Chattaraj P.K. (Ed.) Quantum Trajectories

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184 Quantum Trajectories

of the quantum mechanical phase-space distribution function, another quantity that

possesses a classical ﬂavor and has also provided an alternative route [9] to the for-

mulation of QFD. Besides the phase-space function, the density matrix has also been

used [10] to derive the equations of QFD, both these quantities having the additional

advantage of encompassing the mixed states in contrast to the wavefunction-based

approach of Madelung [1] which can take care of only the pure states of QM. All

the early attempts towards the formulation of QFD, however, dealt with only a single

quantum particle, and thus the corresponding QFD that resulted was automatically in

three-dimensional (3-D) space.

For many-particle systems, Madelung’s wavefunction-based approach leads to

QFD in conﬁguration space [11]. Therefore, there have been attempts to reduce

or convert them to QFD equations in 3-D space. Quite naturally, the framework

of single-particle theories, where the many-particle wavefunction is expressed in

terms of single-particle orbitals, has been chosen to provide the obvious route. Thus,

the Hartree, Hartree–Fock and even natural-orbital based approaches have been

employed [11–13] for this purpose. The QFD approach that emerges from these self-

consistent ﬁeld theories mainly differs from the single-particle QFD in that the overall

velocity ﬁeld becomes irrotational, in contrast to the single-particle case where it is

rotational, although individual orbital contributions to the velocity ﬁeld are irrotational

here as well.Anotherimportant aspect is that the many-particle ﬂuid is a composite one

consisting of ﬂuid-component mixtures which move in a self-consistent ﬁeld consist-

ing of forces resulting from many-body effectsin addition to the external potential.The

quantum potential, the main ingredient of QFD, arises in this case too, although it con-

sists of individual orbital contributions, consistent with the picture of a ﬂuid mixture.

The most remarkable single-particle theory, which is also exact in principle, is the

so-called density functional theory (DFT) [14–16], a well-known theoretical approach

that uses the single-particle electron density [17] as the basic variable and provides a

conceptually simple and computationally economic route to the description of many-

particle systems within a single-particle framework. DFT thus qualiﬁes to be an ideal

vehicle to travel to the land of QFD in 3-D space.

Although originated [14] as a ground-state theory for time-independent systems,

DFT has been extended to include excited states [18] as well. Its extension and gen-

eralization to time-dependent (TD) situations, initially for periodic time-dependence

[19–21] and later for more arbitrary time-dependence [22] and inclusion of magnetic

ﬁelds [23] in addition to the electric ﬁeld, have been instrumental in the application

of TD density functional theory (TDDFT) for the transcription to QFD in 3-D space.

TDDFT employs, in addition to the electron density, another density quantity, viz.

the current density and both TDDFT and the corresponding QFD framework can efﬁ-

ciently handle the TD many-electron correlations, an important issue in many-body

physics.

As is well-known, the implementation of DFT lies in suitable approximation of

the exchange-correlation (XC) and the kinetic energy (KE) functionals. The so-called

local-density approximation (LDA) for the KE with neglect of XC altogether cor-

responds to the well-known Thomas–Fermi (TF) [24] theory, the predecessor of the

modern DFT which was introduced soon after QM came into being. Also, as early

Quantum Fluid Dynamics and Density Functional Theory 185

as 1933, the TD generalization of the TF theory was introduced by Bloch [25] who

applied the corresponding QFD framework for the study of photoabsorption. Many

years later, Mukhopadhyay and Lundqvist [26] developed the QFD corresponding to

the more generalized DFT which included, in principle, the XC contributions. This

route to QFD, where the net-density quantities (and not the orbital densities) are

involved, has also been of considerable interest although the approach is somewhat

phenomenological in nature. A quantum ﬂuid density functional theory (QFDFT)

has also been developed [27] and implemented through a generalized nonlinear

Schrödinger equation.

The TDDFT corresponding to the orbital based Kohn–Sham version has also been

used extensively [20] for the development of QFD in 3-D space. The main differ-

ence again lies in the interpretation of a single pure ﬂuid as against the mixture of

ﬂuid components. In the TDDFT for arbitrary electric and magnetic ﬁelds developed

by Ghosh and Dhara [23], the TDDFT and its QFD version have been extensively

integrated and uniﬁed.

