Chattaraj P.K. (Ed.) Quantum Trajectories

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154 Quantum Trajectories

time-dependent parameter. Thus the approximate wavefunction

ψ(x, X(t), t) =

R(x, X(t), t) exp(i

S(x, X(t), t )/) obeys the Schrödinger equation

i

ψ(x, X(t), t)



−



∂

∂x

+ V (x, X(t))



ψ(x, X(t), t). (10.15)

Note the appearance of the total derivative in the left-hand side of this equation. As

will be shown below, this is important when an adiabatic basis set is used for solving

the quantum problem. Since we have only approximated the equations of motion and

supposed that the initial wavefunction ψ

(x, X) is known, we have

ψ(x, X(t = 0), t = 0) = ψ

(x, X)

X =X(t =0)

(10.16)

as the initial wavefunction in the quantum subspace.

A consistent equation of motion for the classical degrees of freedom is obtained

by taking the total derivative with respect to time of Equation 10.11, noting that this

leads to a term involving the second derivative of S with respect to X, which we have

assumed to be small. Hence, we can use Equation 10.9 to write

P =−

∂



V (x, X) +

Q(x, X)



∂X

x =x(t),X =X(t)

. (10.17)

The fact that at this level of approximation, the quantum potential corresponding

to the quantum subsystem remains in the classical equation of motion, is somewhat

reminiscent of the Pechukas force in the surface hopping method [7]. In both cases, the

classical degrees of freedom are directly affected by changes in the quantum subspace.

In practice, however, since we do not solve Equations 10.8 and 10.9 directly but only

follow a speciﬁc trajectory X(t), we additionally neglect the quantum potential in the

classical degree of freedom. At this level of approximation, the MQCB method is

identical to the one proposed independently by Prezhdo et al. [27–29].

10.3 STOCHASTIC DVR AND MQCS

Considering the same two-dimensional Hamiltonian as above, we choose a one-

dimensional (diabatic) DVR basis |x

 with eigenvalue x

of the position operator,

for the quantum degree of freedom, and expand the total wavefunction according to:

ψ(x, X(t), t) =



(t) |x

, (10.18)

where the wave-packet Ψ(x, t) propagates with time along the x coordinate, and is

a DVR basis function with eigenvalue x

of the position operator, such that c

(t) =

x

|ψ(t)=ψ(x

, X(t), t). The values of the time-dependent coefﬁcients can be

obtained by solving the time-dependent Schrödinger equation using any conventional

method (split operator, etc.).

Initially the wave-packet is a coherent superposition of DVR states. The time

evolution of the coefﬁcients of the wave-packet in these DVR states is given by

Mixed Quantum/Classical Dynamics 155

=−



(t)x

|H |x

=−







(t)x

|T |x

+V (x

, X(t))



(10.19)

where we have used the fact that the nondiagonal matrix elements of the Hamiltonian

reduce to the matrix elements of the kinetic energy operator T in the DVR basis and

the potential energy matrix is diagonal.

In the MQCS method [21], the classical degrees of freedom are calculated using

the force evaluated at one single DVR point,

P =−

∂

(

V (x, X)

)

∂X

x =x

(t),X =X(t)

. (10.20)

The dynamics in the quantum subspace is accounted for by the possibility of quantum

jumps between these points. Hence, x

is a sequence of DVR points, forming a

stochastic quantum trajectory, deﬁned as follows: Equation 10.19 determines the

transition probabilities between DVR states. Let us introduce the diagonal density

matrix elements P

=|c

=|ψ(x

, X(t), t)|

representing the probability for the

system to be in state |x

 at time t. The time evolution of P

(t) is deduced from

Equation 10.19:



(t) (10.21)

where b

is given by







∗

x

|T |x





. (10.22)

In Equation 10.21, the b

coefﬁcients can be interpreted as the population ﬂux from

state |x

 to |x

, with the property that b

(t) =−b

(t). Hence if the system is in

state |x

 at time t, the probability to jump to state |x

 at time t + δt is given by

n→m

δt. (10.23)

In the MQCS method [21], this equation is used as follows. An initial state |x



is selected at random, following a probability distribution given by P

(t = 0) ≡

|x

|Ψ(x,0)|

. For each initial state a sequence of events is selected by testing at

each time step the probability for transition to any other state |x

. Suppose that the

sequence of events has led to state |x

 at time t. A random number g is generated

between 0 and 1. If it is smaller than b

, then at time t + δt the state will be 1; if

<g<b

+ b

then the state will be state 2, etc.; and if



n=m

<g, then

the state at time t + δt will still be |x

. At the end of the propagation (t = t

), the

recombination of the ﬁnal states of all these chains of events gives the wave-packet

at time t

This approach is similar to the way the quantum potential energy surface for the

classical particle dynamics is selected in Tully’s fewest switches method [17] for

156 Quantum Trajectories

mixed quantum–classical dynamics. The MQCS method can thus be thought of as

the implementation of the surface hopping method in the DVR, rather than in the

usual FBR.

