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Gauge independent Wigner function and the laser radiation pressure
Physica 135A (1986) 271-280 North-Holland, Amsterdam
GAUGE INDEPENDENT WIGNER FUNCTION AND THE LASER RADIATION PRESSURE Piotr
BADZIAG
Department of Physics, University of South Africa,
Received Revised
P. 0.
Box 392, Pretoria, 0001, South Africa
18 December
1984
25 September
1985
An analysis of atomic motion in electromagnetic field using gauge independent Wigner function formalism is presented and the expression for the radiation pressure force is rederived using multipole expansion of terms in equations of motion rather than in the interaction Hamiltonian. The lowest order result is then compared with the expression for the radiation pressure force based on E*d approximation of the interaction Hamiltonian. The comparison shows that our approach is more consistent than the standard approach. This is due to the fact that the equation governing the motion of the atomic mass centre does not depend on the Hamiltonian itself but on its derivatives.
1. Introduction There is a long, over 70-year’s history of the study of light pressure upon free atoms and molecules. It starts with pioneer researches by Lebedev’) and Einstein2) and is still continuing, especially in connection with possible applications. For references, see e.g. papers by Stenholml) and Letokhov and Minogin4). Authors usually describe coupling of particles and field by the d - E term obtainable from the minimal coupling by the Power and Zienau transformation”’ ). Because the motion of atoms and molecules is in many aspects classical, the Wigner distribution’) is often used as a convenient tool to describe them. In this paper we follow ideas presented recently’) and describe an atom interacting with an electromagnetic field in terms of the gauge independent Wigner function. As was shown in ref. 8, this approach enables us to avoid both electromagnetic potentials and infinite terms in the interaction Hamiltonian (like the polarization term in the Power and Zienau approach’). When dealing with the radiation pressure, equations we derive contain coupling of the atomic motion to both electric and magnetic fields in the leading term responsible for the pressure. This result, however not new (see e.g. ref. 9), 0378-4371/86/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
272
P. BADZIAG
has not been,
up to our knowledge,
the radiation
pressure
compare Hamiltonian
the result
with one based
and presented
The paper is organized equations
elaborated
In this paper
force.
enough
to include
we perform
calculations
of
the calculations
on E *d approximation
and
of the interaction
e.g. by Stenholm’). as follows:
which in section
in section
3 are applied
2 we present
to the calculation
basic definitions of radiation
and
pressure
force exerted on the two-level atom in a field of a plane monochromatic electromagnetic wave. Comparison of the result with those analyzed in literature together with some concluding remarks are given in section 4. 2. Gauge independent
Wigner
The gauge independent particle in electromagnetic
function
Wigner function operator describing field can be defined as follows:
a charged
f(R,p;i):=&$+:@+(R-r12.i)
da a(R
+ ar, t)
-1.1 Here,
as usual,
:( ): denotes
normal
(2.1)
ordering.
is electromagnetic potential operator. The Wigner function describing a particle expectation value off in this state. The density matrix evolution equation in a electromagnetic Wigner function: t:{$
in a given
equation
t)] + k it, X k[i(R,
field operator,
quantum
for the nonrelativistic
field gives the following
+ i, * V + e{k[k(R,
@is the particle
state
charged
A
is the particle
for the gauge independent
t)]} *
$ jf(R,
p; I):) = 0, (2.2)
where
WIGNER
FUNCTION
AND LASER RADIATION
PRESSURE
273
We have written the integral operations in the form of formal series of differential operators to show their semiclassical limit. In this limit (2.2) gives the classical kinetic equation for the position-kinetic momentum space density of the charged particles in the electromagnetic field. Many particle generalizations of (2.1) and (2.2) are straightforward. In the case of an atom with one optically sensitive electron, one has to consider two particles: a positively charged core and an electron. The Hamiltonian for such a system in a electromagnetic field reads
1 2
A=& ,
+
[
ti(R, -
Ri) +
L @,+
:+&
j+i(R,J)
e
1 2
f$R,,l)
:
e[@(Ri,t>- +(R,, t)l .
(2.3)
Quantities with index “i” describe here a positively charged core and those with index “e” describe an electron. The density matrix evolution equation gives the following equation for the gauge independent Wigner function:
Eq. (2.4) is a direct two particle generalization
of (2.2). The new term:
is responsible for the mutual electron-ion interaction. In the long wave (semiclassical) limit we can put
G[F(Ri, ‘>I= ‘CR,, ‘13 ‘1 i, e,
e,
e =
Pi,
elMi,
e
.
