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Gauge invariant formulation of molecular electrodynamics and the multipolar Hamiltonian
Chemical Physics ELSEVIER
Chemical Physics 198 (1995) 133-143
Gauge invariant formulation of molecular electrodynamics and the multipolar Hamiltonian Vladimir Chernyak, Shaul Mukamel Department of Chemistry, University of Rochester, Rochester, NY 14627, USA Received 23 February 1995; in final form 15 March 1995
Abstract
The multipolar Hamiltonian has many advantages for describing the electrodynamics of nonrelativistic material systems. Usually it is derived by performing a canonical transformation on the minimal coupling Hamiltonian. We show that both the minimal coupling and the multipolar Hamiltonians are two forms of the same Hamiltonian corresponding to two choices of gauge: div A = 0 and r . A ( r ) = 0 respectively. We further discuss the use of the multipolar Hamiltonian in electronically extended systems.
1. I n t r o d u c t i o n A large class of problems in linear and nonlinear optics is connected with the interaction of radiation (i.e., the transverse electromagnetic field) with nonrelativistic material systems. Two forms of the Hamiltonian are most often used in theoretical investigations of these problems: the minimal coupling and the multipolar Hamiltonian [1]. Systems interacting with the radiation field possess a specific kind of symmetry-gauge invariance, which leads to the appearance of constraints when constructing the Hamiltonian [2]. There are different ways to proceed in such a case. One possibility is to fix the gauge with the condition div A ( r ) = 0, (1.1) in the action of the system. We will use the notation A o ( r ) and A ( r ) for the scalar and the vector potentials of the electromagnetic field. After that, the system has no constraints on the Hamiltonian level
and can be quantized canonically [3-5]. This procedure leads to the minimal coupling Hamiltonian. The multipolar Hamiltonian is usually obtained from the minimal coupling Hamiltonian by applying a canonical transformation which mixes the material and the field variables [1,6-9]. An alternative way to obtain the multipolar Hamiltonian involves the Lagrangian formalism in classical mechanics. Power and Thirunamachandran [10] considered a transformation of variables in the classical Lagrangian, whereas Babiker and Loudon [11] used a procedure of gauge fixing and gauge transformation in the Lagrangian formalism. In both approaches transformations affect the classical Lagrangians, and different forms of the classical Lagrangian lead to different forms of classical Hamiltonians written in terms of canonical variables, which in turn lead to different forms of the quantum Hamiltonian after canonical quantization. The quantum Hamiltonians are then connected by a quantum canonical transformation.
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V. Chernyak, S. Mukamel / Chemical Physics 198 (1995) 133-143
In this paper we propose an approach which connects the quantum canonical transformations with gauge transformation. Quint•zing the system before the gauge fixing we obtain a procedure of gauge fixing on the level of the quantum Hamilton formalism; gauge transformations then become canonical transformations, The proposed approached is based on the gauge invariant scheme of quantization introduced by Dirac [12]. This scheme involves an extended space of states with the Hamiltonian defined uniquely on the subspace of physical states denoted the physical subspace. This subspace is determined by the condition that the result of acting with all constraints (which become operators after quantization) on a physical state gives zero. Due to the constraints, a physical state being a wave function depending on the vector potential is determined completely by its values on an infinite-dimensional surface in the space of all possible vector potentials. A choice of such a surface is equivalent to the choice of a gauge. Therefore, different choices of gauge lead to different realizations of the physical subspace, and different forms of the Hamiltonian. We shall show that the minimal coupling and the multipolar Hamilton•arts correspond to two specific choices of gauge. Both gauges are "transverse", the first in the momentum domain, the second in the coordinate space. In Section 2 we present a formulation of quantum electrodynamics with nonrelativistic matter in an arbitrary gauge. Details of the derivation starting with the gauge invariant quantization are given in Append i c e s a , B, and C. In Section 3 we consider"linear" gauges and show that the minimal coupling mulipolar Hamilton•arts correspond to the divA(r) = 0 and r . A ( r ) = 0 respectively. applications of the multipolar Hamilton•an cussed in Section 4.
and the gauges Several are dis-
2. Quantum eleetrodynamics of nonrelativistic matter in an arbitrary gauge In this section we formulate the quantum electrodynamics of nonrelativistic material systems in an arbitrary gauge. In Appendix A we introduce the gauge input•ant quantization scheme based on disregarding the constraints on the first step of quantiza-
tion to obtain a broader space of states ~ . The physical subspace of states ~pn c ~ is obtained by imposing the quantum versions of the constraints. The quantum Hamilton•an /1 is uniquely defined on ~vh" In Appendix B we describe the procedure of gauge fixing and define the gauge transformations on the language of quantum Hamilton•an formalism. For each gauge, denoted by .Z/ we obtain a specific realization ,;U~, of the physical space of states ~ph" The isomorphism P~,:~ph ~ ~7'~" leads to a Hamiltonian H~, acting on ~ , which corresponds to /4 and is called the Hamiltonian in the gauge .Z¢. In Appendix C we introduce the canonical variables for an arbitrary gauge .Z¢ and express the Hamiltonians / ~ in terms of them. The gauge is usually fixed by a set of equations which specify constraints on the vector potential A(r): F~, ( A ) = 0. (2.1) The solutions of Eq. (2.1), i.e., the vector potentials A(r) satisfying the gauge condition can be presented in a form A(r)
~A
±
(r)
+
[
~r ¢~'[ r ' [ A ± ] ) ,
(2.2)
where A ± ( r ) is the transverse part of the vector potential, and the functional q~r is determined from F~r. The canonical field variables are the operators of the transverse vector potential ,4 • (r) and transverse electric displacement /) • (r) which satisfy the commutation relations [/)'(r),
A'(r')]
= -4wi6i(r-r'),
(2.3)
and commute with all material operators. Here 6 • is the transverse &function. The complete vector potential operator A(r) can be expressed in terms of its transverse part A 1 ( r ) by applying Eq. (2.2) in an operator form, which yields: A ( r ) - - A ± ( r ) + ~r q~,(r, [ A • ] ) .
(2.4)
The canonical material variables are the operators ~ ( r ) and ~ + ( r ) (we adopt the second quantized form for the electrons) with the commutation relations l~" t )[';'*tr~, ~l'7"tr'~]+ = 'et (2.5) [ , ]+ denotes the ant•commutator, and we set h = 1.
V. Chernyak,S. Mukamel/ ChemicalPhysics198 (1995)133-143 The operators of the charge density t3(r) and the current j(r) are given by ~(r) = -e~+(r)(b(r), (2.6a)
135
and
zm~(r)A2(r ) .
/~(ntag")~ f dr
f(r) = - i e [ O + ( r ) O~b(r) - & ~ + ( r ) O ( r ) ] .
(2.8f) (2.6b)
The transverse electric field operator /~ ± (r) is connected to the electric displacement operator /9 ± (r) bv the relation /) ± ( r ) =/~ ± ( r ) + 4 rr/3~ ( r ) , (2.7a) with f)(r)
- -f
dr' tS(r')
8 ~ , ( r ' , [A • ]) ~,~• ( r )
We denote H .(magn) the "magnetic interaction" since --lilt it can be expressed in terms of the magnetic field /}(r) using Eq. (C.18) to find A • (r), and Eq. (2.4). This terminology which applies to an arbitrary gauge is somewhat unusual. Using the gauge r.A(r) = 0 related to the multipolar Hamiltonian (see Section 3) it coincides with the conventional definitions. However, in the gauge div A(r) = 0 related to the minimal coupling Hamiltonian the entire interaction is "magnetic".
(2.7b) We will refer to fi~(r) as the microscopic gauge-dependent polarization, where ~" specifies the gauge, The Hamiltonian /4~ in the gauge defined by 1" (or equivalently by q~) has the form (for derivation, see Appendix C) /~a ~'= /-Ie "}-/~rad -~- /~int" (2.8a)
3. Linear gauge and the minimum coupling and multipolar Hamiltonians
In Eq. (2.8a)/4, is the Hamiltonian of electrons with Coulomb interaction ( 01 ~ 0~+ +U(r)(b + ( r ) ~ ( r ) )
In this section we consider the case of "linear" gauges and show that the gauges div A(r) = 0 and r .a(r)= 0 lead to the minimal coupling and the multipolar Hamiltonians respectively. We will call a gauge linear if the gauge fixing conditions Eq. (2.1) are linear in A(r). If F~, is linear in A(r) then q~ islinearin A ± ( r ) and can be presented in a f°rm
f l e = f dr 2m Or
~r(r,[A±])=fq~e(r',r).a±(r')dr'.
if
Or
1
(3.1)
+ 2 dr d r ' l r _ r' ] t3(r)t3(r'), (2.8b) where U(r) is the potential induced by nuclei. /~rad
For "linear" gauges when Eq. (3.1) is satisfied the
is the free field Hamiltonian
/~,¢ ( r )
/~rad -
1 f dr{[b±(r)]Z+ [rot A±(r)]2}, 8.,.r
microscopic polarization adopts a form
= - f dr' dr" 8 ~ ( r - r ' ) ~ , ( r ' ,
r")~(r").
