Matta, Boyd. The quantum theory of atoms in molecules

422 The Lagrangian Approach to Chemistry

2.3

The Action Principle in Quantum Mechanics

2.3.1

Schro¨dinger’s Appeal to the Action

The purpose of this section is to show how Schro¨dinger was led to his derivation

^
of Hc ¼ Ec by a procedure that is equivalent to the variation of a constrained
action integral for an infinitesimal time interval. Schrodinger¨	^
	derived Hc ¼ Ec

by constructing an expression for the energy of a quantum system in a stationary state expressed in terms of a ‘‘wave function cðqÞ’’ whose constrained variation

over	the whole of configuration space using the calculus of variations yields
^	¼ Ec as the Euler equation. He based the form of his functional on the clas-
Hc

sical Hamilton–Jacobi (H–J) equation whose solution Sðq; tÞ is the indefinite time integral of the Lagrangian – the action integral.

For most of classical mechanics – to solve Hamilton’s equations of motion, for example – the Hamiltonian is expressed in terms of the coordinates q and the momenta p, H½q; p&. The Hamiltonian in the H–J equation expresses the momentum in terms of qS=qq, however, and the H–J equation for a time-independent system is of the form H½q; qS=qq& ¼ E. Schro¨dinger argued to replace the solution S(q) by an unknown function cðqÞ such that the H–J equation be expressed as H½q; ðh=cÞqc=qq& ¼ E. Schro¨dinger did not seek solutions to this equation as in the classical case, but instead chose to ‘‘seek a function c such that for any arbitrary variation of it the integral of the said quadratic form, taken over the whole coordinate space, is stationary, . . .’’ Then in his italics ‘‘The quantum conditions are replaced by this variation problem’’ [6].

The result of Schro¨dinger basing his energy integral on the H–J equation is that the momentum is expressed as a spatial derivative of the wave function, a term proportional to qc=qq and, as a consequence, Schro¨dinger’s energy integral is, aside from a trivial change in sign and before the imposition of the normalization constraint, equal to the quantum Lagrangian for a stationary state.

Thus Schrodinger’s¨	^
	variation to obtain Hc ¼ Ec as the Euler equation is equiva-

lent to the variation of a constrained action integral for an infinitesimal time interval.

2.3.2

Schro¨dinger’s Minimization

This section summarizes the steps in the minimization procedure that lead to the

^ ¼

appearance of the Hamiltonian operator and to Hc Ec, the details of the derivation being given in my book [3]. The energy expression that Schro¨dinger minimized by varying c to obtain his equation is given in Eq. (4). Schro¨dinger did not allow c to be complex in his first paper. The integrand is, by analogy with the Hamiltonian in the H–J equation, a function of cðqÞ and qc=qq.

2.3 The Action Principle in Quantum Mechanics 43

ðþy

Gðc; ‘cÞ ¼ drfðh2=2mÞððqc=qxÞ2

y
þ ðqc=qyÞ2 þ ðqc=qzÞ2Þ þ ðV EÞccg	ð4Þ

The first group of terms represents the kinetic energy, equivalently expressed as ðh2=2mÞ‘c ‘c and V is the potential energy, set equal to e2=r for the hydrogen atom in paper I. E is introduced into the expression as an undetermined multiplier to ensure that the wave function remain normalized and is identified with

the total energy in the Euler equation.	If f ðc; ‘cÞ
The variation of Gðc; ‘cÞ is actually quite straightforward.
denotes the integrand in Eq. (4), the variation is given by	dGðc; ‘cÞ ¼
Ð

drfðqf =qcÞdc þ ðqf =q‘cÞd‘cg, where f ðc; ‘cÞ is varied with respect to both c and ‘c. To obtain the Euler equation one must clear the expression of the term involving d‘c a step accomplished using an integration by parts employing the identity ‘c d‘c ¼ ‘2cdc þ ‘ ð‘cdcÞ. This step introduces the usual kinetic energy operator ðh2=2mÞ‘2c into the variational integral. The resulting expression for the variation is given in Eq. (5).

dGðc; ‘cÞ ¼ ð drfH^ c Ecgdc
þ þ dSðrsÞfðh2=2mÞ‘c nðrÞdcg ¼ 0	ð5Þ

and the integration by parts leads to the appearance of the Hamiltonian operator

^	2	=2mÞ‘	2	þ V. The variation consists of a contribution from the variation
H ¼ ð h

of c over the entire system all multiplied by dc and another from its variation on the surface bounding the system. The surface term arises by applying Gauss’ theorem to the final term of the identity used to rid the expression of ‘dc. Because the surface in this case resides at infinity where, as Schro¨dinger points out, one requires dc ¼ 0, the surface term vanishes and the variation dGðc; ‘cÞ reduces to:

ð

ð ; Þ ¼ f ^ g ¼ ð Þ dG c ‘c dr Hc Ec dc 0 6

The variation in Eq. (6) will making Gðc; ‘cÞ stationary [13].

