Baer M., Billing G.D. (eds.) - The role of degenerate states in chemistry (Adv.Chem.Phys. special issue, Wiley, 2002)

262	r. englman and a. yahalom

by [72] and was later rederived by others. [In the original formulation of [72] only the contracted form of the field Abc (appearing in the definition of F)

Akcm ¼ AcbðtbÞmk	ð131Þ
enters. This has the form
Akcm ¼ tkcmðRÞ	ð132Þ
If the intermediate summations are over a complete set, then
ðFbcÞmk ¼ FabcðtaÞmk ¼ 0&	ð133Þ

This result extends the original theorem [72] and is true due to the linear independence of the t-matrices [67]. The meaning of the vanishing of F is that, if wkðRÞ is the partner of the electronic states spanning the whole Hilbert space, there is no indirect coupling (via a gauge field) between the nuclear states; the only physical coupling being that between the electronic and nuclear coordinates, which is given by the potential energy part of Hðr; RÞ. When the electronic N set is only part of the Hilbert space (e.g., N is finite), then the underlying electron– nuclei coupling gets expressed by an additional, residual coupling between the nuclear states. Then Fabc 6¼ 0 and the Lagrangean has to be enlarged to incorporate these new forces.

We further make the following tentative conjecture (probably valid only under restricted circumstances, e.g., minimal coupling between degrees of freedom): In quantum field theories, too, the YM residual fields, A and F, arise because the particle states are truncated (e.g., the proton-neutron multiplet is an isotopic doublet, without consideration of excited states). Then, it is within the truncated set that the residual fields reinstate the neglected part of the interaction. If all states were considered, then eigenstates of the form shown in Eq. (90) would be exact and there would be no need for the residual interaction negotiated by A and F.

VI. LAGRANGEANS IN PHASE-MODULUS FORMALISM

A.Background to the Nonrelativistic and Relativistic Cases

The aim of this section is to show how the modulus-phase formulation, which is the keytone of our chapter, leads very directly to the equation of continuity and to the Hamilton–Jacobi equation. These equations have formed the basic building blocks in Bohm’s formulation of non-relativistic quantum mechanics [318]. We begin with the nonrelativistic case, for which the simplicity of the derivation has

complex states of simple molecular systems

263

mainly pedagogical merits, but then we go over to the relativistic case that involves new results, especially regarding the topological phase. Our conclusions (presented in VI.H) are that, for a broad range of commonly encountered situations, the relativistic treatment will not affect the presence or absence of the Berry phase that arises from the Schro¨dinger equation.

The earliest appearance of the nonrelativistic continuity equation is due to Schro¨dinger himself [2,319], obtained from his time-dependent wave equation. A relativistic continuity equation (appropriate to a scalar field and formulated in terms of the field amplitudes) was found by Gordon [320]. The continuity equation for an electron in the relativistic Dirac theory [134,321] has the wellknown form [322]:

qnJn ¼ 0				ð134Þ
where the four-current Jn is given by
J	n	n	c	ð135Þ
		¼ cg

(The symbols in this equation are defined below). It was shown by Gordon [323], and further discussed by Pauli [104] that, by a handsome trick on the four current,

this can be broken up into two parts Jn ¼ Jðn0Þ þ Jðn1Þ (each divergence-free), representing, respectively, a conductivity current (Leitungsstrom):

Jðn0Þ	¼ 2mc nh hqn i c An				cic ch hqn þ i c An
		i		e	e

and a polarization current [324]

n		ih		m n
Jð1Þ	¼		qmðcg g		cÞ n ¼6 m
		2mc

io c

ð136Þ

ð137Þ

Again, the summation convention is used, unless we state otherwise. As will appear below, the same strategy can be used upon the Dirac Lagrangean density to obtain the continuity equation and Hamilton–Jacobi equation in the modulusphase representation.

Throughout, the space coordinates and other vectorial quantities are written either in vector form x, or with Latin indices xk ðk ¼ 1; 2; 3Þ; the time (t) coordinate is x0 ¼ ct. A four vector will have Greek lettered indices, such as xn ðn ¼ 0; 1; 2; 3Þ or the partial derivatives qn. m is the electronic mass, and e the charge.