The QFD equations correspond to a continuous ﬂuid that can however be dis-

cretized to have a discrete-particle picture which, although ﬁctitious, can lead to

the concept of quantum trajectory [28] in analogy to the well-established concept of a

classical trajectory. The concept of quantum trajectories within QFD, their numerical

evaluation through solution of the QFD equations, and the corresponding application

areas have recently been discussed in detail by Wyatt [5].

In this work, our objective is to provide a brief account of QFD within the level

of a single-particle theory, viz. the TDDFT. In what follows, we ﬁrst present QFD

for a single-particle system in Section 12.2, followed by a very brief discussion on

many-particle systems in Section 12.3. We then discuss the single-particle framework

of DFT and TDDFT in Section 12.4 and its application to QFD in 3-D space in Sec-

tion 12.5. Some miscellaneous aspects of DFT and QFD based theories are discussed

in Section 12.6. Finally, in Section 12.7, we offer a few concluding remarks.

12.2 QUANTUM FLUID DYNAMICAL TRANSCRIPTION

OF QUANTUM MECHANICS (QM) OF A SINGLE PARTICLE

The quantum mechanical description of a single-electron moving in a scalar potential

V (r, t ), is based on the TD Schrödinger equation given by

∇

ψ(r, t ) − (1/2m)V (r, t )ψ(r, t) =−iD(∂/∂t)ψ(r, t), (12.1)

where D = /2m. Madelung [1] was the ﬁrst to use the polar form for the complex

wavefunction ψ(r, t) = R(r, t ) exp[S(r, t)] in terms of the real quantities, viz. the

amplitude R(r, t) and the phase S(r, t ) and obtained the two ﬂuid dynamical equations

given by the continuity equation

(∂/∂t)ρ(r, t) +∇.[ρ(r, t)v(r, t )]=0 (12.2)

and the Euler equation

(d/dt)v(r, t ) ≡[(∂/∂t) + v(r, t).∇]v(r, t) =−(1/m)∇[V (r, t) +Q(r, t)], (12.3)

186 Quantum Trajectories

where ρ(r, t) = R

(r, t ) is the density variable, v(r, t ) = 2D∇S(r, t) the velocity

ﬁeld and j(r, t) = ρ(r, t)v(r, t) is the current density. These equations of QFD denote

the motion of the electron ﬂuid in the force ﬁeld of the classical potential V (r, t),

augmented by the force arising from the quantum potential Q(r, t ), given by

Q(r, t ) =−2mD

∇

R(r, t )/R(r, t). (12.4)

The transformation of the Schrödinger equation into the pair of QFD equations is

thus clear. It is, however, possible to consider the reverse transformation as well and,

as has been shown [29] recently, for arbitrary values of the constant D (other than

D = /2m assumed here) and the presence of a potential of the form of Equation 12.4

as arising here, the ﬂuid dynamical equations lead to a macroscopic Schrödinger-like

equation.

The velocity ﬁeld v(r, t), as deﬁned here, is in fact only the real part of a more

general complex velocity ﬁeld [3, 28] expressed as

u(r, t ) =−2iD∇ ln ψ(r, t) (12.5)

which is also irrotational in nature.

The QFD formalism corresponding to the Schrödinger equation in a vector poten-

tial K(r, t )(= (e/mc)A(r, t) for an electromagnetic ﬁeld), viz.

[(1/2)(−2iD∇−K(r, t))

+ (1/m)V (r, t)]ψ(r, t) = 2iD(∂/∂t)ψ(r, t), (12.6)

has also been developed through the use of the polar form of the wavefunction,

leading to the continuity Equation 12.2 with a modiﬁed expression for the velocity

ﬁeld v(r, t) = 2D∇S(r, t) −K(r, t) and the Euler equation given by

[(∂/∂t) +v.∇]v(r, t) =−(∂K/∂t) +v ×curl K−(1/m)∇[V (r, t) +Q(r, t )]. (12.7)

For an electromagnetic ﬁeld, the velocity ﬁeld consists of paramagnetic and dia-

magnetic contributions and the terms −(∂K/∂t) = (−(e/mc)(∂A/∂t)) and v ×curl K

correspond respectively to the magnetic contribution to the TD electric ﬁeld E(r, t) =

−(1/c)(∂A/∂t) −∇V (r, t) and the magnetic force (e/c)v × curl A. The QFD equa-

tions in this situation also are again classical-like and are governed by the quantum

force in addition to the classical force.