^ ¼

vanish for arbitrary dc only if Hc Ec 0. Thus, yields Schro¨dinger’s equation for a stationary state

2.3.2.1Two Ways of Expressing the Kinetic Energy

Gðc; ‘cÞ expresses the kinetic energy in the form þðh2=2mÞ‘c ‘c rather than as ðh2=2mÞc‘2c, as it appears when obtained from Schro¨dinger’s equation [14]. It is a consequence of the presence of this term that the extension of the variation of the action to an open system leads to its unique definition in terms

44 2 The Lagrangian Approach to Chemistry

of the zero-flux boundary condition [2, 3]. The di erence between the two forms of the kinetic energy is locally proportional to the Laplacian of the electron density r, as given without loss of generality in Eq. (7) for a one-electron system [3, 15]:

ð

h2

=4m

Þð

c ‘2c

þ

c‘2c

Þ ð

h2

=2m

Þ

‘c

‘c

¼ ð

h2=4m ‘2r

7

Þ

ð Þ

a result alternatively expressed as K(r) G(r) ¼ L(r). Integration of Eq. (7) over a region of space W bounded by a surface SðW; rsÞ, yields:

KðWÞ GðWÞ ¼ ðh2=4mÞ ðW ‘2rðrÞ dt
¼ ðh2=4mÞ þ dSðW; rÞ‘rðrÞ nðrÞ	ð8Þ

identifying the average kinetic energies as KðWÞ and GðWÞ, respectively. The volume integral of ‘2r ¼ ‘ ‘r, the divergence of a vector in Eq. (8), is replaced by the surface integral of the flux in ‘r through the surface of the region W using Gauss’ theorem. When the region W refers to all space then KðWÞ ¼ GðWÞ, because of the vanishing of r and its gradients at infinity, but for a region with finite boundaries the two quantities di er by the flux in ‘r through the surface of W and the kinetic energy is ill-defined. If, however, the surface SðWÞ is one of zeroflux in ‘r as defined in Eq. (9):

‘rðrÞ nðrÞ ¼ 0 for all points rs on the surface SðW; rsÞ ð9Þ

where nðrÞ is a unit vector normal to the surface, then KðWÞ ¼ GðWÞ and the kinetic energy is a well defined quantity [1]. Equation (9) defines a surface that is not crossed by any trajectories traced out by the vector ‘rðrÞ and is consequently referred to as a ‘‘zero-flux surface’’. It is this surface that forms the boundary condition for an open quantum system when the action principle is extended to an open system [2, 3].

2.3.3

Obtaining an Atom from Schro¨dinger’s Variation

What would happen if one were to repeat Schro¨dinger’s derivation of his wave equation for an atom ‘‘in a molecule’’ rather than for an isolated atom? If one wishes to follow Schro¨dinger in the search for an atom in a molecule, one has no choice but to replace the infinite boundaries he placed on his energy expression in Eq. (4) by a set of finite limits that define a region W. That is, the atom will be defined as a bounded region of real space [2, 3]. The energy expression obtained by placing finite limits on Gðc; ‘cÞ in Eq. (4) to define a region W is denoted by Gðc; ‘c; WÞ. (c is now allowed to be complex, with the variations in

2.3 The Action Principle in Quantum Mechanics 45

c and c being independent). Its variation must include a term corresponding to a variation of the boundary if it is to be determined by the variational procedure rather than being arbitrarily assigned. The variation of Gðc; ‘c; WÞ yields the same terms as given in Eq. (5) for dGðc; ‘cÞ together with a second contribution

^ ¼

to the surface integral from the variation of the surface. Because Hc Ec still applies to the total system of which W is a part, the volume integral of the term