B.Nonrelativistic Electron

The phase-modulus formalism for nonrelativistic electrons was discussed at length by Holland [324], as follows.

264

r. englman and a. yahalom

The Lagrangean density L for the nonrelativistic electron is written as

h2

eV

ieh

A

1

ih

_ _

L ¼

2m $c

$c

cÞ þ

cÞ

c

c 2cm

ðc

$c $c

2 ðc c c

ð138Þ

Here dots over symbols designate time derivatives. If now the modulus a and phase f are introduced through

c ¼ aeif

the Lagrangean density takes the form

h2

2

2

2

2

eh 2

2 qf

L ¼

½ð$aÞ

þ a

ð$fÞ

& ea

V þ

a

$f A ha

2m

cm

qt

ð139Þ

ð140Þ

The variational derivative of this with respect to f yields the continuity equation

dL

0

qr

v

0

141

df

¼

! qt

Þ ¼

ð

Þ

þ $ ðr

in which the charge density is defined as: r ¼ ea2 and the velocity is

1

e

v ¼

h$f

A

m

c

Variationally deriving with respect to a leads to the Hamilton–Jacobi equation

dL 0 qS 1 S e A

eV h

2

2

a

2

2

142

2

r

e

A

¼ !

þ

$

þ ¼

þ

ð Þ

da

qt

2m

c

2ma

2mc2

in which the action is defined as: S ¼ hf. The right-hand side of Eq. (142) contains the ‘‘quantum correction’’ and the electromagnetic correction. These results are elementary, but their derivation illustrates the advantages of using the two variables, phase and modulus, to obtain equations of motion that are substantially different from the familiar Schro¨dinger equation and have straightforward physical interpretations [318]. The interpretation is, of course, connected to the modulus being a physical observable (by Born’s interpretational postulate) and to the phase having a similar though somewhat more problematic status. (The ‘‘observability’’ of the phase has been discussed in the literature by various sources, e.g., in [28] and, in connection with a recent development, in [31,33]. Some of its aspects have been reviewed in Section II.)

complex states of simple molecular systems

265

Another possibility to represent the quantum mechanical Lagrangian density is using the logarithm of the amplitude l ¼ ln a; a ¼ el. In that particular representation, the Lagrangean density takes the following symmetrical form

L ¼ e2l	h2			qf			eh	$f A
		½ð$lÞ2	þ ð$fÞ2	& h		eV þ			ð143Þ
	2m				qt		cm

C.Similarities Between Potential Fluid Dynamics

and Quantum Mechanics

In writing the Lagrangean density of quantum mechanics in the modulus-phase representation, Eq. (140), one notices a striking similarity between this Lagrangean density and that of potential fluid dynamics (fluid dynamics without vorticity) as represented in the work of Seliger and Whitham [325]. We recall briefly some parts of their work that are relevant, and then discuss the connections with quantum mechanics. The connection between fluid dynamics and quantum mechanics of an electron was already discussed by Madelung [326] and in Holland’s book [324]. However, the discussion by Madelung refers to the equations only and does not address the variational formalism which we discuss here.

If a flow satisfies the condition of zero vorticity, that is, the velocity field v is such that $ v ¼ 0, then there exists a function n such that v ¼ $n. In that case, one can describe the fluid mechanical system with the following Lagrangean density

qn			1
L ¼					ð$nÞ2 eðrÞ r	ð144Þ
	qt		2

in which r is the mass density, e is the specific internal energy and is some arbitrary function representing the potential of an external force acting on the fluid. By taking the variational derivative with respect to n and r, one obtains the following equations

qr	þ $ ðr$nÞ ¼ 0						ð145Þ
qt
qn	¼	1		ð$nÞ	2	h	ð146Þ
qt		2

in which h ¼ qðreÞ=qr is the specific enthalpy. The first of those equations is the continuity equation, while the second is Bernoulli’s equation.