12.3 QUANTUM FLUID DYNAMICAL EQUATIONS FOR

MANY-PARTICLE SYSTEMS IN CONFIGURATION SPACE

For an N -electron system, with its electrons moving in the ﬁeld of their mutual

repulsion as well as the nuclei and external ﬁelds due to scalar as well as vector

potentials, the corresponding TD Schrödinger equation is given by



(1/2)



(−2iD∇

− K(r

, t))

+ (1/m)V (r

, t)



ψ(r

, t) = 2iD(∂/∂t)ψ(r

, t)

(12.8)

Quantum Fluid Dynamics and Density Functional Theory 187

with the potential V (r

, t) =



) − eφ(r

, t)]+U (r

, t), consisting of con-

tributions from the potential V

due to the nuclei, mutual inter-electronic Coulomb

repulsion U(r

, t), and the external TD scalar potential φ(r, t). For convenience, we

will be using D = /2m and the vector potential K(r, t ) = (e/mc)A(r, t) for an

electromagnetic ﬁeld in all the discussion that will now follow. The TD Schrödinger

equation can now be transformed in a manner similar to that for a single particle, i.e.,

by using the polar form ψ(r

, t) = R(r

, t) exp[S(r

, t)] to obtain [11] the similar

looking QFD equations, viz. the continuity equation

(∂/∂t)ρ

, t) +



∇

, t) = 0 (12.9)

and an Euler equation that can be written [11] as

(∂/∂t)v

, t) +



, t).∇

, t)



(1 − δ

, t) × (∇

×v

, t))

=−(eE(r

, t) +(e/c)v

, t) × B(r

, t))

− (1/m)∇[V

, t) +U (r

, t) +Q(r

, t)]. (12.10)

Here, the notation r

denotes all the N-electron coordinates and ∇

represents the

gradient operator involving the coordinate r

of the k-th electron, and the external

electric and magnetic ﬁelds corresponding to the TD scalar and vector potentials are

given by E(r, t ) =−∇φ(r, t) − (1/c)(∂/∂t)A(r, t) and B(r, t ) = curl A(r, t).

The resulting QFD equations are, however, in conﬁguration space, thus involving

the N-particle density ρ

, t) = R

, t) and the conﬁguration space current

density j

, t) = ρ

, t)v

, t), with the 3N-D velocity ﬁeld (corresponding

to the k-th electron) given by

, t) = (/m)∇

S(r

, t) −(e/mc)A(r

, t). (12.11)

Although these density variables in 3N-dimensional conﬁguration space can easily

be projected to 3-D space to obtain the single-particle electron density ρ(r, t ) and the

current density j(r, t ) quantities deﬁned as

ρ(r, t ) =



, t)



δ(r −r

) (12.12)

j(r, t ) =





δ(r −r

, t) (12.13)

and the continuity Equation 12.9 can also easily be transformed into 3-D space to

obtain

(∂/∂t)ρ(r, t) +∇.j(r, t) = 0, (12.14)

188 Quantum Trajectories

the same is not true for the Euler Equation 12.10 which is in conﬁguration space.

Here, the integrations correspond to 3N-dimensional integrals and the delta function

in Equations 12.12 and 12.13 helps one to essentially represent the (N − 1)-electron

coordinate integral and multiplication by N.

The QFD equations are, however, appealing only if they are in 3-D space in terms

of the basic variables ρ(r, t) and j(r, t). For many-electron systems, to obtain an Euler

equation of QFD in 3-D space, one can resort to the framework of single-particle

theories which employ an orbital partitioning of the electron-density and the current-

density variables.