(2.8c)
(3.2)
and /~int describing the electron interaction with the
We now turn to specific gauges. First, we take for Eq. (2.1) the condition div A = 0. We then have
transverse electromagnetic field is
/-lint = /~(2 ) -j- /~{nt agn),
(2.8d)
q~, - O,
(3.3a)
with
which by using Eq. (3.2) leads to
i-~i~)=_½fdr[15±(r)fi,~,(r)+t;~(r)~)±(r)]
13~(r)=-O,
+ 2'rrj¢ dr [ f ~ ( r ) ] 2,
(2.8e)
/)± ( r ) =/~± ( r )
and from Eq. (2.4) yields fi,(r) = A ± ( r ) .
(3.3b) (3.3c)
136
V. Chernyak, S. Mukamel / Chemical Physics 198 (1995) 133-143
We then have /~.(0) lnt = 0,
(3.3d)
and we obtain the Hamiltonian
find q~r(r, r'). It follows from Eqs. (2.4) and (3.5) that ~e satisfies the following relation A ± ( r ) + ~rr q~r ( r , [A ±]
/~,~?"=/~e "[- /~rad + /~i(n~agn),
(3.4)
• r = 0.
Solving Eq. (3.9) we get
with ,~ • (r) substituted into /~.(magn) instead of A(r). This is the minimal coupling-~intamiltonian" To obtain the multipolar Hamilton±an we fix the gauge (Eq. (2.1)) with the condition
Combining Eqs. (3.10a) and (3.1) results in
r.A(r)
~ , ( r ' , r") = _ £ 1 dA
=0,
(3.5)
which means that the vector potential is transverse in coordinate space. Since Eq. (3.5) is linear in A(r), / ~ (r) becomes a purely material operator (see Eq. (3.2)) and ~'lnt ~(0) can be recast in a form
/~i (0, = --
f dr
(r)] 2
(3.6)
We can also express A(r) in terms of the magnetic field /~(r) by making use of the gauge fixing condition of Eq. (3.5). Taking into account Eq. (3.5), Eq. (C.18) can be written as r 0A(r) A(r) - + -]r[ Or ]r]
Irl r × / ~ ( r ) ,
(3.7)
1 fc dr' r'--[r'× 1 Irl • I r'l --
o
/?(r')],
(3.8a)
where C r is a segment of a straight line [0, r], or alternatively fo f A(r)=-ldAAjdr'8(r'-ar)[r×B(r')]. (3.8b) Substituting Eq. (3.8b) into Eq. (2.8) and taking ~.(0) in the form of Eq. (3.6), the Hamiltonian /4,¢ assumes the multipolar second quantized form, with /~.(magn) representing the magnetic terms. The only remaining point is to show that the expression for the polarization Eq. (3.2) is the same as the one used in the multipolar Hamiltonian. To that end we need to I/It
(3.10a)
r
r"8(r'-Ar").
(3.10b)
Substituting q~(r', r") from Eq. (3.10b) into Eq. (3.2) we obtain the expression for the polarization as used in the multipolar Hamilton±an [1,8].
The multipolar Hamilton±an has been successfully used to describe the interaction of molecules with the transverse electromagnetic field [1]. Calculations using this form of the Hamilton±an become particularly simple when it is possible to neglect the magnetic terms (/_~.(magn)'} in the interaction, which leads to an -
1
whose solution is
mt
q~,(r, [ A ±]) = - ,~f- d r ' . Z • ( r ' ) .
4. Discussion
+
A(r)-
(3.9)
--lnt
i
interaction which is linear in the transverse field. The magnetic terms are smaller in nonrelativistic systems by a factor of v/c, where c is the speed of light and v is a typical electron speed. In the expansion in multipoles the expansion parameter is L/A where L is the size of a system, and A is the optical wavelength. For a hydrogen atom these parameters coincide, and are equal the fine structure constant e2/hc. Therefore, when neglecting the magnetic terms we can also invoke the dipole approximation. However, in large molecules the parameter L/A can be much larger than v / c so that we can neglect the magnetic terms, yet need to keep higher terms in the multipolar series. This justifies using the multipolar Hamiltonian while neglecting the magnetic terms. In the remainder of the section we will comment on applying the multipolar Hamiltonian without the magnetic terms. The "self-interaction" (the last term in the rhs of Eq. (3.6)) which looks complicated in the Hamiltonian is very important, and leads to a compact form of the interaction terms when solving
V. Chernyak, S. Mukamel/ ChemicalPhysics 198 (1995) 133-143 dynamical problems. It has been shown in [13] that the equation of motion for the Heisenberg operator Q(t) related to an arbitrary material operator Q assumes the form dO i d t = i[/4c' Q] + -2
f dr {[f" (r)' O] "~l(r)
+/~ l ( r ) . [ f • ( r ) , Q] ).
(4.1a)
Note that / ~ • ( r ) is a mixed (field and material) operator which does not commute with [ fi - (r), Q]. Making use of Eq. (3.10) we can recast Eq. (4.1a) in a form dO "'e[t~ ~1,5] i / . j L \L/l -i dr dt ' 2
Q]
-}- (Peff( r ) [ /3( r ) , Q] ),
Co.(r)
(4. lb)
with ~¢ff(r) ~ - f
dr'./~(r').
(4.1c)
Finally we show how to apply the multipolar Hamiltonian for systems with size larger than the wavelength and with extended electron states such as semiconductors. In such systems, it is not possible to apply the dipole approximation to the entire system; perh°Wever' m cell oit is[15]. mpossible e nto introduce t u na dipole i t We consider a one-dimensional tight binding model for the system in the z-direction [16]. The charge density is given by ~( r) = - e ~., 6 ( r - R,) c,^+c~, (4.3) ,, where c, ^+ (c,) is the creation (annihilation) operator of an electron on the site n. We assume that R, form a one-dimensional regular lattice along the z-axis, with two sites per unit cell. Treating the transverse field classically for the sake of simplicity we obtain the interaction term in the Hamiltonian /4int = f dz t3(z)q~ff(z),
S = S e + Srad + Sint, where the interaction term is [14] dr d~- P • ( r , ~-)E , ( r ,
(4.2a)
r),
(4.2b)
with z q~eff(z)= - f
Ez
0~"
dr d~-p(r, "/')~eff(r, 7").
O~Oeff ~z
(4.4c)
-ef dq qgeff(--q)
(4.2c)
The interaction term can be represented also as the interaction of the charge density with the effective potential: Sint = - f
(4.4b)
For a long chain, we can switch to a basis set of the Bloch functions determined by momentum k and a two-dimensional vector wi(k) (i = c, v denotes the conduction and the valance bands), which describes the Bloch function in a unit cell. Switching to the momentum domain we obtain /~int =
OA±(r, T )
dz' Ez(z' ).
zo It follows from Eq. (4.4b), that
and E ± ( r , T) =
(4.4a)
~c r
Eq. (4.1b) shows that when the magnetic terms are neglected, the second term has a form of the interaction of the charge density ~(r) with the effective potential ~eff(r) induced by the transverse field according to Eq. (4.1c). The interaction terms also assume a compact form if we adopt the path-integral approach [12], using the action S which has a form related to the partition given by Eq. (2.8a):
s,n, = f
137
(4.2d)
× f dk ( wi( k + q), wj( k ) )~'i( k + q)6j( k ), (4.5) where (wi, w ) is the scalar product of the Bloch functions in a unit cell and ~-(k), ~i(k) are the electron creation and annihilation operators in the
V. Chernyak, S. Mukamel / Chemical Physics 198 (1995) 133-143
138
Bloch basis set. Considering the cross terms in Eq. (4.5) ( i = c, j = v ) w e have (wc(k), w v ( k ) ) = 0 ,
(4.6)
since the states in different bands are orthogonal, and for qa << 1 (a is the lattice constant) we have ( w c ( k + q ) , wv(k)) = i e - l t z ( k ) q ,
(4.7a)
with
Switching to the Hamilton formalism by means of the canonical procedure, we obtain the Poisson bracket for the canonical variables A, A 0, E, E o, t~, q, { E ( r ) , a ( r ) } = 4rr/~(r - r'),
(A.Za)
{Eo(r), a o ( r ) } = 4xr6(r - r),
(A.Zb)
{tp(r), q,(r')} = - i 6 ( r - r ' ) .
(A.2c)
The Hamiltonian has the form
/z(k) ~ - - i e
0 ~ - - - - , wv(k) .
(4.7b)
Substituting Eq. (4.7a) into Eq. (4.5) and making use of Eq. (4.4c) we obtain Hi,t = - f
n=fdr
Otp0~O+U ~ 0 + ~---~w[E2 + (rot A) 2]
e 1 + j . A - 2----~pA 2 - 47r A°( a "E - 4"rrp)), (A.3)
dk f d q E z ( q ) t z ( k ) e + ( k + q ) e v ( k ) .