Going back to the quantum mechanical system described by Eq. (140), we introduce the following variables ^n ¼ hf=m; r^ ¼ ma2. In terms of these new

266

r. englman and a. yahalom

variables the Lagrangean density in Eq. (140) will take the form

2

$

p

2

^

1

h

^

e

L ¼ "

qn

r

Þ

#r^

ð$n^Þ2

ð

V

ð147Þ

qt

2

2m2

r^

m

in which we assumed that no magnetic fields are present and thus A ¼ 0. When compared with Eq. (144) the following correspondence is noted

2

p

2

^

, n

^

h

ð$

rÞ

, e

e

V

,

ð

148

Þ

n

r , r

2m2

r^

m

p

The quantum ‘‘internal energy’’ ðh2=2m2Þð$ r^Þ2=r^ depends also on the derivative of the density, unlike in the fluid case, in which internal energy is a function of the mass density only. However, in both cases the internal energy is a positive quantity.

Unlike classical systems in which the Lagrangean is quadratic in the time derivatives of the degrees of freedom, the Lagrangeans of both quantum and fluid dynamics are linear in the time derivatives of the degrees of freedom.

D.Electrons in the Dirac Theory

(Henceforth, for simplicity, the units c ¼ 1, h ¼ 1 will be used, except at the end, when the results are discussed.) The Lagrangean density for the particle is in the presence of external forces

		i	m	m
L ¼			½cg	ðqm þ ieAmÞc ðqm ieAmÞcg	c& mcc	ð149Þ
	2

Here, c is a four-component spinor, Am is a four potential, and the 4 4 matrices gm are given by

g

0

¼

I

0

g

k

¼

0

sk

ðk ¼ 1; 2; 3Þ

ð150Þ

0

I

sk

0

where we have the 2 2 matrices

		!			!
I ¼	1	0	s1 ¼	0	1
	0	1		1	0	!
		!
s2 ¼	0	i	s3 ¼	1	0	ð151Þ
	i	0		0	1

complex states of simple molecular systems

267

By following [323], we substitute in the Lagrangean density, Eq. (149), from the Dirac equations [322], namely, from

	i		n					i		n
c ¼		g		ðqn þ ieAnÞc		c ¼			ðqn ieAnÞcg
	m							m
and obtain
		1				n m
L ¼			m		ðqn ieAnÞcg g		ðqm þ ieAmÞc mcc

ð152Þ

ð153Þ

We thus obtain a Lagrangean density, which is equivalent to Eq. (149) for all solutions of the Dirac equation, and has the structure of the nonrelativistic

Lagrangian density, Eq. (140). Its variational derivations with respect to and c c

lead to the solutions shown in Eq. (152), as well as to other solutions. The Lagrangean density can be separated into two terms

L ¼ L0 þ L1

ð154Þ

according to whether the summation symbols n and m in (149) are equal or different. The form of L0 is

	0	1		ðq	m	m
L		¼	m			ieA Þcðqm þ ieAmÞc þ mcc		ð155Þ

Contravariant Vm and covariant Vn four vectors are connected through the metric gmn ¼ diag ð1; 1; 1; 1Þ by

Vm ¼ gmnVn

ð156Þ

The second term in Eq. (154), L1 will be shown to be smaller than the first in the near nonrelativistic limit.

Introducing the moduli ai and phases fi for the four spinor components ci (i ¼ 1; 2; 3; 4), we note the following relations (in which no summations over i are implied):

ci ¼ aieifi
	0 a		e ifi
ci ¼ gii		i
	2	0		ð157Þ
cici ¼ ai		gii

The Lagrangean density eqaution (153) rewritten in terms of the phases and moduli takes a form that is much simpler (and shorter), than that which one

268				r. englman and a. yahalom
would obtain by substituting from Eq. (139) into Eq. (149). It is given by
L0	1		Xi	gii0 ½qnaiqnai þ ai2ððqnfi þ eAnÞðqnfi þ eAnÞ m2Þ&
	¼				ð158Þ
		m

When one takes its variational derivative with respect to the phases fi, one obtains the continuity equation in the form

	dL0	dL1		159
	dfi	¼ dfi	ð		Þ

The right-hand side will be treated in a following section VI.E, where we shall see that it is small in the nearly nonrelativistic limit and that it vanishes in the absence of an electromagnetic field. The left-hand side can be evaluated to give

dfi

¼ m qn½

i

ðq

fi þ

Þ&

qn i

ð

Þ ð

Þ

dL0

2

a2

n

eAn

2

Jn

no summation over i

160

The above defined currents are related to the conductivity current by the relation

X

Jðn Þ ¼ g0 Jn ð161Þ

0 ii i i

Although the conservation of Jin separately is a stronger result than the result obtained in [104], one should bear in mind that the present result is only approximate.

