eulerian rate equation for the density is

Analytical Studies of Complex Quantum Trajectories 297

recovered and the imaginary part of the complex total energy density determines the

local rate of change in the density. If the grid point velocity is equal to the velocity

of a complex quantum trajectory, we obtain the Lagrangian rate equation describ-

ing the rate of change in the density along the complex quantum trajectory and the

complex quantum Lagrangian density determines the rate of change along this tra-

jectory. Furthermore, when the grid point velocity is equal to the velocity of a Pólya

trajectory, the rate of change in the probability density originates completely from

the local contribution. In particular, only the complex total potential energy density

contributes to the rate of change in the density along a complex path with the velocity

equal to the QMF divided by 2m where m is the mass of the particle. The effect of

this speciﬁc velocity is to cancel the contribution from the kinetic energy to the rate

of change in the density. For stationary states, the rate of change in the probability

density along a Pólya trajectory is equal to zero, and this result reﬂects the fact that

the PVF is parallel to contours of the Born probability density [42,43].

The rate of change in the Born probability density along an arbitrary path can

be described by the ALE rate equation; moreover, this density is closely related to

the trajectory dynamics in the complex plane. The trajectory dynamics for the wave

function is determined through the QMF by the dynamical equation, dz(t)/dt =

p(z, t)/m. The PVF of the QMF contains exactly the same information as the QMF

itself; hence, it also correctly reﬂects the information of stagnation points and poles.

Furthermore, the PVF associated with the QMF is shown to be parallel to contours of

the Born density. Therefore, the complex-extended Born probability density correctly

reﬂects information on trajectory dynamics in complex space.

18.4 CONCLUDING REMARKS AND FUTURE DIRECTIONS

Analytical studies of the complex quantum trajectory formalism are based on a pre-

computed wave function or the exact analytical form of the wave function. Within the

framework of the complex quantum Hamilton–Jacobi formalism, this brief review

describes one-dimensional stationary scattering problems, quantum vortices, quantum

streamlines for the QMF and its PVF around stagnation points and poles, wave-packet

interference and quantum caves, and the complex-extended Born probability density.

These relevant analytical studies indicate that the complex quantum trajectory method

can provide a novel and insightful understanding of quantum phenomena.

Most of the previously mentioned analytical studies concentrate on one-

dimensional problems. Thus, multidimensional processes in chemical physics deserve

further investigation. Moreover, some fertile areas (including multimode system-bath

dynamics, mixed quantum-classical dynamics, and density matrix evolution in dis-

sipative systems) could be further explored within the complex quantum trajectory

formalism.

ACKNOWLEDGMENTS

We thank the Robert Welch Foundation (grant no. F–0362) for their ﬁnancial support

of this research.

298 Quantum Trajectories

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Modiﬁed de Broglian

Mechanics and

the Dynamical Origin

of Quantum Probability

Moncy V. John

CONTENTS

19.1 Historical Introduction................................................................................301

19.2 Complex Quantum Trajectories..................................................................303

19.3 Probability from the Velocity Field.............................................................305

19.4 Probability Distribution for the Subnests....................................................309

19.5 Discussion...................................................................................................310

Bibliography ..........................................................................................................311

19.1 HISTORICAL INTRODUCTION

A well-known formulation of classical mechanics of system of particles is based on

solving the Hamilton–Jacobi equation

∂S

∂t

+ H



∂S

∂q

, t



= 0, (19.1)

where S(q

, α

, t) is Hamilton’s principal function, H is the Hamiltonian, q

are the

conﬁguration space variables, α

are constants of integration, and t denotes time. If

the Hamiltonian is independent of t and is a constant of motion equal to the total

energy E, then S and Hamilton’s characteristic function W are related by

S = W − Et. (19.2)

Since W is independent of time, the surfaces with W = constant have ﬁxed locations

in conﬁguration space. However, the S = constant surfaces move in time and may

be considered as wavefronts propagating in this space [1]. For a single particle one

can show that the wave velocity at any point is given by

u =

|∇W |

. (19.3)

301

302 Quantum Trajectories

This shows that the velocity of a point on this surface is inversely proportional to

the spatial velocity of the particle. Also one can show that the trajectories of the

particle must always be normal to the surfaces of constant S. The momentum of the

particle is obtained as [1]

p =∇W. (19.4)

It is not clear whether this wave picture was known to classical physicists. However,

in hindsight, we may now observe that there exists a correspondence between classical

mechanics and geometrical optics. That is, the particle trajectories orthogonal to

surfaces of constant S in classical mechanics, as we have discussed above, are similar

to light rays traveling orthogonal to Huygens’ wavefronts.