(4.8) Eq. (4.8) has a form of the dipole approximation with respect to a unit cell, /x(k) being the k-dependent dipole moment of a unit cell.
with the constraints Eo(r ) = 0 ,
a . E ( r ) + 4rrp(r) = 0 .
(A.4)
In Eqs. (A.3) and (A.4) we use the notation p ( r ) = -e~O(r)~b(r), (A.5a) is the charge density and
Appendix A. Gauge invariant quantization
j ( r ) = - i e ( t~0~b- O~b),
We start with presenting the gauge invariant scheme of quantization. Consider a system containing nonrelativistic electrons in an external potential, interacting with the electromagnetic field. The Coulomb interaction of electrons is taken into account by the longitudinal field. We start with the classical action in the second quantized form [17] S[ 2, ~0, A0, A]
is the current. The basic idea of the gauge invariant quantization scheme [12] is to use the canonical quantization, disregarding the constraints at first, and obtaining the commutation relations for the canonical operators from the Poisson bracket. The second step is to form the space of states, providing an irreducible representation of the operator algebra. The third step is to introduce the physical states subspace, which includes the states which are nullified by the quantized version of the constraints Eqs.
=
j
)
The commutation relations for the operators, quantized canonical variables (we denote by Q the operator corresponding to a classical variable Q) have the form
dr dt i~b ~t ieAo 0 1 __ -U~bVipV~b 2m
1 2 1 ] + --8,rr( ~ - oa°) - ~ - (rot A) 2J ,
(A.5b)
(A.la)
[/~(r), A ( r ) ] = - 4 r r i 6 ( r -
r'),
(A.6a)
[/~0(r), (A0) ] = - - 4 ~ r i 6 ( r - r ' ) ,
(A.6b)
[ ~ t ( r ) , ~ ( r ' ) ] += - 6 ( r - r ' ) .
(A.6c)
with Vm = ~m -- ieAm
(A.lb)
Vm = ~m + ieAm"
(A.lc)
[ , ]+ denotes the anticommutator, and we set h = 1. The algebra defined by Eqs. (A.6) can be realized on
V. Chernyak, S. Mukamel / Chemical Physics 198 (1995) 133-143
the space of wavefunctions ~0[A, A0] depending on the scalar and vector potentials (we use the square brackets for listing functional variables), and taking values in the space of material states with the natural action of the material operators. The action of the field operators on the wavefunctions is defined as follows: (A(r)O)[A, ao]
=A(r)~b[A, A0],
(A.7a)
~ b [ A , A0]
(E(r)~t)[A'a°]=-a'rri
~A(r)
,
(A.7b)
139
for/~0, we see that Eq. (A.8) means that a physical wavefunction does not depend on A 0. ~b[a, a0] = qJ[A]. (a.10) Introducing the longitudinal and the transverse parts of the vector potential A = A II + A ± = 0~ + A ± , (A.11) we obtain for a variation of a wavefunction ~ with respect to q~(r) 0 ~b 6t~= - f dr 6~(r) Jr ~A(r)' (a.12)
with the action of the operators /~0 and A0 defined in the same manner, The physical states are those satisfying the conditions
which after taking Eq. (A.7b) into account leads to ~ ~b 3./~(r) ~b= 4-rri- q~(r) (A.13)
/~0(r) tb = 0, (3./~(r) - 4'rr/3(r))~b = 0.
Making use of Eqs. (A.3b), (A.gb) and (A.10) we obtain the general form of a physical state
(A.ga) (g.gb)
The Hamiltonian /~ is the quantum version of ( A . 3 ) O = e x p ( - i f d r q ~ ( r ) ~ ( r ) ) ~ [ A ± ] . (A.14) and has the form 1 The results of this appendix will be used in ApI~=t~m+~--~ f dr[/~2 + (rot /]) 2] pendix B to introduce the gauge fixing and transformations and in Appendix C to obtain Hamiltonians
+
f
(
)
dr f(r) .A(r) - ~ m t3(r)A2(r) ,
at fixed gauges.
(A.9a)
Appendix B. Gauge fixing and transformations with (1
O~t O~
f H,, =
]
+ U(r)(U(r)(b(r) . d r \ 2m ~r Jr
/ (A.9b)
We omitted the last terms of Eq. (A.3) since due to Eq. (A.gb) this term vanishes on the physical subspace. It is well known [2] that one can add to the Hamiltonian terms which vanish on the physical subspace; therefore, in the present formalism we have a unique Hamiltonian on the physical subspace, Note, that the physical subspace is the correct space of states of the system, and the physical quantities correspond to gauge invariant operators, i.e., operators for which the physical subspace is invariant [ 1 2 ] . The constraints of Eqs. (A.8) can be solved, leading to a convenient description of the physical subspace. Making use of the analogue of Eq. (A.7b)
In this appendix we turn to the problem of gauge fixing and transformations. Let 77` be the space of vector potential A(r). It can be decomposed according to Eq. (A.11) ~-= TI I ~ ~7-±, with
(B.la)
All ~ TII, A±~ T±. (B.lb) It follows from Eq. (a.14) that a physical wavefunction ~, which is a mapping of ~b: ~ 7~, 7f" being the space of the material system, is determined by its values on a submanifold ~ c ~ with the following property: the projection p: .Z~'~ ~ 1 is a one-to-one mapping. The latter condition implies that ~ can be described by a mapping %r: ~ 1 ~ ~il' .~r then consists of vector potentials A(r) of the form
a(r) =a" (r) + O~oA(r, [ a i ] )
(B.2)
V. Chernyak, S. Mukamel / Chemical Physics 198 (1995) 133-143
140
for all possible transverse vector potentials A ± (r). The choice of Mr is the gauge fixing procedure, Usually the gauge is fixed by a system of equations. F~(A) = 0, (B.3) Mr being the space of solutions of Eq. (B.3). The gauge fixing (i.e., the choice of . g ) allows us to describe the physical space of wavefunctions ~b: 7 / ~ 7 f satisfying Eq. (A.14) as a space of functions ~b~,: . ~ T W , confining a function ~b on Mr to obtain ~9~,.We can return to ~b starting with ~b~ making use of Eq. (A.14). We will denote the space of all functions ~b: 7/'-~ 7W with ~'~, its physical subspace of functions satisfying Eq. (A.14) Yg'ph,and the space of functions ~bl: Mr ~ 7~ with Y~e, we have an isomorphism p~: Jfph ~ ~¢~" We call the space Y~, the space of states of the system at the gauge given by Mr. We can choose the transverse part of the vector potential A • as a set of coordinates on Mr, then ,,~, can be described as the space of functions ~O~r[A ± (r)] and the isomorphism between Xph and Y~, can be given by
transformations are canonical (unitary) transformations of a special form given by Eq. (B.6). In Section 3 we show that the minimal coupling and the multipolar Hamiltonians are two forms of H corresponding to the gauges div A = 0 and r.A(r) = 0. This means that the canonical transformation used to obtain the multipolar Hamiltonian from the minimal coupling Hamiltonian [1,8] is a gauge transformation.
Appendix C. Itamiltonian for a fixed gauge and canonical variables In Appendix B we have defined the procedure for gauge fixing and transformations. In this appendix we will find the form /)~, of the Hamiltonian H of Eqs. (A.9) related to a gauge given by Mr. Making use of the facts that /) acts on Y'C~phand p~ is an isomorphism between •ph, and ,ge~, given by Eq. (B.4), it is natural to define H,e acting on ~T¢~,as H~r ---P~'-1/~p~,.
~ e [ a • ( r ) ] = ~b[A±(r) + 0~p~,(r, [ a ± l ) ] .
(C.1)
(B.4) Let Mr and Mr' define two different gauges It follows from Eqs. (B.4) and (A.14) that
It is convenient to express/4~ in terms of canonical variables, i.e. operators acting on the space of functions ~b~,[A1 (r)]. We choose the canonical variables to be the operators A± (r) and /9±(r) defined as
qj~,[ a ~-(r)]
A • (r)~h~,,[ a ± ]
=exp - i f
dr(q~(r, [a±])
-~of(r,[A±]))fg(r))~bf[A±(r)].
/)±(r)~9~[ a ± ] - = - 4 r r i (B.5)
(B.6a)
where/)~,¢, is a unitary operator which has the form / ) ~ , f - = e x p ( - i f ~J d r ( ~ ( r ,
~a±(r) ,
(C.2b)
= f6 ±( r - r')A(r') dr',
~ ~A±(r)
f dr' 8 ~ ( r - r ') ~ J ~Z(r'-----~'
(C.3a) (C.3b)
and 6 ± is the transverse &function. It follows from Eqs. (C.2)that ,4± and /)± commute with all material operators and
[,~1])
- q~j,(r, [ A ± ])) t3(r) ).