The variational derivatives of L0 with respect to the moduli ai give the following equations:

dL0

2

½qnq

na

a

iððqnfi þ

eA

n

fi þ

eAn

m2

162

dai

¼ m

i

Þ

Þ&

ð

Þ

nÞðq

The result of interest in the expressions shown in Eqs. (160) and (162) is that, although one has obtained expressions that include corrections to the nonrelativistic case, given in Eqs. (141) and (142), still both the continuity equations and the Hamilton–Jacobi equations involve each spinor component separately. To the present approximation, there is no mixing between the components.

E.The Nearly Nonrelativistic Limit

In order to write the previously obtained equations in the nearly nonrelativistic limit, we introduce phase differences si that remain finite in the limit c ! 1. Then

fi ¼ g0iið mx0 þ siÞ q0fi ¼ g0iið m þ q0siÞ $fi ¼ g0ii$si ð163Þ

complex states of simple molecular systems

269

We reinstate the velocity of light c in this and in Section VI.F in order to appreciate the order of magnitude of the various terms. When contributions from L1 are neglected, the expression in Eq. (162) equated to zero gives the following equations, in which the large ði ¼ 1; 2Þ and small ði ¼ 3; 4Þ components are separated.

1

e

$2ai

e2

1

2ai

qtsi þ

ð$

si

AÞ2 þ eA0 ¼

þ

A2

þ

qt

þ ðqtsiÞ2

2m

c

2mai

2mc2

2mc2

ai

þ

2eA0qtsi þ e2A02 e2A2

ði ¼ 1; 2Þ

ð164Þ

s

i þ

1

ð$

s

i

eÞ

A

Þ

2

e

A

¼

r2ai

þ

e2

A2

þ

1

qt2ai

qt

2m

ð c

þ ð Þ

0

2mai

2mc2

2mc2

ai

þ

ðqtsiÞ2 þ 2ð eÞA0qtsi þ e2A02 e2A2

ði ¼ 3; 4Þ

ð165Þ

In the same manner, we obtain the following equations from Eq. (160)

qtri þ $ ðri

iÞ ¼ c2 qt ri

m

v

1

qtsi

þ eA0

Þ

ri ¼

ma2

$si ce A

i

1

;

2

v

i ¼

m

ð ¼

i

qtri þ $ ðri

iÞ ¼ c2 qt ri

þm

v

1

qtsi

ð eÞA0

Þ

ri ¼

ma2

ðceÞ

A

i

3

;

4

v

i ¼

$si

ð ¼

i

m

ð166Þ

ð167Þ

The terms before the square brackets give the nonrelativistic part of the Hamilton–Jacobi equation and the continuity equation shown in Eqs. (142) and (141), while the term with the square brackets contribute relativistic corrections. All terms from L0 are of the nonmixing type between components. There are further relativistic terms, to which we now turn.

F.The Lagrangean-Density Correction Term

As noted above, L1 in Eq. (154) arises from terms in which m ¼6 n. The corresponding contribution to the four current was evaluated in [104,323] and was shown to yield the polarization current. Our result is written in terms of the magnetic field H and the electric field E, as well as the spinor four-vector c and the vectorial 2 2 sigma matrices given in Eq. (151).

e

I

0

ie

0

I

L1

¼

cðH rÞ

0

I

c þ

c

ðE rÞ I

0

c

ð168Þ

mc

mc

270	r. englman and a. yahalom

These terms are analogous to those on p. 265 of [7]. It will be noted that the symbol c has been reinstated as in Section VI.F, so as to facilitate the order of magnitude estimation in the nearly nonrelativistic limit. We now proceed based on Eq. (168) as it stands, since the transformation of Eq. (168) to modulus and phase variables and functional derivation gives rather involved expressions and will not be set out here.