^
ðHc EcÞdc vanishes and only the surface terms remain in the variation of
Gðc; ‘c; WÞ, Eq. (10):
dGðc; ‘c; WÞ ¼ þ dSðW; rsÞfðh2=2mÞ‘c nðrsÞdc
þ dSðW; rsÞf ðc; ‘cÞg þ cc	ð10Þ

The first surface term, as detailed in the variation of Gðc; ‘cÞ, arises by ridding the expression of variations in ‘c. The second term is the result of varying the surface of W by varying c. The function f ðc; ‘cÞ denotes the integrand in Schro¨- dinger’s functional, Eq. (4), and when evaluated in the surface and multiplied by the infinitesimal shift in the surface dSðWÞ, the term gives the contribution to the variation resulting from the variation of c on the surface. Because functions are not necessarily Hermitian over an open system W, the variation must include contributions from the complex conjugate (cc) terms. Equation (10), which does not define any particular surface, does not seem promising. Clearly the requirement of the principle of least action that the variation vanish must be discarded and the concept of stationarity broadened. This result is initially quite surprising if one lacks the knowledge that it forms the basis for the generalization of the principle of least action introduced by Schwinger.

2.3.3.1The Role of Laplacian in the Definition of an Atom

If an atom is to be a bounded region of space, one might expect its physical description would require the presence of a surface integral that would describe contributions to its properties arising from the flux in currents through its surface. It is the purpose of this section to demonstrate that imposition of the zeroflux boundary condition, Eq. (9), during the variation of Gðc; ‘c; WÞ leads to this very result. This is most important, for it brings the definition of an atom in a molecule into the realm of physics – the physics of an open system.

Equation (10) is transformed into a statement of physics by two rather remarkable consequences of the properties of the Laplacian of the electron density, ‘2rðrÞ. The first is that when Schro¨dinger’s equation is satisfied, the integrand f ðc; ‘cÞ of Schro¨dinger’s energy integral reduces to LðrÞ, the term proportional to ‘2rðrÞ defined in Eq. (7). This same property obtains for the many-electron case, and for the Lagrangian density of the action integral in the general timedependent case, persisting even in the presence of an electromagnetic field. So the term involving the variation of the surface may be re-expressed as ðh2=4mÞdSðW; rsÞ‘2rðrÞ. The second of the remarkable consequences of the Lap-

46 2 The Lagrangian Approach to Chemistry

lacian is its appearance in the constraint that determines the surface of the open system. This constraint is presented in detail in several places [3, 16], and is outlined only briefly here. Imposing the zero-flux surface condition expressed in Eq. (9) at every stage of the variation is equivalent to requiring that the variation of the integral of ‘2rðrÞ over the region W vanish. This condition in

turn enables one to replace the term ðh2=4mÞdSðW; rsÞ‘2rðrÞ with the termðh2=4mÞfð‘c Þdc þ c d‘cg nðrÞ, a step outlined in my book (pp. 158 ).

Combining this result with the first of the surface terms in Eq. (10) results in a most remarkable transformation of the surface contribution, from one without any discernable physical content for a surface of unspecified form into an integral describing the flux in the variation of the quantum mechanical current density j(r) through a surface of zero-flux in ‘r, as shown in Eq. (11) for a zero-flux surface.

þ

dGðc; ‘c; WÞ ¼ ðih=2Þ dSðW; rsÞdjðrÞ nðrÞ þ cc ð11Þ

The variation in j(r) is caused by variations in the wave function, as shown in Eq. (12):

djðrÞ ¼ ðh=2miÞfc ‘ðdcÞ ð‘c Þdcg

ð12Þ

Thus imposition of the zero-flux boundary condition on the variation of Schro¨- dinger’s functional causes it to be ‘‘stationary’’, requiring its variation to equal the surface flux in the current generated by the variations in c. Although unknown to us at the time of its derivation, Eq. (11) is a result of Schwinger’s principle of stationary action for a time-independent system.