~,,e[ a ± ]
(C.2a)
where A ± (r)
We can recast Eq. (B.5) in the form ~b~r=/)~,,~b~,,
-a±(r)~O~[a±],
(B.6b)
[19±(r),A±(r')]=-a~ri~•(r-r').
(C.4)
]
Eqs. (B.6) imply that wavefunctions in two different gauges are connected with a unitary (canonical) transformation. This means that quantum gauge
We will also consider the full vector potential A(r)-,4±(r)
+ ~rr~O,,(r, [
])
(C.5)
V. Chernyak,S. Mukamel/ChemicalPhysics198 (1995)133-143
141
and according to Eq. (C.5) the operator A(r) is a functional of the transverse vector potential A j- . We can now express the Hamilton•an /4~r in terms of the canonical variables. From Eqs. (A.9), we see that /4 can be partitioned as
The dependence of a physical wavefunction on the longitudinal part of the vector potential All= 0~o is given by Eq. (A.14) which follows from the constraint Eq. (A.bb), and we obtain on the physical subspace
/ l = H x +/42,
(C.6a)
/~" =0/x
(C.6b)
where /x i is the operator form of the Green function of the Laplace operator ,~. Substituting Eq. (CAD into Eq. (C.9) and taking into account the fact that ts(r) is a material operator we obtain
with 1 f /4~ =- ~ dr/~2(r),
and /~2 contains other terms which can be expressed
in terms of A(r) and material operators. Both Ham•ltonians /ta and /~2 are gauge invariant, i.e., they commute with the constraints (Eqs. (A.8)), hence the physical subspace of states is invariant with respect to both. To obtain the Hamilton•an (/~2)~ related to /42 by means of Eq. (C.1) we make use of Eq. (B.4) connecting ~ and ~0. (/~)~e has the same form a s /~2 where the operators A(r) are defined b~y Eqs. (C.2a) and (C.5) and for a material operator Q, (Q~0~,)[A± (r)]--- Q(~,,e[a± ( r ) ] ) . (C.7)
1(4~rt5),
1
(Hll)~, =
(C,11)
1
dr dr'lr_r,-----~lts(r)ts(r'),
(C.12)
i.e., (H~l)~r represents the Coulomb energy. The most interesting term is H11 . To find (/~±)~, we need to connect the operators /~ • (r) to /) • (r). To that end we note that we know how the operators /~• (r) act on •ph and how D • ( r ) act on ~ , . Let qJ,r ~ Y~e and q/~ ~,TC~phrepresent the same state, i.e., ~0~,=p~(qD. This means that tp and qJ~, satisfy Eq. (B.4), which when combined with Eq. (C.2b) yields
This means that (/~2).av has the form (/42)A=/4m +
+far
~ + (r), = -4"rri--
~a'(r)
rot A ( r ) ] 2 dr
(
f(r)'a(r)-
e
+ 0q~r(r', [ A ± ])].
~---mts(r)A2(r ) . (C.8)
To obtain (HI)A it is convenient to represent /41 in a form of a sum of its longitudinal and transverse parts
qJ [ A • ( r ') (C.13)
For a physical function ~b we have /~± (r)~--- -4~ri ~A ( r ~ ±~
,
(C.14a)
/41 =/411 +/~1 • ,
(C.9a)
- ~O-- - i t 5 ( r ) ~p. 8q~(r)
/411- ~ 1 f dr [/~ll(r)] 2,
(C.9b)
Combining Eqs. (C.13) and (C.14) we obtain
1 /11•-- 8--~-f dr [/~± ( r ) ] 2,
(C.9c)
b • (r)~b
where E"(r) and /~ ± (r) are given by E ( r ) =/~ ± ( r ) + Ell(r), 8 /~• ( r ) = - 4 7 r i S( A r -•- -~- -
=/~± ( r ) ~b- 4-rrf dr' 8q~e(r"[Al 8A± ( r ) ]) t s ( r ' ) ~ (C.10a)
(C.15)
(C.10b)
for an arbitrary physical state ~ ~ ~TC'ph.This means that a relation between / ) • ( r ) and / ~ • ( r ) can be written in the form
' /~ll(r) = - 4 - r r i - ~AII( r ) "
(C.14b)
(C.10c)
/~ l ( r ) = E ± ( r ) + 4 xr/3~ ( r ) ,
(C.16a)
142
V. Chernyak, S. Mukamel/ Chemical Physics 198 (1995) 133-143
with / ~ (r) = -
f
dr' /3(r')
~q~,(r', [ ~ l ] ) ~A±(r) , (C.16b)
is the microscopic polarization in the gauge related to ~¢'. Finally we can present the Hamiltonian H~ in the gauge defined by ,g¢ (or equivalently by q~,e) in the form (C.17a) In Eq. (C.17a) /~e is the Hamiltonian of electrons with the Coulomb interaction included:
/~.K =/-Ie q'-/~rad q'- /~int"
f
I1e =
( 1 0~+ 0~
dr
~+
2m-- Or Or + U(r)
)
(r)(b(r)
!/ drdr'lr-r'lD(r)[~(r')'
with all material operators and with commutation relations given by Eq. (C.4). The operators A(r) are expressed in terms of a~(r) by Eq. (C.5). We will call /~i(ntagn) the magnetic part of interaction (magnetic terms) since it can be expressed in terms of the magnetic field /~(r) = rot A(r), (C.18) by finding A" (r) from Eq. (C. 18) and applying Eq. (C.5). The transverse electric field /~'(r) can be expressed in terms of the canonical variables using Eqs. (C.16). In a general case, the expression for the microscopic polarization / ~ (r) given by Eq. (C.16b) contains the material variables /3(r) as well as the field variables A±(r). The polarization becomes apurelymaterialvariablewhenifthegauge condition given by Eq. (B.3) F~r is linear in A, or equivalently ,~/is a vector subspace of W', or equiv-
1
+2
(C.17b)
/Qrad is the free field Hamiltonian: /-Irad = ~-~ f dr{[19±(r)]2+[rota±(r)]2},
and the H a m i l t o n i a n /~int describing electron interaction with the transverse electromagnetic field is given by
,.,., .~.2.,.:~. l~,.'~
4~pi
(C.17d)
/~int ----"'intO(O)--'4-.]-~.(magn ).mt ,
with o.(o)= "'mt --
2
f dr [D±(r)t3;(r)
M
+ 2=f d, and
[
~...~(3agn)= f dr f ( r ) . A ( r )
e
-
(r)] (C.17e)]
-~---~m/~(r)A2(r)
~
~
.
(c.17f) In Eqs. (C.17) /) • (r) and A • (r) defined by Eqs. (C.2) are the canonical field variables commuting
Fig. 1. Geometrical representation of Eq. (C.16a). D ± is the
derivativein the directiontangent to .~'. E j- and 4~P ± transverseand longitudinal components.
are
its
V. Chernyak, S. Mukamel/ ChemicalPhysics 198 (1995) 133-143 alently ~ e is a linear functional of A ± ( r ) , i.e., qLr can be written as f ~,rr,( [A ±]) =
q~,(r', r ) . a l ( r
') dr', (C.19)
which leads to the following expression for the polarization _--/~"~( r ) = -- f d r ' dr" ~ • ( r -
r ' ) ~p~r(r' r " ) / 3 ( r " ) ' " (C.20)
Our f o r m a l i s m points out a geometric origin of the m i c r o s c o p i c polarization /:;7- This is illustrated in Fig. 1. In the derivation of Eqs. (C.16) we noticed that on the space •ph, /~± ( r ) acts as a variational derivative in a transverse direction, and / ) • ( r ) acts as a derivative in a direction tangent to 1". Thererepresented by 4'rrrA~'± fore, the difference 15 3_ is given by a derivative in the longitudinal direction, i.e., a functional derivative with respect to the longitudinal part of the vector potential which is defined by its transverse part (see Eq. (2.2)), which yields '.~,fi3_ proportional to /3, in agreement with Eqs. (A.13) and (A.14).
Acknowledgement W e gratefully a c k n o w l e d g e the support of the National Science Foundation and the Air Force Office of Scientific Research.