As has already been mentioned in the Introduction, DFT provides a rigorous

single-particle based framework for the description of many-particle systems in 3-D

space. With its TD generalization, it now enables one to investigate the dynami-

cal properties, thus extending the scope of DFT enormously. Since TDDFT already

involves the current density variable, in addition to the electron density, it is ide-

ally suited for providing a route to QFD. In the following section, we discuss the

framework of DFT and TDDFT in particular and explore, in the next section, their

versatility for obtaining the QFD in 3-D space.

12.4 DENSITY FUNCTIONAL THEORY OF MANY-ELECTRON

SYSTEMS: A SINGLE-PARTICLE FRAMEWORK

FOR QFD IN 3-D SPACE

As is well-known, DFT has established itself as a versatile tool for describing many-

particle systems within a single-particle framework. Unlike other single-particle theo-

ries like Hartree or Hartree–Fock theories, DFT is formally exact and can incorporate

the important effect of many-electron correlations effectively and economically (also

in principle, exactly). Although it existed earlier in the form of TF theory and its

variants, its rigorous foundation was laid by the pioneering Hohenberg–Kohn the-

orem [14], demonstrating a unique mapping between the electron density ρ(r)ofa

many-electron system in its ground state and the external potential V

(r) (say, due to

the nuclei) which characterizes the system. In the Kohn–Sham version, the ground-

state DFT consists of a set of single-particle Schrödinger-like equations, the so-called

Kohn–Sham equations, expressed as

{−(

/2m)∇

+ V

(r) +v

SCF

(r;[ρ])}ψ

(r) = ε

(r), (12.15)

which are to be solved self-consistently to obtain the one-electron orbitals {ψ

(r)},

which in turn, determine the density as ρ(r) =



∗

(r)ψ

(r) and an effective

self-consistent potential given by

SCF

(r;[ρ]) = V

COUL

(r) +v

(r); V

COUL

= (δU

int

/δρ); V

= (δE

/δρ).

(12.16)

Here, U

int

[ρ]=(1/2)e



ρ(r)ρ(r



)/|r − r



| is the classical Coulomb energy and

[ρ] is the XC energy, components of the well-known Hohenberg–Kohn–Sham

energy density functional

Quantum Fluid Dynamics and Density Functional Theory 189

E[ρ]=T

[ρ]+



drv

(r)ρ(r) +U

int

[ρ]+E

[ρ], (12.17)

where T

[ρ](=−(

/2m)



ψ

∗

(r)|∇

|ψ

(r)) represents the KE of a ﬁctitious

non-interacting system of a density the same as that of the actual interacting sys-

tem. Clearly, the framework of the Kohn–Sham version of DFT employs an orbital

partitioning of the density and the picture that emerges is that of a ﬁctitious non-

interacting system of particles moving in the ﬁeld of the external potential V

(r) and

an additional density-dependent effective potential V

SCF

(r;[ρ]). For practical imple-

mentation, the XC energy density functional E

[ρ] is usually to be approximated

since an exact expression for this functional is still unknown.

The ground-state DFT was later extended to excited states, and oscillating TD

potentials and, mainly through the work of Peuckert [19], Bartolotti [21] and Deb

and Ghosh [20], a formal TDDFT was born for oscillating time-dependence. A gen-

eralized TDDFT for an arbitrary TD scalar potential was subsequently developed by

Runge and Gross [22] which was later extended to TD electric and magnetic ﬁelds

with arbitrary time-dependence by Ghosh and Dhara [23]. For many-electron systems

characterized by a potential V

(r) due to the nuclei and subjected to an additional oscil-

lating external potential v

ext

(r, t ), a Hohenberg–Kohn like theorem was proved by

using the Hamiltonian for the steady state, viz. (H (t)−i(∂/∂t))and the minimal prop-

erty of a quasi-energy quantity deﬁned as a time-average of its expectation value over

a period. The TD analog of the Kohn–Sham equation, as derived by them, is given by