In modern physics, the discovery of the wave nature of matter is credited to L. de

Broglie who famously postulated this phenomenon in 1923 just by reﬂecting upon

nature’s love for symmetry. On its basis, he made the astonishing prediction in the

history of physics that a beam of electrons would exhibit wave-like phenomena of

diffraction and interference upon passing through sufﬁciently narrow slits, for which

he was awarded the Nobel Prize in 1929.

In spite of this, the fact that Broglie’s 1924 Ph.D. thesis [2] contained the seed of a

new mechanics, which can replace classical mechanics, did not get due attention for a

long time. In it, and later in a paper published in 1927 [3], he proposed that Newton’s

ﬁrst law of motion be abandoned, and replaced by a new postulate, according to which

a freely moving body follows a trajectory that is orthogonal to the surfaces of equal

phase of an associated guiding wave. He presented his new theory in the 1927 Solvay

Conference too, but it was not well-received at that time [4].

The Schrödinger equation, which is the cornerstone of quantum mechanics, can

be recast in a form reminiscent of the Hamilton–Jacobi equation by substituting

Ψ = R exp(iS/) in it. The guidance relation for particle trajectories, as proposed by

de Broglie, was

=∇

S. (19.5)

Solving this, we get a particle trajectory for each initial position. In 1952, D. Bohm

rediscovered this formalism [5], but in a slightly different way. He noted that if we take

the time derivative of the above equation, Newton’s law of motion may be obtained

in the form

=−∇

(V +Q). (19.6)

Here

Q =−Σ



∇

|Ψ|

(19.7)

is called the “quantum potential.” This results in the same mechanics as that of

de Broglie, but in the language of Newton’s equation with an unnatural quantum

potential. We shall note that the ﬁrst order equation of motion Equation 19.5 sug-

gested by de Broglie represented a uniﬁcation of the principles of mechanics and

optics and this should be distinguished from Bohm’s second order dynamics based

Modiﬁed de Broglian Mechanics 303

on Equations 19.6 and 19.7 [4]. It may also be mentioned that Bohm’s revival of

de Broglie’s theory in pseudo-Newtonian form has led to a mistaken notion that de

Broglie–Bohm (dBB) theory constituted a return to classical mechanics. In fact, de

Broglie’s theory was a new formulation of dynamics in terms of wave–particle duality.

It is well-known that neither the particle picture nor the trajectories have any role

in the standard Copenhagen interpretation of quantum mechanics and that several

physicists, including Albert Einstein, were not happy with this interpretation for var-

ious reasons. Einstein was not very happy with the de Broglian mechanics either [4].

The reason for this was speculated to be the problem it faced with real wave func-

tions, where the phase gradient ∇S = 0. For the wide class of bound state problems,

the time-independent part of the wave function is real and hence while applying the

equation of motion 19.5, the velocity of the particle turns out to be zero everywhere.

This feature, that the particles in bound eigenstates are at rest irrespective of their

position and energy, is counter-intuitive and is not a satisfactory one.

A deterministic approach to quantum mechanics, which claims the absence of this

problem is the Floyd–Faraggi–Matone (FFM) trajectory representation [6–8]. In one

dimension, they use a generalized Hamilton–Jacobi equation which is equivalent to

that used by dBB. But FFM differs from dBB in that the equation of motion in the

domain [x, t] is rendered by Jacobi’s theorem. For stationarity, the equation of motion

for the trajectory time t, relative to its constant coordinate τ, is given as a function

of x by

t −τ =

∂W

∂E

, (19.8)

where τ speciﬁes the epoch. Thus the dBB and FFM trajectory representations differ

signiﬁcantly in the use of the equation of motion, though they are based on equivalent

generalized Hamilton–Jacobi equations.

19.2 COMPLEX QUANTUM TRAJECTORIES

By modifying the de Broglian mechanics, complex quantum trajectories were ﬁrst

obtained and drawn for the case of some fundamental problems in quantum mechanics,

such as the harmonic oscillator, potential step, wave packets etc., in 2001 [9]. This

modiﬁed dBB approach also attempts to unify the principles of mechanics and optics,

but in a different way. An immediate application of the formalism was in solving the

above problem of stationarity of quantum particles in bound states, in a natural way.

For obtaining this representation, we shall substitute Ψ = e

S/

in the Schrödinger

equation to obtain the quantum Hamilton–Jacobi equation (QHJE) [1]

∂

∂t







∂

∂x



+ V





i

∂

∂x

, (19.9)

and then postulate an equation of motion similar to that of de Broglie:

m ˙x ≡

∂

∂x



∂Ψ

∂x

, (19.10)