Equation 10.21 can be rewritten to give:



(

− T

)

(10.24)

with



(2)





∗

if T

> 0 (10.25)

where we have denoted by D

(2)

=x

|∂

/∂x

the matrix elements of the second

derivative.

10.4 DISCRET BEABLES

In this section, we discuss the connection between a continuous quantum trajectory

and stochastic hopping between discrete grid points ﬁrst proposed by Vink in 1993

[36]. To this end, it is enough to consider only one quantum degree of freedom.

10.4.1 V

INK’S FORMULATION

We consider a particle of mass M in a one-dimensional space extending over −∞ <

x<+∞. This space is discretized as x

= n; n =−∞, +∞ where  is the grid

spacing. If we denote by P

=|ψ(x

, t)|

=|ψ

the probability to ﬁnd the particle

at x

, the evolution in time of P

is given by

∂P

∂t

=−





∗

∂

∂x

. (10.26)

Using the three-point centered expansion of the second derivative,

∂

∂x

n−1

− 2ψ

+ ψ

n+1



(10.27)

it is found that

∂P

∂t

=−



M



∗

n−1

+ ψ

∗

n+1

. (10.28)

This equation can be written as

∂P

∂t



(10.29)

Mixed Quantum/Classical Dynamics 157

with

=−



M

{ψ

∗

n+1

m,n+1

+ ψ

∗

n−1

m,n−1

}. (10.30)

Writing

n±1

= ψ

± ψ



(10.31)

where the prime denotes a ﬁrst derivative with respect to x, and using the form

= A

exp(iS

/) with A

and S

real, we obtain



M





m,n−1

− S



m,n+1



(10.32)

and we note that J

=−J

. Comparing Equation 10.29 with the rate equation

∂P

∂t



(

− T

)

(10.33)

and using Bell’s minimal choice [37], we get for n = m,

P

; if J

≥ 0 (10.34)

and 0 otherwise. Thus using Equation 10.32, we obtain, for n = m,



M

m,n−1

; if S



≥ 0

=−



M

m,n+1

; if S



≤ 0. (10.35)

Finally T

is determined by the normalization

dt = 1 −



n=m

dt (10.36)

with

dt ≤

M



n−1

− S



n+1

(10.37)

in order to have T

dt ≥ 0. Vink has shown [36] that if one uses Equations 10.35 and

10.36 it is possible to recover the causal Bohmian trajectories through a stochastic

algorithm with minimal choice for the transition density. The proof is straightforward.

For a trajectory starting in point n at t = 0 the probability to jump to the position

n + 1 is, if we assume that S



= 0,

n+1



M

dt. (10.38)

The mean average displacement after dt will then be given by dx = P

n+1

which

using Equation 10.38 gives in the limit  → 0

∂x

∂t

∂S

∂x

(10.39)

which is precisely the Bohmian equation for the quantum trajectories.

158 Quantum Trajectories

What remains to be done is to show that the second moment of the distribution,

i.e., the dispersion in the average displacement, vanishes in the limit  → 0. The

mean-squared dispersion for one jump is given by

= 

n+1

− (P

n+1

)

= 





1 −





(10.40)

which indeed vanishes for  → 0.

10.4.2 N-P

OINT FORMULA

If instead of a three-point ﬁnite difference formula for the second derivative as in

Equation 10.27, we use a ﬁve-point centered expression [38], we shall have

∂

∂x

−ψ

n−2

+ 16 ψ

n−1

− 30 ψ

+ 16 ψ

n+1

− ψ

n+2

12

(10.41)

and we get

=−



12M

{−ψ

∗

n−2

m,n−2

+ 16ψ

∗

n−1

m,n−1

(10.42)

+ 16ψ

∗

n+1

m,n+1

− ψ

∗

n+2

m,n+2

}. (10.43)

Writing

n±ν

= ψ

± νψ



(10.44)

and following the line of reasoning of the three-point case, we ﬁnally get, for n = m,



6M

(δ

m,n+2

+ 8δ

m,n−1

); if S



≥ 0

=−



6M

(δ

m,n−2

+ 8δ

m,n+1

); if S



≤ 0. (10.45)

The difference with respect to the three-point case is that now, even for a given

sign of S



, we can have jumps to m ± 1 and also to m ∓ 2 with probabilities

n±1



dt; P

n±2



dt. (10.46)

However if we calculate the mean average distance for a dt we get

dx = P

n+1

− 2P

n−2

P

n+1



dt (10.47)

i.e., the same Bohmian trajectory equation. As for the mean-squared dispersion

= 

n+1

+ (−2)

n−2

−



P

n+1



= 2





1 −





(10.48)

which again vanishes for  → 0.