2.3.4

Getting Chemistry from dG(c,‘c; W)

Whereas Eq. (11), the expression for the variation in Gðc; ‘c; WÞ, links the definition of an atom with physics, it is not in a useable form. How does one describe the variation in the wave function that causes the change in j(r), Eq. (12)? Transforming Eq. (11) into an operational form is accomplished by replacing the variations in c with the action of quantum mechanical operators on c. All physical changes in the properties of a system, can be described by the action of the appro-

		^
priate quantum mechanical operator G on c. Thus, one makes the substitution
^	where e	denotes an infinitesimal change resulting from the
dc ¼ eði=hÞGc
	^	^

action of the operator G on c, with ði=hÞ present for dimensional reasons. GðrÞ

may be any linear Hermitian operator constructed from the electronic position and/or momentum coordinates of a single electron and is referred to as the generator of the change in the system. Its application to a wave function causes an infinitesimal unitary transformation in c. As explained below, it is this step that

2.3 The Action Principle in Quantum Mechanics 47

completes the linking of the derivation Schwinger’s principle of stationary action. comes:

of an atom in a molecule with With this substitution, Eq. (11) be-

dGðc; ‘c; WÞ ¼ ðe=2Þþ dSðW; rsÞjGðrÞ nðrÞ þ cc	ð13Þ
	^
where jGðrÞ is the current density for the property determined by the generator G:
jGðrÞ ¼ ðh=2miÞfc ‘ðG^ cÞ ð‘c ÞðG^ cÞg	ð14Þ

Comparison of Eqs. (12) and (14) shows that the variation in c appearing in

^	ðrÞ. Thus the vari-
djðrÞ is replaced by the action of G on c in the expression for jG

ation in Schro¨dinger’s energy functional for an open system bounded by a zeroflux surface – an atom in a molecule – is proportional to the surface flux in the current density of the infinitesimal generator causing the change in the system. Fluxes in currents normal to the bounding surface are the features that distinguish the physics of an open system from that of a total system, for which all surface terms vanish. There are, of course, no current flows in a stationary state in the absence of a magnetic field but, following Feynman: ‘‘We generalize the word ‘‘flux’’ to mean the surface integral of the normal component of a vector. Although it is not the flow of anything, we still call it the ‘‘flux’’ ’’ [12]. A recent publication [17] claims that the surface flux term in Eq. (11), and hence in the general case, Eq. (13), vanishes for a stationary state, appearing to confuse the absence of a current in a stationary state with its ‘‘flux’’ through a surface, a claim that invalidates Schwinger’s principle and the physics of an open system, and Eq. (16) that follows directly for Schro¨dinger’s equation for a stationary state. A response to that paper has recently appeared [18].

The final, encompassing statement of the principle of stationary action for an open system in a stationary state is obtained by use of the equation of motion

^

for the generator G. In the general time-dependent case, the time derivative of

^ h^ i=

the average value of G, d G dt is determined by the average of the commutator

ð= Þ½ ^ ; ^ &

i h H G . This average vanishes for a molecule in a stationary state:

^ ^	ð15Þ
hc; ½H; G&ci ¼ 0

^

because of the Hermiticity of H, a property that does not usually apply to an open system [3]. The same commutator average does not however, vanish for an atom in a molecule, the contribution from the commutator being balanced by the sur-

^	equa-
face flux in the current of G, a result readily obtained from Schrodinger’s¨
tion [3] and given in Eq. (16):
ði=hÞhc; ½H^ ; G^ &ciW þ cc ¼ þ dSðW; rsÞjGðrÞ nðrÞ þ cc	ð16Þ

48	2 The Lagrangian Approach to Chemistry
	Equation (16) yields the same surface term obtained in the	variation of
	Gðc; ‘c; WÞ and substitution of this result into Eq. (13) yields the atomic state-
	ment of the principle of stationary action for a stationary state:
^ ^		ð17Þ
	dGðc; ‘c; WÞ ¼ ðe=2Þfði=hÞhc; ½H; G&ciW þ ccg

This statement determines the physics of an atom in a molecule [3]. All of the theorems of quantum mechanics are obtained by appropriate choice of the gener-

^

ator G and thus it applies to every property determined by a quantum mechanical operator – to all of the measurable properties of a system. The atomic statement

^ ¼ ^ ^

of the virial theorem for example, is obtained by setting G r p, the product of the electronic position and momentum coordinates. Equation (17) applies to any system bounded by a zero-flux surface and thus a single principle provides the quantum mechanical description of the total system and of its constituent atoms. Indeed one may regard the physics of some total system – of the entire molecule – as a special limiting case of the more general expression pertaining to an open system given in Eq. (17). Thus when W refers to the total molecule, the commutator average equals zero, Eq. (15), and the variation in Schro¨dinger’s functional becomes stationary in the usual sense that dGðc; ‘c; WÞ ¼ dGðc; ‘cÞ ¼ 0. It is important to note that the derivation of Eq. (17) yields Schro¨dinger’s equation and all of the theorems of quantum mechanics. Thus a single principle serves to completely determine the physics of a stationary state, of the total system, and of its constituent atoms, all as a consequence of Schwinger’s principle [3].