143
References [1] D.P. Craig and T. Thirunamachandran, Molecular quantum electrodynamics (Academic Press, London, 1984). [2] P.A.M. Dirac, Lectures on Quantum Mechanics, Belfer Graduate School of Science, Yeshiva University, New York (1964). [3] R. Loudon, The quantum theory of light, 2rid ed. (Clarendon Press, Oxford, 1983). [4] C. Cohen-Tannoudji, I. DuPont-Roc and G. Grynberg, Photons and atoms: Introduction to quantum elcctrodynamics (Wiley, New York, 1989). [5] C. Cohen-Tannoudji, J. Dupont-Roc and G. Grynberg, Atom-photon interactions (Wiley, New York, 1992). [6] M. G6ppert-Mayer, Ann. Physik (Leipzig) (1931) 273. [7] E.A. Power and S. Zienau, Phil. Trans. Roy. Soc. London A 251 (1959) 427. [8] E.A. Power, Introductory quantum electrodynamics (Longmans, London, 1964). [9] w.P. Healy, Phys. Rev. A 22 (1980) 2891; Nonrelativistic quantum electrodynamics (Academic Press, New York, 1982). [10] E.A. Power and T. Thirunamachandran, Phys. Rev. A 28 (1983) 2649. [11] M. Babiker and R. Loudon, Proc. Roy. Soc. Lond. A 385 (1983) 439. [12] P.A.M. Dirac, The principles of quantum mechanis, 4th ed. (Oxford Univ. Press, Oxford, 1967). [13] J. Jenkins and S. Mukamel, J. Chem Phys. 98 (1993) 7046. [14] v. Chernyak and S. Mukamel, J. Chem. Phys. 100 (1994) 2953. [15] H. Haug and S.W. Koch, Quantum theory of the optical and electronic properties of semiconductors (World Scientific, New Jersey, 1990). [16] M. Hartmann, V. Chernyak and S. Mukamel, Phys. Rev. B, submitted. [17] V.N. Popov, Functional integrals in quantum field theory and statistical physics (Reidel, Dordrecht, 1983).
Chemical Physics 198 (1995) 133-143
Gauge invariant formulation of molecular electrodynamics and the multipolar Hamiltonian Vladimir Chernyak, Shaul Mukamel Department of Chemistry, University of Rochester, Rochester, NY 14627, USA Received 23 February 1995; in final form 15 March 1995
Abstract
The multipolar Hamiltonian has many advantages for describing the electrodynamics of nonrelativistic material systems. Usually it is derived by performing a canonical transformation on the minimal coupling Hamiltonian. We show that both the minimal coupling and the multipolar Hamiltonians are two forms of the same Hamiltonian corresponding to two choices of gauge: div A = 0 and r . A ( r ) = 0 respectively. We further discuss the use of the multipolar Hamiltonian in electronically extended systems.
1. I n t r o d u c t i o n A large class of problems in linear and nonlinear optics is connected with the interaction of radiation (i.e., the transverse electromagnetic field) with nonrelativistic material systems. Two forms of the Hamiltonian are most often used in theoretical investigations of these problems: the minimal coupling and the multipolar Hamiltonian [1]. Systems interacting with the radiation field possess a specific kind of symmetry-gauge invariance, which leads to the appearance of constraints when constructing the Hamiltonian [2]. There are different ways to proceed in such a case. One possibility is to fix the gauge with the condition div A ( r ) = 0, (1.1) in the action of the system. We will use the notation A o ( r ) and A ( r ) for the scalar and the vector potentials of the electromagnetic field. After that, the system has no constraints on the Hamiltonian level
and can be quantized canonically [3-5]. This procedure leads to the minimal coupling Hamiltonian. The multipolar Hamiltonian is usually obtained from the minimal coupling Hamiltonian by applying a canonical transformation which mixes the material and the field variables [1,6-9]. An alternative way to obtain the multipolar Hamiltonian involves the Lagrangian formalism in classical mechanics. Power and Thirunamachandran [10] considered a transformation of variables in the classical Lagrangian, whereas Babiker and Loudon [11] used a procedure of gauge fixing and gauge transformation in the Lagrangian formalism. In both approaches transformations affect the classical Lagrangians, and different forms of the classical Lagrangian lead to different forms of classical Hamiltonians written in terms of canonical variables, which in turn lead to different forms of the quantum Hamiltonian after canonical quantization. The quantum Hamiltonians are then connected by a quantum canonical transformation.
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134
V. Chernyak, S. Mukamel / Chemical Physics 198 (1995) 133-143
In this paper we propose an approach which connects the quantum canonical transformations with gauge transformation. Quint•zing the system before the gauge fixing we obtain a procedure of gauge fixing on the level of the quantum Hamilton formalism; gauge transformations then become canonical transformations, The proposed approached is based on the gauge invariant scheme of quantization introduced by Dirac [12]. This scheme involves an extended space of states with the Hamiltonian defined uniquely on the subspace of physical states denoted the physical subspace. This subspace is determined by the condition that the result of acting with all constraints (which become operators after quantization) on a physical state gives zero. Due to the constraints, a physical state being a wave function depending on the vector potential is determined completely by its values on an infinite-dimensional surface in the space of all possible vector potentials. A choice of such a surface is equivalent to the choice of a gauge. Therefore, different choices of gauge lead to different realizations of the physical subspace, and different forms of the Hamiltonian. We shall show that the minimal coupling and the multipolar Hamilton•arts correspond to two specific choices of gauge. Both gauges are "transverse", the first in the momentum domain, the second in the coordinate space. In Section 2 we present a formulation of quantum electrodynamics with nonrelativistic matter in an arbitrary gauge. Details of the derivation starting with the gauge invariant quantization are given in Append i c e s a , B, and C. In Section 3 we consider"linear" gauges and show that the minimal coupling mulipolar Hamilton•arts correspond to the divA(r) = 0 and r . A ( r ) = 0 respectively. applications of the multipolar Hamilton•an cussed in Section 4.
and the gauges Several are dis-
2. Quantum eleetrodynamics of nonrelativistic matter in an arbitrary gauge In this section we formulate the quantum electrodynamics of nonrelativistic material systems in an arbitrary gauge. In Appendix A we introduce the gauge input•ant quantization scheme based on disregarding the constraints on the first step of quantiza-
tion to obtain a broader space of states ~ . The physical subspace of states ~pn c ~ is obtained by imposing the quantum versions of the constraints. The quantum Hamilton•an /1 is uniquely defined on ~vh" In Appendix B we describe the procedure of gauge fixing and define the gauge transformations on the language of quantum Hamilton•an formalism. For each gauge, denoted by .Z/ we obtain a specific realization ,;U~, of the physical space of states ~ph" The isomorphism P~,:~ph ~ ~7'~" leads to a Hamiltonian H~, acting on ~ , which corresponds to /4 and is called the Hamiltonian in the gauge .Z¢. In Appendix C we introduce the canonical variables for an arbitrary gauge .Z¢ and express the Hamiltonians / ~ in terms of them. The gauge is usually fixed by a set of equations which specify constraints on the vector potential A(r): F~, ( A ) = 0. (2.1) The solutions of Eq. (2.1), i.e., the vector potentials A(r) satisfying the gauge condition can be presented in a form A(r)
~A
±
(r)
+
[
~r ¢~'[ r ' [ A ± ] ) ,
(2.2)
where A ± ( r ) is the transverse part of the vector potential, and the functional q~r is determined from F~r. The canonical field variables are the operators of the transverse vector potential ,4 • (r) and transverse electric displacement /) • (r) which satisfy the commutation relations [/)'(r),
A'(r')]
= -4wi6i(r-r'),
(2.3)
and commute with all material operators. Here 6 • is the transverse &function. The complete vector potential operator A(r) can be expressed in terms of its transverse part A 1 ( r ) by applying Eq. (2.2) in an operator form, which yields: A ( r ) - - A ± ( r ) + ~r q~,(r, [ A • ] ) .
(2.4)
The canonical material variables are the operators ~ ( r ) and ~ + ( r ) (we adopt the second quantized form for the electrons) with the commutation relations l~" t )[';'*tr~, ~l'7"tr'~]+ = 'et (2.5) [ , ]+ denotes the ant•commutator, and we set h = 1.
V. Chernyak,S. Mukamel/ ChemicalPhysics198 (1995)133-143 The operators of the charge density t3(r) and the current j(r) are given by ~(r) = -e~+(r)(b(r), (2.6a)
135
and
zm~(r)A2(r ) .
/~(ntag")~ f dr
f(r) = - i e [ O + ( r ) O~b(r) - & ~ + ( r ) O ( r ) ] .
(2.8f) (2.6b)
The transverse electric field operator /~ ± (r) is connected to the electric displacement operator /9 ± (r) bv the relation /) ± ( r ) =/~ ± ( r ) + 4 rr/3~ ( r ) , (2.7a) with f)(r)
- -f
dr' tS(r')
8 ~ , ( r ' , [A • ]) ~,~• ( r )
We denote H .(magn) the "magnetic interaction" since --lilt it can be expressed in terms of the magnetic field /}(r) using Eq. (C.18) to find A • (r), and Eq. (2.4). This terminology which applies to an arbitrary gauge is somewhat unusual. Using the gauge r.A(r) = 0 related to the multipolar Hamiltonian (see Section 3) it coincides with the conventional definitions. However, in the gauge div A(r) = 0 related to the minimal coupling Hamiltonian the entire interaction is "magnetic".