Mixed Quantum/Classical Dynamics 159

These results can be generalized to an arbitrary number of points. According to

Fornberg [38], the expression for the second derivative in the (p + 1)-point centered

ﬁnite difference approximation is

∂

∂x



p/2



j=−p/2

(p)ψ

n+j

(10.49)

with

j=0

(p) =

(−)

j+1

2(p/2!)

(p/2 + j)! (p/2 −j )!

; C

(p) =−2

p/2



k =1

. (10.50)

We note that in the limit p →∞

j=0

(∞) =

(−)

j+1

; C

(∞) =−

. (10.51)

Using these expressions and Equation 10.44 we obtain for S



> 0 and n = m the

jumping density



M

p/2



j=−p/2

(p)δ

m,n−j



M

p/2



j=−p/2

(−)

j+1

2(p/2!)

j(p/2 + j)! (p/2 − j)!

m,n−j

(10.52)

Using this equation we get

n+j



M

(p) dt =



(−)

j+1

2(p/2!)

j(p/2 + j)! (p/2 − j)!

dt (10.53)

from which

dx = 



n+j



M

dt2



(−)

j+1

(p/2!)

(p/2 + j)! (p/2 −j )!

(10.54)

and since



k=1

(−)

k+1

(n!)

(n + k)! (n − k)!

= 1/2 (10.55)

the Bohmian equation is found.

Similarly, the mean-squared dispersion is given by

= 







n+j

−



j,l=j

jlP

n+j

n+l





(10.56)

which presumably also vanishes for  → 0.

160 Quantum Trajectories

10.4.3 DVR POINTS

If in Equation 10.25, the linearization approximation Equation 10.44 is used, we can

write

− x

(2)





∗



if T

> 0 (10.57)

and we note that imposing c

(t) = R

(t) exp iS

(t)/, we have

{c

∗



/|c

}=S



(10.58)

and therefore Equation 10.57 becomes

− x

(2)



if T

> 0. (10.59)

Comparing this expression with those found in the N-point formulation we note

that indeed there is a clear correspondence between them. They are all proportional

to S



/M and if in the limit of inﬁnite number of DVR points



− x

)

(2)

= 1 (10.60)

they all give the Bohmian trajectories.

10.5 DISCUSSION

Comparing MQCB and MQCS, we see that both methods calculate the back-reaction

of the quantum system onto the classical system not by averaging the force over the

quantum wavefunction (as does the mean-ﬁeld method), but by choosing to evaluate

this force only at one single point. The time evolution of this point, however, is

different in both methods: the MQCB method uses the deterministic Bohm trajectory

associated with the quantum wavefunction; the MQCS method generated these points

by stochastic jumps between discrete DVR points of an underlying DVR basis. On the

other hand, generalizing Vink’s formulation, we have shown that Bohmian trajectories

can be obtained by stochastic jumps between discrete points in the limit of a large

number of points. Clearly, the MQCS method is based on the same idea except that

the discrete points are chosen according to the DVR representation. In the limit of

a large number of DVR points the MQCS trajectories should converge toward the

Bohmian MQCB trajectories.

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A Hybrid

Hydrodynamic–Liouvillian

Approach to

Non-Markovian Dynamics

Keith H. Hughes and Irene Burghardt

CONTENTS

11.1 Introduction.................................................................................................163

11.2 Theory.........................................................................................................165

11.2.1 System-Bath Hamiltonian.............................................................165

11.2.2 Effective-Mode Representation ....................................................166

11.2.3 Effective-Mode Chain Including Markovian Closure ..................167

11.2.4 Spectral Densities..........................................................................168

11.2.5 Reduced-Dimensional Effective-Mode Chain Including

Markoviant Closure ......................................................................169

11.2.6 Partial Moments............................................................................170

11.2.7 Lagrangian Trajectories ................................................................172

11.3 Results and Discussion ...............................................................................173

11.4 Conclusion ..................................................................................................178

Acknowledgments..................................................................................................179

Bibliography ..........................................................................................................179

11.1 INTRODUCTION

Trajectory based approaches have proved invaluable in most areas of dynamics. In

chemistry, trajectory based classical molecular dynamics has made it possible to study

the dynamics of up to 10

–10

atoms, and its success in this ﬁeld is well documented.

In ﬂuid dynamics, Lagrangian trajectory based methods have also proved popular and

are the method of choice for many ﬁelds in the ﬂuid dynamics community. Trajectory

based approaches have the advantages that they provide an intuitive insight into

a dynamical process and, as witnessed in classical molecular dynamics, trajectory

based approaches can be very computationally efﬁcient.

Against this backdrop of success of trajectory based approaches it is no surprise

that there has been a drive toward developing such approaches in quantum mechanics.

163