{−(

/2m)∇

(r) +V

ext

(r, t ) +V

SCF

(r, t )}ψ

(r, t ) = i(∂/∂t)ψ

(r, t ) (12.18)

which is to be solved for the one-electron orbitals ψ

(r, t ) self-consistently, with the

electron-density and the current-density obtained by using the relations

ρ(r, t ) =



∗

(r, t )ψ

(r, t ); (12.19)

j(r, t ) =−(i/2m)



[ψ

∗

(r, t )∇ψ

(r, t ) − ψ

(r, t )∇ψ

∗

(r, t )]. (12.20)

The effective potential V

SCF

(r, t ), like its time-independent counterpart (Equation

12.16), consists of the classical Coulomb as well as XC contributions which now

depend on the TD density. The explicit form of the XC functional may, however, be

different and may also be current-density dependent as well. An elegant analysis of

the Floquet formulation of TDDFT for oscillating time-dependence has recently been

discussed by Samal and Harbola [30].

The restriction to the case of oscillating TD scalar potentials was overcome ﬁrst by

Runge and Gross [22] who formulated a TDDFTfor arbitrary TD scalar potentials [22]

leading to TD Kohn–Sham equations very similar to the ones described above. Their

approach was later used by Ghosh and Dhara who formulated [23] a TDDFT for TD

scalar as well as TD vector potentials, with arbitrary time-dependence. This broadened

the scope and applicability of TDDFT to include the study of the interaction with

electromagnetic radiation, a magnetic ﬁeld, and various other problems.

190 Quantum Trajectories

Considering the TD-Schrödinger equation (i(∂/∂t)−

H )ψ = 0 for an N -electron

system moving initially in the ﬁeld of an external potential V

(r) due to the nuclei

besides their mutual Coulomb interaction and then subjected to an additional TD-

scalar potential φ(r, t) and a TD-vector potential A(r, t) and the Taylor series expan-

sion of these two potentials with respect to time around the initial time, it was proved

that both the potentials are uniquely (apart from only an additive TD function) deter-

mined by the single-particle current density j(r, t) of the system. It is also obvious

from the continuity equation that the current density j(r, t) uniquely determines the

electron density as well. This enables one to treat the energy and other quantities

for the TD problems as functionals of the current density or more conveniently as

functionals of both j(r, t) and ρ(r, t ), in contrast to the stationary ground state where

one has only the electron-density functionals.

Based on the mapping to a ﬁctitious system of non-interacting particles and

hence an orbital partitioning along the lines of the Kohn–Sham [15] version of time-

independent DFT, Ghosh and Dhara [23] showed that the TD one-electron orbitals

(r, t ) obtained through self-consistent solution of the effective one-particle TD

Schrödinger-like equations (TD Kohn–Sham type equations) given by





−i∇+

eff

(r, t )



+ v

eff

(r, t )

(r, t ) =



i

∂

∂t



(r, t ), (12.21)

determine the exact densities ρ(r, t ) and j(r, t) through the relations

ρ(r, t ) =



∗

(r, t )ψ

(r, t ) (12.22)

and

j(r, t ) =−

i



[ψ

∗

(r, t )∇ψ

(r, t ) − ψ

(r, t )∇ψ

∗

(r, t )]+

ρ(r, t )A

eff

(r, t ),

(12.23)

where {n

}represents the occupation numbers of the orbitals and is unity for occupied

and zero for unoccupied orbitals at zero temperature. The ﬁnite temperature effects

can, however, be incorporated by assuming the n

’s to be governed by the Fermi–Dirac

distribution. The picture that emerges here is that the density and current density of the

actual system of interacting electrons characterized by the given external potentials are

obtainable by calculating the same for a system of non-interacting particles moving in

the ﬁeld of effective scalar and vector potentials V

eff

(r, t ) and A

eff

(r, t ) respectively,

which can be expressed as

eff

(r, t ) = V

(r) −eφ(r, t) +



δU

int

δρ





δE

δρ





2mc





eff

− A



(12.24)

eff

(r, t ) = A(r, t ) +



δU

int

δj





δE

δj



, (12.25)

Quantum Fluid Dynamics and Density Functional Theory 191

having contributions from the external potentials augmented by internal contributions

determined by the density variables. The energy quantities U

int

[ρ, j] and E

[ρ, j]

again represent the classical Coulomb energy, and the XC energy density functional

respectively, and are in general both electron-density and current-density dependent.