(2.7b) We will refer to fi~(r) as the microscopic gauge-dependent polarization, where ~" specifies the gauge, The Hamiltonian /4~ in the gauge defined by 1" (or equivalently by q~) has the form (for derivation, see Appendix C) /~a ~'= /-Ie "}-/~rad -~- /~int" (2.8a)
3. Linear gauge and the minimum coupling and multipolar Hamiltonians
In Eq. (2.8a)/4, is the Hamiltonian of electrons with Coulomb interaction ( 01 ~ 0~+ +U(r)(b + ( r ) ~ ( r ) )
In this section we consider the case of "linear" gauges and show that the gauges div A(r) = 0 and r .a(r)= 0 lead to the minimal coupling and the multipolar Hamiltonians respectively. We will call a gauge linear if the gauge fixing conditions Eq. (2.1) are linear in A(r). If F~, is linear in A(r) then q~ islinearin A ± ( r ) and can be presented in a f°rm
f l e = f dr 2m Or
~r(r,[A±])=fq~e(r',r).a±(r')dr'.
if
Or
1
(3.1)
+ 2 dr d r ' l r _ r' ] t3(r)t3(r'), (2.8b) where U(r) is the potential induced by nuclei. /~rad
For "linear" gauges when Eq. (3.1) is satisfied the
is the free field Hamiltonian
/~,¢ ( r )
/~rad -
1 f dr{[b±(r)]Z+ [rot A±(r)]2}, 8.,.r
microscopic polarization adopts a form
= - f dr' dr" 8 ~ ( r - r ' ) ~ , ( r ' ,
r")~(r").
(2.8c)
(3.2)
and /~int describing the electron interaction with the
We now turn to specific gauges. First, we take for Eq. (2.1) the condition div A = 0. We then have
transverse electromagnetic field is
/-lint = /~(2 ) -j- /~{nt agn),
(2.8d)
q~, - O,
(3.3a)
with
which by using Eq. (3.2) leads to
i-~i~)=_½fdr[15±(r)fi,~,(r)+t;~(r)~)±(r)]
13~(r)=-O,
+ 2'rrj¢ dr [ f ~ ( r ) ] 2,
(2.8e)
/)± ( r ) =/~± ( r )
and from Eq. (2.4) yields fi,(r) = A ± ( r ) .
(3.3b) (3.3c)
136
V. Chernyak, S. Mukamel / Chemical Physics 198 (1995) 133-143
We then have /~.(0) lnt = 0,
(3.3d)
and we obtain the Hamiltonian
find q~r(r, r'). It follows from Eqs. (2.4) and (3.5) that ~e satisfies the following relation A ± ( r ) + ~rr q~r ( r , [A ±]
/~,~?"=/~e "[- /~rad + /~i(n~agn),
(3.4)
• r = 0.
Solving Eq. (3.9) we get
with ,~ • (r) substituted into /~.(magn) instead of A(r). This is the minimal coupling-~intamiltonian" To obtain the multipolar Hamilton±an we fix the gauge (Eq. (2.1)) with the condition
Combining Eqs. (3.10a) and (3.1) results in
r.A(r)
~ , ( r ' , r") = _ £ 1 dA
=0,
(3.5)
which means that the vector potential is transverse in coordinate space. Since Eq. (3.5) is linear in A(r), / ~ (r) becomes a purely material operator (see Eq. (3.2)) and ~'lnt ~(0) can be recast in a form
/~i (0, = --
f dr
(r)] 2
(3.6)
We can also express A(r) in terms of the magnetic field /~(r) by making use of the gauge fixing condition of Eq. (3.5). Taking into account Eq. (3.5), Eq. (C.18) can be written as r 0A(r) A(r) - + -]r[ Or ]r]
Irl r × / ~ ( r ) ,
(3.7)
1 fc dr' r'--[r'× 1 Irl • I r'l --
o
/?(r')],
(3.8a)
where C r is a segment of a straight line [0, r], or alternatively fo f A(r)=-ldAAjdr'8(r'-ar)[r×B(r')]. (3.8b) Substituting Eq. (3.8b) into Eq. (2.8) and taking ~.(0) in the form of Eq. (3.6), the Hamiltonian /4,¢ assumes the multipolar second quantized form, with /~.(magn) representing the magnetic terms. The only remaining point is to show that the expression for the polarization Eq. (3.2) is the same as the one used in the multipolar Hamiltonian. To that end we need to I/It
(3.10a)
r
r"8(r'-Ar").
(3.10b)
Substituting q~(r', r") from Eq. (3.10b) into Eq. (3.2) we obtain the expression for the polarization as used in the multipolar Hamilton±an [1,8].
The multipolar Hamilton±an has been successfully used to describe the interaction of molecules with the transverse electromagnetic field [1]. Calculations using this form of the Hamilton±an become particularly simple when it is possible to neglect the magnetic terms (/_~.(magn)'} in the interaction, which leads to an -
1
whose solution is
mt
q~,(r, [ A ±]) = - ,~f- d r ' . Z • ( r ' ) .
4. Discussion
+
A(r)-
(3.9)
--lnt
i
interaction which is linear in the transverse field. The magnetic terms are smaller in nonrelativistic systems by a factor of v/c, where c is the speed of light and v is a typical electron speed. In the expansion in multipoles the expansion parameter is L/A where L is the size of a system, and A is the optical wavelength. For a hydrogen atom these parameters coincide, and are equal the fine structure constant e2/hc. Therefore, when neglecting the magnetic terms we can also invoke the dipole approximation. However, in large molecules the parameter L/A can be much larger than v / c so that we can neglect the magnetic terms, yet need to keep higher terms in the multipolar series. This justifies using the multipolar Hamiltonian while neglecting the magnetic terms. In the remainder of the section we will comment on applying the multipolar Hamiltonian without the magnetic terms. The "self-interaction" (the last term in the rhs of Eq. (3.6)) which looks complicated in the Hamiltonian is very important, and leads to a compact form of the interaction terms when solving
V. Chernyak, S. Mukamel/ ChemicalPhysics 198 (1995) 133-143 dynamical problems. It has been shown in [13] that the equation of motion for the Heisenberg operator Q(t) related to an arbitrary material operator Q assumes the form dO i d t = i[/4c' Q] + -2
f dr {[f" (r)' O] "~l(r)
+/~ l ( r ) . [ f • ( r ) , Q] ).
(4.1a)
Note that / ~ • ( r ) is a mixed (field and material) operator which does not commute with [ fi - (r), Q]. Making use of Eq. (3.10) we can recast Eq. (4.1a) in a form dO "'e[t~ ~1,5] i / . j L \L/l -i dr dt ' 2
Q]
-}- (Peff( r ) [ /3( r ) , Q] ),
Co.(r)
(4. lb)
with ~¢ff(r) ~ - f
dr'./~(r').
(4.1c)
Finally we show how to apply the multipolar Hamiltonian for systems with size larger than the wavelength and with extended electron states such as semiconductors. In such systems, it is not possible to apply the dipole approximation to the entire system; perh°Wever' m cell oit is[15]. mpossible e nto introduce t u na dipole i t We consider a one-dimensional tight binding model for the system in the z-direction [16]. The charge density is given by ~( r) = - e ~., 6 ( r - R,) c,^+c~, (4.3) ,, where c, ^+ (c,) is the creation (annihilation) operator of an electron on the site n. We assume that R, form a one-dimensional regular lattice along the z-axis, with two sites per unit cell. Treating the transverse field classically for the sake of simplicity we obtain the interaction term in the Hamiltonian /4int = f dz t3(z)q~ff(z),
S = S e + Srad + Sint, where the interaction term is [14] dr d~- P • ( r , ~-)E , ( r ,
(4.2a)
r),
(4.2b)
with z q~eff(z)= - f
Ez
0~"
dr d~-p(r, "/')~eff(r, 7").
O~Oeff ~z
(4.4c)
-ef dq qgeff(--q)
(4.2c)
The interaction term can be represented also as the interaction of the charge density with the effective potential: Sint = - f
(4.4b)
For a long chain, we can switch to a basis set of the Bloch functions determined by momentum k and a two-dimensional vector wi(k) (i = c, v denotes the conduction and the valance bands), which describes the Bloch function in a unit cell. Switching to the momentum domain we obtain /~int =
OA±(r, T )
dz' Ez(z' ).
zo It follows from Eq. (4.4b), that
and E ± ( r , T) =
(4.4a)
~c r
Eq. (4.1b) shows that when the magnetic terms are neglected, the second term has a form of the interaction of the charge density ~(r) with the effective potential ~eff(r) induced by the transverse field according to Eq. (4.1c). The interaction terms also assume a compact form if we adopt the path-integral approach [12], using the action S which has a form related to the partition given by Eq. (2.8a):
s,n, = f
137
(4.2d)
× f dk ( wi( k + q), wj( k ) )~'i( k + q)6j( k ), (4.5) where (wi, w ) is the scalar product of the Bloch functions in a unit cell and ~-(k), ~i(k) are the electron creation and annihilation operators in the
V. Chernyak, S. Mukamel / Chemical Physics 198 (1995) 133-143
138
Bloch basis set. Considering the cross terms in Eq. (4.5) ( i = c, j = v ) w e have (wc(k), w v ( k ) ) = 0 ,
(4.6)
since the states in different bands are orthogonal, and for qa << 1 (a is the lattice constant) we have ( w c ( k + q ) , wv(k)) = i e - l t z ( k ) q ,
(4.7a)
with
Switching to the Hamilton formalism by means of the canonical procedure, we obtain the Poisson bracket for the canonical variables A, A 0, E, E o, t~, q, { E ( r ) , a ( r ) } = 4rr/~(r - r'),
(A.Za)
{Eo(r), a o ( r ) } = 4xr6(r - r),
(A.Zb)
{tp(r), q,(r')} = - i 6 ( r - r ' ) .