To compare L1 with L0 we rewrite the latter in terms of the phase variables introduced in Eq. (163)

0

X

0

1

2

2

2

0 2

0

2 qsi

L

¼ 2

gii

½ð$aiÞ

þ ai

ð$siÞ

& ðgiieÞai A

ai

i

2m

qt

2e

X

1

þ

ai2

$si

A þ O

ð169Þ

mc

i

c2

which contains terms independent of c as well as terms of the order Oð1=cÞ and

Oð1=c2Þ.

In Eq. (168), the first, magnetic-field term admixes different components of the spinors both in the continuity equation and in the Hamilton–Jacobi equation. However, with the z axis chosen as the direction of H, the magnetic-field term does not contain phases and does not mix component amplitudes. Therefore, there is no contribution from this term in the continuity equations and no amplitude mixing in the Hamilton–Jacobi equations . The second, electric-field term is nondiagonal between the large and small spinor components, which fact reduces its magnitude by a further small factor of Oðparticle velocity=cÞ. This term is therefore of the same small order Oð1=c2Þ, as those terms in the second line in Eqs. (164) and (166) that refer to the upper components.

We conclude that in the presence of electromagnetic fields the components remain unmixed, correct to the order Oð1=cÞ.

G.Topological Phase for Dirac Electrons

The topological (or Berry) phase [9,11,78] has been discussed in previous sections. The physical picture for it is that when a periodic force, slowly (adiabatically) varying in time, is applied to the system then, upon a full periodic evolution, the phase of the wave function may have a part that is independent of the amplitude of the force. This part exists in addition to that part of the phase that depends on the amplitude of the force and that contributes to the usual, ‘‘dynamic’’ phase. We shall now discuss whether a relativistic electron can have a Berry phase when this is absent in the framework of the Schro¨dinger equation, and vice versa. (We restrict the present discussion to the nearly nonrelativistic limit, when particle velocities are much smaller than c.)

complex states of simple molecular systems

271

The following lemma is needed for our result. Consider a matrix Hamiltonian h coupling two states, whose energy difference is 2m

m	E1 cos ot	a	Þ	E2 sin	ot		ð170Þ
h ¼	þ E2 sin ððotÞþ			m E1cosð	ðoÞt þ aÞ

The Hamiltonian contains two fields, periodically varying in time, whose intensities E1 and E2 are nonzero. The parameter o is their angular frequency and is (in appropriate energy units) assumed to be much smaller than the field strengths. This ensures the validity of the adiabatic approximation [33]. The parameter a is an arbitrary angle. It is assumed that initially, at t ¼ 0, only the component with the positive eigenenergy is present. Then after a full revolution the initially excited component acquires or does not acquire a Berry phase (i.e., returns to its initial value with a changed or unchanged sign) depending on whether jE1j is greater or less than m (¼ half the energy difference).

Proof: When the time-dependent Schro¨dinger equation is solved under adiabatic conditions, the upper, positive energy component has the coefficient: the dynamic phase factor C, where

C ¼ cos	2 arctan m				E1cos	ðotÞ	a		ð171Þ
	1				E2 sin	ot
				þ		ð þ	Þ

Tracing the arctan over a full revolution by the method described in Section IV and noting the factor 1=2 in Eq. (171) establishes our result. (The case that jE1j ¼ m needs more careful consideration, since it leads to a breakdown of the adiabatic theorem. However, this case will be of no consequence for the results.)

We can now return to the Dirac equations, in which the time varying forces enter through the four-potentials ðA0; AÞ. [The ‘‘two states’’ in Eq. (28) refer now to a large and to a small (positive and negative energy) component in the solution of the Dirac equation in the near nonrelativistic limit.] In the expressions (164 and 165) obtained for the phases si and arising from the Lagrangean L0, there is no coupling between different components, and therefore the small relativistic correction terms will clearly not introduce or eliminate a Berry phase. However, terms in this section supply the diagonal matrix elements in Eq. (28). Turning now to the two terms in Eq. (168), the first, magnetic field term again does not admix the large and small components, with the result that for either of these components previous treatments based on the Schro¨dinger or the Pauli equations [321,324] should suffice. Indeed, this term was already discussed by Berry [9]. We thus need to consider only the second, electric-field term that admixes the two types of components. These are the source of the offdiagonal matrix elements in Eq. (28). However, we have just shown that in order to introduce a new topological phase, one needs field strengths matching the