While the basic equations of DFT and TDDFT are exact in principle, for practical

implementation, the XC energy functional has to be approximated using a suitable

form. The standard LDA or other gradient corrected forms or non-local approaches

are often used widely since an exact form for this quantity is still not in sight. For

TD potentials, the approximations to suitable XC potentials has however been less

well-developed and further work in this area is essential.

12.5 QUANTUM FLUID DYNAMICAL EQUATIONS WITHIN

A DENSITY FUNCTIONAL FRAMEWORK

The discussion on DFT and TDDFT makes it clear that one can solve the TD Kohn–

Sham like Equations 12.21 self-consistently to obtain the one-electron orbitals and

hence the orbital densities and current densities as well as the corresponding total

densities. This approach, based on an orbital partitioning scheme, can be used to

derive the QFD equations by transforming Equation 12.21 through the use of the

polar form of each orbital wavefunction, viz. ψ

(r, t ) = R

(r, t ) exp[S

(r, t )]. The

real and imaginary parts of the resulting equation can then be separated and after alge-

braic manipulations, one obtains, for individual orbitals (k-th orbital), the continuity

equation

(∂/∂t)ρ

(r, t ) +∇.j

(r, t ) = 0 (12.26)

and the Euler equation

(d/dt)v

(r, t ) ≡[(∂/∂t) + v

(r, t ).∇]v

(r, t )

=−(e/m)[E

eff

(r, t ) + (1/c)v

(r, t ) × B

eff

(r, t )]

− (1/m)∇[V

eff

(r, t ) + Q

(r, t )], (12.27)

where the effective electric and magnetic ﬁelds are given respectively by

eff

(r, t ) =−∇φ(r, t) − (1/c)(∂/∂t)A

eff

(r, t ) (12.28)

and

eff

(r, t ) = curl A

eff

(r, t ), (12.29)

and the quantum potential is given by

(r, t ) = (−

/2m)∇

(r, t )/R

(r, t )

= (

/8m)∇ρ

(r, t ).∇ρ

(r, t )/ρ

(r, t ) − (

/4m)∇

(r, t )/ρ

(r, t ),

(12.30)

which is clearly orbital-dependent.

192 Quantum Trajectories

Here, the orbital density and current density (for the k-th orbital) are respectively

given by

(r, t ) = R

(r, t ) (12.31)

and

(r, t ) = ρ

(r, t )v

(r, t ), (12.32)

with the corresponding velocity ﬁeld expressed as

(r, t ) = (/m)∇S

(r, t ) + (e/mc)A

eff

(r, t ). (12.33)

The net electron and current densities are respectively given by ρ(r, t) =



(r, t ) and j(r, t) =



(r, t ). It may be noted that the picture is that

of a ﬂuid mixture with components corresponding to each orbital, moving in the

force ﬁeld of effective electric and magnetic ﬁelds which are the same for each ﬂuid

component and an additional quantum force which is orbital dependent and is thus

different for different ﬂuid components, as is obvious from Equations 12.28 through

12.30.

It may be noted that while the continuity equations for the individual orbitals

can easily be summed up to result in the overall continuity Equation 12.14 given by

(∂/∂t)ρ(r, t) +∇.j(r, t) = 0, one cannot express the Euler Equation 12.27 in terms

of the overall density quantities. The Euler Equation 12.27 can, however, be recast

into other standard forms of ﬂuid dynamical equations, such as the Navier–Stokes

equation, given by

(∂/∂t)j

(r, t ) =−(e/m)[ρ

(r, t )E

eff

(r, t ) + (1/c)j

(r, t )×B

eff

(r, t )]

− (1/m)ρ

(r, t )∇V

eff

(r, t ) +∇.T

(r, t ), (12.34)

where the stress tensor T

(r, t ) which consists of quantum contribution [3, 31] corre-

sponding to the quantum potential Q

(r, t ) as well as contributions from the current

density of the k-th orbital is expressed as

(r, t ) = (/2m)