(A.2c)
The Hamiltonian has the form
/z(k) ~ - - i e
0 ~ - - - - , wv(k) .
(4.7b)
Substituting Eq. (4.7a) into Eq. (4.5) and making use of Eq. (4.4c) we obtain Hi,t = - f
n=fdr
Otp0~O+U ~ 0 + ~---~w[E2 + (rot A) 2]
e 1 + j . A - 2----~pA 2 - 47r A°( a "E - 4"rrp)), (A.3)
dk f d q E z ( q ) t z ( k ) e + ( k + q ) e v ( k ) .
(4.8) Eq. (4.8) has a form of the dipole approximation with respect to a unit cell, /x(k) being the k-dependent dipole moment of a unit cell.
with the constraints Eo(r ) = 0 ,
a . E ( r ) + 4rrp(r) = 0 .
(A.4)
In Eqs. (A.3) and (A.4) we use the notation p ( r ) = -e~O(r)~b(r), (A.5a) is the charge density and
Appendix A. Gauge invariant quantization
j ( r ) = - i e ( t~0~b- O~b),
We start with presenting the gauge invariant scheme of quantization. Consider a system containing nonrelativistic electrons in an external potential, interacting with the electromagnetic field. The Coulomb interaction of electrons is taken into account by the longitudinal field. We start with the classical action in the second quantized form [17] S[ 2, ~0, A0, A]
is the current. The basic idea of the gauge invariant quantization scheme [12] is to use the canonical quantization, disregarding the constraints at first, and obtaining the commutation relations for the canonical operators from the Poisson bracket. The second step is to form the space of states, providing an irreducible representation of the operator algebra. The third step is to introduce the physical states subspace, which includes the states which are nullified by the quantized version of the constraints Eqs.
=
j
)
The commutation relations for the operators, quantized canonical variables (we denote by Q the operator corresponding to a classical variable Q) have the form
dr dt i~b ~t ieAo 0 1 __ -U~bVipV~b 2m
1 2 1 ] + --8,rr( ~ - oa°) - ~ - (rot A) 2J ,
(A.5b)
(A.la)
[/~(r), A ( r ) ] = - 4 r r i 6 ( r -
r'),
(A.6a)
[/~0(r), (A0) ] = - - 4 ~ r i 6 ( r - r ' ) ,
(A.6b)
[ ~ t ( r ) , ~ ( r ' ) ] += - 6 ( r - r ' ) .
(A.6c)
with Vm = ~m -- ieAm
(A.lb)
Vm = ~m + ieAm"
(A.lc)
[ , ]+ denotes the anticommutator, and we set h = 1. The algebra defined by Eqs. (A.6) can be realized on
V. Chernyak, S. Mukamel / Chemical Physics 198 (1995) 133-143
the space of wavefunctions ~0[A, A0] depending on the scalar and vector potentials (we use the square brackets for listing functional variables), and taking values in the space of material states with the natural action of the material operators. The action of the field operators on the wavefunctions is defined as follows: (A(r)O)[A, ao]
=A(r)~b[A, A0],
(A.7a)
~ b [ A , A0]
(E(r)~t)[A'a°]=-a'rri
~A(r)
,
(A.7b)
139
for/~0, we see that Eq. (A.8) means that a physical wavefunction does not depend on A 0. ~b[a, a0] = qJ[A]. (a.10) Introducing the longitudinal and the transverse parts of the vector potential A = A II + A ± = 0~ + A ± , (A.11) we obtain for a variation of a wavefunction ~ with respect to q~(r) 0 ~b 6t~= - f dr 6~(r) Jr ~A(r)' (a.12)
with the action of the operators /~0 and A0 defined in the same manner, The physical states are those satisfying the conditions
which after taking Eq. (A.7b) into account leads to ~ ~b 3./~(r) ~b= 4-rri- q~(r) (A.13)
/~0(r) tb = 0, (3./~(r) - 4'rr/3(r))~b = 0.
Making use of Eqs. (A.3b), (A.gb) and (A.10) we obtain the general form of a physical state
(A.ga) (g.gb)
The Hamiltonian /~ is the quantum version of ( A . 3 ) O = e x p ( - i f d r q ~ ( r ) ~ ( r ) ) ~ [ A ± ] . (A.14) and has the form 1 The results of this appendix will be used in ApI~=t~m+~--~ f dr[/~2 + (rot /]) 2] pendix B to introduce the gauge fixing and transformations and in Appendix C to obtain Hamiltonians
+
f
(
)
dr f(r) .A(r) - ~ m t3(r)A2(r) ,
at fixed gauges.
(A.9a)
Appendix B. Gauge fixing and transformations with (1
O~t O~
f H,, =
]
+ U(r)(U(r)(b(r) . d r \ 2m ~r Jr
/ (A.9b)
We omitted the last terms of Eq. (A.3) since due to Eq. (A.gb) this term vanishes on the physical subspace. It is well known [2] that one can add to the Hamiltonian terms which vanish on the physical subspace; therefore, in the present formalism we have a unique Hamiltonian on the physical subspace, Note, that the physical subspace is the correct space of states of the system, and the physical quantities correspond to gauge invariant operators, i.e., operators for which the physical subspace is invariant [ 1 2 ] . The constraints of Eqs. (A.8) can be solved, leading to a convenient description of the physical subspace. Making use of the analogue of Eq. (A.7b)
In this appendix we turn to the problem of gauge fixing and transformations. Let 77` be the space of vector potential A(r). It can be decomposed according to Eq. (A.11) ~-= TI I ~ ~7-±, with
(B.la)
All ~ TII, A±~ T±. (B.lb) It follows from Eq. (a.14) that a physical wavefunction ~, which is a mapping of ~b: ~ 7~, 7f" being the space of the material system, is determined by its values on a submanifold ~ c ~ with the following property: the projection p: .Z~'~ ~ 1 is a one-to-one mapping. The latter condition implies that ~ can be described by a mapping %r: ~ 1 ~ ~il' .~r then consists of vector potentials A(r) of the form
a(r) =a" (r) + O~oA(r, [ a i ] )
(B.2)
V. Chernyak, S. Mukamel / Chemical Physics 198 (1995) 133-143
140
for all possible transverse vector potentials A ± (r). The choice of Mr is the gauge fixing procedure, Usually the gauge is fixed by a system of equations. F~(A) = 0, (B.3) Mr being the space of solutions of Eq. (B.3). The gauge fixing (i.e., the choice of . g ) allows us to describe the physical space of wavefunctions ~b: 7 / ~ 7 f satisfying Eq. (A.14) as a space of functions ~b~,: . ~ T W , confining a function ~b on Mr to obtain ~9~,.We can return to ~b starting with ~b~ making use of Eq. (A.14). We will denote the space of all functions ~b: 7/'-~ 7W with ~'~, its physical subspace of functions satisfying Eq. (A.14) Yg'ph,and the space of functions ~bl: Mr ~ 7~ with Y~e, we have an isomorphism p~: Jfph ~ ~¢~" We call the space Y~, the space of states of the system at the gauge given by Mr. We can choose the transverse part of the vector potential A • as a set of coordinates on Mr, then ,,~, can be described as the space of functions ~O~r[A ± (r)] and the isomorphism between Xph and Y~, can be given by
transformations are canonical (unitary) transformations of a special form given by Eq. (B.6). In Section 3 we show that the minimal coupling and the multipolar Hamiltonians are two forms of H corresponding to the gauges div A = 0 and r.A(r) = 0. This means that the canonical transformation used to obtain the multipolar Hamiltonian from the minimal coupling Hamiltonian [1,8] is a gauge transformation.
Appendix C. Itamiltonian for a fixed gauge and canonical variables In Appendix B we have defined the procedure for gauge fixing and transformations. In this appendix we will find the form /)~, of the Hamiltonian H of Eqs. (A.9) related to a gauge given by Mr. Making use of the facts that /) acts on Y'C~phand p~ is an isomorphism between •ph, and ,ge~, given by Eq. (B.4), it is natural to define H,e acting on ~T¢~,as H~r ---P~'-1/~p~,.
~ e [ a • ( r ) ] = ~b[A±(r) + 0~p~,(r, [ a ± l ) ] .
(C.1)
(B.4) Let Mr and Mr' define two different gauges It follows from Eqs. (B.4) and (A.14) that
It is convenient to express/4~ in terms of canonical variables, i.e. operators acting on the space of functions ~b~,[A1 (r)]. We choose the canonical variables to be the operators A± (r) and /9±(r) defined as
qj~,[ a ~-(r)]
A • (r)~h~,,[ a ± ]
=exp - i f
dr(q~(r, [a±])
-~of(r,[A±]))fg(r))~bf[A±(r)].