2.4

From Schro¨dinger to Schwinger

2.4.1

From Dirac to Feynman and Schwinger

The generalization of the variation of Schro¨dinger’s energy functional to obtain the physics of an open system, Eq. (17), necessitated the generation and retention of terms resulting from the variation of the surface and of the variation of the wave function on the surface. Although such terms are discarded in the principle of least action when applied to a total system, they play a crucial role in the physics of an open system. It is this step that transforms Schro¨dinger’s approach into Schwinger’s principle of stationary action. We can only briefly hint at the beauty of the chain of reasoning that leads from Dirac to Feynman and Schwinger and ultimately to an atom in molecule. Dirac introduced transformation theory into quantum mechanics. This is the underlying mathematical formulation of the new physics which consists of the general mathematical scheme of linear operators and state vectors with its associated probability interpretation. In doing so, he stressed how the theory of infinitesimal unitary transformations in

2.4 From Schro¨dinger to Schwinger 49

quantum mechanics parallels the infinitesimal canonical transformations of classical theory thereby ‘‘providing the mathematical foundation of the analogy between classical and quantum equations of motion’’. Thus identification of the vari-

^

ations in c with the action of infinitesimal generators eG to obtain Eq. (17) has deep implications beyond yielding operational expressions for the mechanics of an open system. In 1933 such thinking led Dirac to write a paper [19] wherein he posed the question of what would correspond to the limiting classical expression for the quantum transition amplitude that determines the dynamic behavior of the system with time. Dirac was asking for the correspondence of quantum mechanics with the Lagrangian method of classical mechanics, a formulation he considered to be more fundamental than that based on Hamiltonian theory. Dirac proposed that the transition amplitude be given by an exponential of ði=hÞW12, where W12 is the classical action evaluated along the unique path that causes it to be stationary, a proposal that set the stage for the reformulations of quantum mechanics proposed by Feynman in his path integral approach [8] and by Schwinger in his quantum dynamical principle [4]. Both approaches enable one to derive the commutation relations, as opposed to the need to postulate them in the Hamiltonian approach [20].

Schwinger realized that by retaining the variations on the boundary of the space–time volume of the action integral swept out by the temporal evolution of a system, followed by their identification with the generators of infinitesimal unitary transformations, he could combine the action principle which yields Schro¨- dinger’s equation of motion with Dirac’s transformation theory and thus recover ‘‘all of physics’’ from a single dynamical principle – the principle of stationary action [4]. The retention of the variations both of and on the zero-flux surface at a single time t followed by their identification with the action of infinitesimal generators is precisely the step made in the generalization of Schro¨dinger’s variation of his energy functional leading to Eq. (17). These variations give rise to the same surface terms that are found in Schwinger’s formulation and are dealt with in precisely the same manner. While we proceeded out of necessity, Schwinger did so by choice.

2.4.2

From Schwinger to an Atom in a Molecule

In 1978 it was realized that the generalization of the variation of Schro¨dinger’s functional was but a special case of the general statement of physics provided by Schwinger’s principle of stationary action and the theory of an atom in a molecule was readily extended to the general time-dependent case [16, 21]. Criticisms of the extension have been responded to in full [22, 23]. The operational statement of Schwinger’s principle of stationary action is given in terms of the variation of the Lagrangian, Eq. (18):

^ ^	ð18Þ
dLðC; ‘C; t; WÞ ¼ ðe=2Þfði=hÞhC; ½H; G&CiW þ ccg

50 2 The Lagrangian Approach to Chemistry

The form of this expression is to be compared with that obtained from variation of Schro¨dinger’s energy functional for a stationary state, Eq. (17), from which it di ers only by the expected change in sign. On evaluation of the variation in the Lagrangian, Eq. (18) yields Eq. (19) for the equation of motion for a generator

^ ð Þ

G r , a function of the coordinate r of a single electron [24].

ð	ð
ð1=2ÞN	W dr dr0qfC G^ ðrÞC þ ccg=qt
	^ ^
¼ ð1=2Þfði=hÞhCj½H; GðrÞ&jCiW þ ccg
ð1=2Þ þ dSðrs; WÞ fJGðrsÞ þ ccg		ð19Þ

The most recent derivation of this result is given in terms of an augmented Lagrange-function operator, termed a proper operator, because only its variation recovers the equations of motion predicted by Schro¨dinger’s field equation [25].