∇∇ρ

(r, t )

+ (1/ρ

(r, t ))[j

(r, t )j

(r, t ) − (/2m)

(∇ρ

(r, t ))(∇ρ

(r, t ))]. (12.35)

Thus, in this QFD formulated through orbital based TDDFT, the electron ﬂuid is

interpreted as a mixture of N components and each ﬂuid component is described by

Euler type equations characterized by common effective electric and magnetic ﬁelds,

and an orbital-dependent quantum force or stress tensor.

While the hydrodynamical scheme mentioned above involves the orbital partition-

ing of the density quantities, it may be noted that besides the orbital based DFT, there

is also the net-density based orbital-free DFT, the TD version of which can also be

used for the description of QFD. In this scheme, one has the continuity Equation 12.14

and the Euler equation expressed as

(∂/∂t)j(r, t) = P

{V }

[ρ(r, t ), j(r, t)] (12.36)

Quantum Fluid Dynamics and Density Functional Theory 193

where the vector P

{V }

[ρ(r, t ), j(r, t)] is a functional of the two densities ρ(r, t) and

j(r, t ) for a speciﬁed external potential denoted by {V }, and can be written formally as

{V }

[ρ(r, t ), j(r, t)]=(i)

−1

ψ(t)|[

]|ψ(t)

− (1/m)ρ(r, t)[∇V

(r) +eE(r, t)]−(e/mc)j(r,t)× B(r,t)

(12.37)

where E(r, t) and B(r, t) denote the TD electric and magnetic ﬁelds deﬁned by

the scalar and vector potentials as E(r, t) =−∇φ(r, t) − (1/c)(∂/∂t)A(r, t ) and

B(r, t ) = curl A(r, t). The operators

and

are the paramagnetic current oper-

ator and unperturbed Hamiltonian operator respectively. In Equation 12.37, the total

force is contributed by the classical forces (last two terms on the right-hand side of the

equation) due to various external potentials, as well as the forces of quantum origin

(the ﬁrst term on the right-hand side) which includes the XC contribution, which are

to be expressed as density functionals of ρ(r, t) and j(r, t) so that the hydrodynamical

Equations 12.36 through 12.37 can be used along with the continuity Equation 12.14

for the direct calculation of these density and current-density quantities.

Explicit forms of the functional P

{V }

[ρ(r, t ), j(r, t)] have been employed in Equa-

tions 12.36 and 12.37 by Deb and coworkers [32] and Chattaraj and coworkers [33],

leading to the so-called QFDFT [27] and a generalized nonlinear Schrödinger-type

equation. Applications to collision processes and interaction of atomic and molecular

systems with radiation ﬁelds have been considered by them in detail.

12.6 MISCELLANEOUS ASPECTS OF THE QUANTUM

FLUID DYNAMICAL APPROACH

The QFD of many-electron systems in 3-D space presented here corresponds to the

TDDFT framework. The exact XC potential corresponding to TDDFT is, however,

unknown although various approximations have been introduced. Other routes [3]

to QFD via the phase space distribution function or the reduced density matrix have

become available. Also discussed are the QFD for non-local potentials [3] as well as

for mixed states, dissipation, and corresponding to a new hybrid quantum-classical

approach [34].

The picture of the electron ﬂuid has been further strengthened through a local ther-

modynamic transcription [35] within a DFT based framework. Its TD generalization

has also been proposed [36]. Other equations of the equilibrium properties for the

electron ﬂuid have been derived [37]. Various concepts of an interpretive nature such

as local temperature, local hardness and softness, local virial and hypervirial relations,

have been introduced within the DFT framework. Various other local force laws have

also been obtained [38]. Different applications of QFD and quantum potential to quan-

tum transport [39], tunneling, interaction with intense laser ﬁelds [40] and various

strong ﬁeld processes [41] have been reported. Various TD processes have also been

studied [42] through QFDFT. It will be of interest to develop the QFD equations of

change [43] within the framework of DFT using the transition density.