/)±(r)~9~[ a ± ] - = - 4 r r i (B.5)
(B.6a)
where/)~,¢, is a unitary operator which has the form / ) ~ , f - = e x p ( - i f ~J d r ( ~ ( r ,
~a±(r) ,
(C.2b)
= f6 ±( r - r')A(r') dr',
~ ~A±(r)
f dr' 8 ~ ( r - r ') ~ J ~Z(r'-----~'
(C.3a) (C.3b)
and 6 ± is the transverse &function. It follows from Eqs. (C.2)that ,4± and /)± commute with all material operators and
[,~1])
- q~j,(r, [ A ± ])) t3(r) ).
~,,e[ a ± ]
(C.2a)
where A ± (r)
We can recast Eq. (B.5) in the form ~b~r=/)~,,~b~,,
-a±(r)~O~[a±],
(B.6b)
[19±(r),A±(r')]=-a~ri~•(r-r').
(C.4)
]
Eqs. (B.6) imply that wavefunctions in two different gauges are connected with a unitary (canonical) transformation. This means that quantum gauge
We will also consider the full vector potential A(r)-,4±(r)
+ ~rr~O,,(r, [
])
(C.5)
V. Chernyak,S. Mukamel/ChemicalPhysics198 (1995)133-143
141
and according to Eq. (C.5) the operator A(r) is a functional of the transverse vector potential A j- . We can now express the Hamilton•an /4~r in terms of the canonical variables. From Eqs. (A.9), we see that /4 can be partitioned as
The dependence of a physical wavefunction on the longitudinal part of the vector potential All= 0~o is given by Eq. (A.14) which follows from the constraint Eq. (A.bb), and we obtain on the physical subspace
/ l = H x +/42,
(C.6a)
/~" =0/x
(C.6b)
where /x i is the operator form of the Green function of the Laplace operator ,~. Substituting Eq. (CAD into Eq. (C.9) and taking into account the fact that ts(r) is a material operator we obtain
with 1 f /4~ =- ~ dr/~2(r),
and /~2 contains other terms which can be expressed
in terms of A(r) and material operators. Both Ham•ltonians /ta and /~2 are gauge invariant, i.e., they commute with the constraints (Eqs. (A.8)), hence the physical subspace of states is invariant with respect to both. To obtain the Hamilton•an (/~2)~ related to /42 by means of Eq. (C.1) we make use of Eq. (B.4) connecting ~ and ~0. (/~)~e has the same form a s /~2 where the operators A(r) are defined b~y Eqs. (C.2a) and (C.5) and for a material operator Q, (Q~0~,)[A± (r)]--- Q(~,,e[a± ( r ) ] ) . (C.7)
1(4~rt5),
1
(Hll)~, =
(C,11)
1
dr dr'lr_r,-----~lts(r)ts(r'),
(C.12)
i.e., (H~l)~r represents the Coulomb energy. The most interesting term is H11 . To find (/~±)~, we need to connect the operators /~ • (r) to /) • (r). To that end we note that we know how the operators /~• (r) act on •ph and how D • ( r ) act on ~ , . Let qJ,r ~ Y~e and q/~ ~,TC~phrepresent the same state, i.e., ~0~,=p~(qD. This means that tp and qJ~, satisfy Eq. (B.4), which when combined with Eq. (C.2b) yields
This means that (/~2).av has the form (/42)A=/4m +
+far
~ + (r), = -4"rri--
~a'(r)
rot A ( r ) ] 2 dr
(
f(r)'a(r)-
e
+ 0q~r(r', [ A ± ])].
~---mts(r)A2(r ) . (C.8)
To obtain (HI)A it is convenient to represent /41 in a form of a sum of its longitudinal and transverse parts
qJ [ A • ( r ') (C.13)
For a physical function ~b we have /~± (r)~--- -4~ri ~A ( r ~ ±~
,
(C.14a)
/41 =/411 +/~1 • ,
(C.9a)
- ~O-- - i t 5 ( r ) ~p. 8q~(r)
/411- ~ 1 f dr [/~ll(r)] 2,
(C.9b)
Combining Eqs. (C.13) and (C.14) we obtain
1 /11•-- 8--~-f dr [/~± ( r ) ] 2,
(C.9c)
b • (r)~b
where E"(r) and /~ ± (r) are given by E ( r ) =/~ ± ( r ) + Ell(r), 8 /~• ( r ) = - 4 7 r i S( A r -•- -~- -
=/~± ( r ) ~b- 4-rrf dr' 8q~e(r"[Al 8A± ( r ) ]) t s ( r ' ) ~ (C.10a)
(C.15)
(C.10b)
for an arbitrary physical state ~ ~ ~TC'ph.This means that a relation between / ) • ( r ) and / ~ • ( r ) can be written in the form
' /~ll(r) = - 4 - r r i - ~AII( r ) "
(C.14b)
(C.10c)
/~ l ( r ) = E ± ( r ) + 4 xr/3~ ( r ) ,
(C.16a)
142
V. Chernyak, S. Mukamel/ Chemical Physics 198 (1995) 133-143
with / ~ (r) = -
f
dr' /3(r')
~q~,(r', [ ~ l ] ) ~A±(r) , (C.16b)
is the microscopic polarization in the gauge related to ~¢'. Finally we can present the Hamiltonian H~ in the gauge defined by ,g¢ (or equivalently by q~,e) in the form (C.17a) In Eq. (C.17a) /~e is the Hamiltonian of electrons with the Coulomb interaction included:
/~.K =/-Ie q'-/~rad q'- /~int"
f
I1e =
( 1 0~+ 0~
dr
~+
2m-- Or Or + U(r)
)
(r)(b(r)
!/ drdr'lr-r'lD(r)[~(r')'
with all material operators and with commutation relations given by Eq. (C.4). The operators A(r) are expressed in terms of a~(r) by Eq. (C.5). We will call /~i(ntagn) the magnetic part of interaction (magnetic terms) since it can be expressed in terms of the magnetic field /~(r) = rot A(r), (C.18) by finding A" (r) from Eq. (C. 18) and applying Eq. (C.5). The transverse electric field /~'(r) can be expressed in terms of the canonical variables using Eqs. (C.16). In a general case, the expression for the microscopic polarization / ~ (r) given by Eq. (C.16b) contains the material variables /3(r) as well as the field variables A±(r). The polarization becomes apurelymaterialvariablewhenifthegauge condition given by Eq. (B.3) F~r is linear in A, or equivalently ,~/is a vector subspace of W', or equiv-
1
+2
(C.17b)
/Qrad is the free field Hamiltonian: /-Irad = ~-~ f dr{[19±(r)]2+[rota±(r)]2},
and the H a m i l t o n i a n /~int describing electron interaction with the transverse electromagnetic field is given by
,.,., .~.2.,.:~. l~,.'~
4~pi
(C.17d)
/~int ----"'intO(O)--'4-.]-~.(magn ).mt ,
with o.(o)= "'mt --
2
f dr [D±(r)t3;(r)
M
+ 2=f d, and
[
~...~(3agn)= f dr f ( r ) . A ( r )
e
-
(r)] (C.17e)]
-~---~m/~(r)A2(r)
~
~
.
(c.17f) In Eqs. (C.17) /) • (r) and A • (r) defined by Eqs. (C.2) are the canonical field variables commuting
Fig. 1. Geometrical representation of Eq. (C.16a). D ± is the
derivativein the directiontangent to .~'. E j- and 4~P ± transverseand longitudinal components.
are
its
V. Chernyak, S. Mukamel/ ChemicalPhysics 198 (1995) 133-143 alently ~ e is a linear functional of A ± ( r ) , i.e., qLr can be written as f ~,rr,( [A ±]) =
q~,(r', r ) . a l ( r
') dr', (C.19)
which leads to the following expression for the polarization _--/~"~( r ) = -- f d r ' dr" ~ • ( r -
r ' ) ~p~r(r' r " ) / 3 ( r " ) ' " (C.20)
Our f o r m a l i s m points out a geometric origin of the m i c r o s c o p i c polarization /:;7- This is illustrated in Fig. 1. In the derivation of Eqs. (C.16) we noticed that on the space •ph, /~± ( r ) acts as a variational derivative in a transverse direction, and / ) • ( r ) acts as a derivative in a direction tangent to 1". Thererepresented by 4'rrrA~'± fore, the difference 15 3_ is given by a derivative in the longitudinal direction, i.e., a functional derivative with respect to the longitudinal part of the vector potential which is defined by its transverse part (see Eq. (2.2)), which yields '.~,fi3_ proportional to /3, in agreement with Eqs. (A.13) and (A.14).
Acknowledgement W e gratefully a c k n o w l e d g e the support of the National Science Foundation and the Air Force Office of Scientific Research.
143
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