The resulting open systems are termed proper open systems.										^
The term N		dr 0	f	C G^	ð	Þ	C	þ	g
					r				cc , whose time derivative appears in Eq. (19),
											ð Þ
defines the	density of the property G associated with the operator G r . The inte-
		Ð

gration symbol denotes that the operator is averaged over the coordinates of all the electrons except those denoting the position r, the coordinate of the electron to be integrated over W. All open-system properties, including the energy, are thus defined in terms of a real-space density distribution, in the same manner as is the electron density where the integrand is simply C C, with the result that the value of some property for the total system equals the sum of its atomic contributions. The symbol h iW implies the same averaging. The LHS vanishes for a stationary state as does the surface term for a total isolated system. The variational result given in Eq. (19) determines the physics of all measurable properties, the properties of a total system being a special limiting case of those for an open system.

QTAIM is predicated on the premise that a theory of an atom in a molecule must predict what can be measured in the laboratory. It is this philosophy that underlies Hans Bethe’s view of science: ‘‘its great advantage is you can prove something is true or something is false’’, a statement he further paraphrased as ‘‘In science, you know you know’’ [26]. Therefore, the single necessary and su - cient criterion for determining the relevance of QTAIM atoms to chemistry is agreement of the predicted additive atomic and group contributions with their experimental values, agreement with observation being the only test of theory. QTAIM has demonstrably satisfied this experimental criterion, with examples from heats of formation, electric and magnetic susceptibilities, and molar volumes, and from the recent demonstration of the recovery of the measured atomic contributions to infrared intensities obtained from the atomic polar tensor [27, 28]. This agreement is but a necessary consequence of quantum mechanics predicting all measurable properties, Eq. (19). A recent paper provides a comprehensive summary of the recovery of measured properties by QTAIM [29].

The interested reader is referred to the original papers and my book for a complete description of the extension of Schwinger’s principle of stationary action to

2.4 From Schro¨dinger to Schwinger 51

an open system. There are numerous important nuances that one must take into account before one attempts a critique. An important example of this stems from the fundamental step in Schwinger’s approach – the variation of the state vector on the space–time boundary of the open system with the variation of the surface, followed by identification of these variations with the generators of infinitesimal unitary transformations. Thus the theory requires the use of a special class of trial functions whose variation will correspond to continuous changes in the dynamic variables of the physical system. This is the very requirement needed to ensure the applicability of the zero-flux surface condition as the defining constraint of a proper open system.

Equation (19) is the bridge that links the Lagrangian and Heisenberg representations of quantum mechanics thereby linking theory with experiment. Table 2.1

Table 2.1 The atomic theorems for molecules and crystals obtained for several important generators.

Atomic force theorem		^
		G ¼ p^
m ð	drqJðrÞ=qt ¼ ð	dr ð dt0C ð ‘V^ÞC þ þ dSðrÞsðrÞ nðrÞ

WW

Atomic virial theorem		^	¼ ^r p^
		G
m ð	drr qJðrÞ=qt ¼ 2TðWÞ þ ð		dr ð dt0C ð r ‘V^ÞC þ þ dSðrÞr sðrÞ nðrÞ

WW

Atomic torque theorem		^	¼ ^r	p^
		G
m ð	drr qJðrÞ=qt ¼ ð	dr ð dt0C ð r ‘VÞC þ dSsðrÞ r n

WW

Atomic current theorem	^
	G ¼ ^r

ðð þ

drrqrðrÞ=qt ¼ dr JðrÞ dSnðrÞ JðrÞr

WW

Atomic continuity theorem	^	^
	G ¼ N

ðþ

	W drqrðrÞ=qt ¼ dSJðrÞ nðrÞ
Atomic power theorem		^	2	=2m, written without 1/2m
		G ¼ p^
ð	drqrp2 ðrÞ=qt ¼ ð	dr ð dt0ðh=iÞfðC‘C C ‘CÞ ‘V^g þ þ dS RefJp2 ðrÞg

WW

Quantum stress tensor density

ð

sðrÞ ¼ ðh2=4mÞ dt0fð‘‘C ÞC ‘C ‘C ‘C‘C þ C ‘‘Cg

Quantum vector current density

ð

JðrÞ ¼ ðh=2miÞ dt0fC ‘C C‘C g