Leaving only the terms of the first order in pi, J(Mi,e.c) and in spatial derivatives of fields (dipole approximation) and neglecting MelM, in comparison
274
P. BADZIAG
with 1 we obtain
:($ +u -7 x
from +
(2.4):
ti[-o,ti(r)] * ; -
eji(R, t)(l
+
PR*r)
+ f (V + u)
&R, t)
xf((r,p;R,P;t):=O. Here
lower
case
letters
(2.5) describe
relative
motion
of the
electron
and
the
positively charged core whereas the capital ones describe the center of mass motion. If we now describe relative motion in the representation given by the atom’s stationary states, we get the following equation instead of (2.5):
(2.6) We have neglected functions
&
here (1 /c)V
are defined
X Up(rl)*ucy(r2)f( where
uct(r) is the
x B and (r,t/j - V)E
in comparison
with E. The
as follows:
electron
it-, + wave
ir,. p;R, P; t) , function
in the
stationary
state
ICY) and
r
WIGNER
FUNCTION
AND
LASER
RADIATION
PRESSURE
27.5
radiation pressure, because, in principle, it is of the same order as the one arising from the electric field and responsible for the radiation pressure (it follows from the Maxwell equations that (1 /c) a B,l at is of the same order as VjEk). To see how it influences the final results, let us examine the two-level atoms with the dipole moments induced along electric field, in the field of the electromagnetic wave given by: E(R, t) = E cos(k.R B(R, t) = kl;r
-
wt) ,
(2.7a)
-
(2.7b)
cos(k.R
wt) ,
where k=(O,O,k),
E=(E,O,O),
w=ck.
3. Two-level atom in a plane monochromatic
electromagnetic
wave
In this case eq. (2.6) with E(R, t) and B(R, t) given by (2.7a) and (2.7b) respectively, together with the phenomenological terms responsible for spontaneous emission gives the following equations for functions fij:
- iih,,Ecos(k.R-wt)
{p2,fi2-p12f21}=0,
(3.la)
f2?+iEcos(k.R--Ot)(P21fi2-P1Zf21) -fib%
Ecos(k*R-ot)
-$ +iw,, + ir+ -iih%
~J*V
Ecos(k*R-wt)
{p21fi2-p12f21}=0,
f2, +iEcos(k*R-ot){p,,f,,
(3.lb) -p2,fz2}
{pzlf,l+p21f22}=0.
These equations give the following set of equations for the functions:
(3.lc)
P. BADZIAG
276
i
l
f+E%hcos(k.R-wt)
-$+d’
$
+ v - V + l-
(3.2~1)
k C
(3.2b)
g + 2E cos(k - R - wt)< - rf=o,
I
(3.2~)
~-w~,~~-~E(P~,~~cos(~.R-
wt)g=O.
(3.2d)
Here, in (3.2d), the term proportional to hk - df/dP is neglected, because in the final equation for f (f describes the atom’s probability density) this term gives a contribution of the order (hkia,,)’ and such quantities are neglected in (2.6). One can point out appropriate equations Instead
here that, although eqs. (3.2b-d) are identical in, for example, Stenholm’s pape?), eq. (3.2a)
as the is not.
of our
(w,,/w)cos(k.R Stenholm
- wt).
has
sin(k - R - wt) . The difference between our equations and those of Stenholm is due to the fact that our equations do not neglect terms with the magnetic field. which are usually disregarded in the first order approximation. These terms should have already
been included
in the first order
the semiclassical
level, where
of the interaction
Lagrangian
approximation.
the second includes
order
One can see that even at
term in the multipole
expansion
the expression
fr.[VxB(R,r)]. In the equations ; Because
of motion
$ [r x B(R, of Maxwell
er*E(R,
this gives the following
for R:
t)] equations
this term is of the same order as those due to the
t)
part of the interaction
term in the equation
Lagrangian
-
WIGNER
FUNCTION
AND
LASER
RADIATION
PRESSURE
277
To show how the presence of the magnetic field in the equations of motion influences the final formula for the radiation pressure force, we hereafter calculate c[ f] from (3.2b-d) and then substitute the resulting expression for 5 in (3.2a). Thus we arrive at the equation for f(R, P; t) in the form:
[
I
i+“.v+F.$
f=O.
(3.3)
The resulting expression for F is given in (3.6). As usual we perform our calculations in the limit:
(3.4)
We also assume that f is a slowly varying function of R and t. To simplify the equation we rewrite them in the variables R,=R-vt,
t,=t.
In these variables we obtain
(& 1 +r
g = rf-
2E cos(kz, - LZt)l ,
(3Sb) (3Sc)
(3.5d) where O=w-k.v.
Zero order approximation g
in (3Sb) gives
(0) -f,
so that in the first order solution we can put 5, = a cos(kz, - Ot) + c sin(kz, - L?t) , q = b cos(kz,
- Ot) + d sin(kz, - L?t) .
P. BADZIAG
278
This and (3.5b)
gives
g=f-(Ely)aS where
y = 1 r and g2 is a rapidly
oscillating
term of the order:
(aE/r)(rln) less than (&Z/T),
This is one order can write:
so that in the first order
(Ely)a
g =f-
This and (3.5c-d)
gives the following
Ru + yc - q,d
set of equations
for a, b, c, d:
= 0,
fJb+wz,c+yd=O, w?,u + yb - f..d = 0, (y + E’~p,,~*/~)u Simple
algebra
- q,b
- fjc = 2E(p,,l’f.
gives now
a =2yElp,,12
;
,
z b=
w,I
“f,
R -
Y &[(I-
where
In the limit
-
fl’+
y*
2 cd*, L
I_ i
L
Y
c=
approximation
w,,
2 -
o2
-
y2
R’
U
w;’ +
-0’
1+ “5’ i w,, +a’
” ) ;Rwzl lu, w;, + R2 w2, + 0’
R’
il
f12
+
wt,
’
U
Jl fl*+w;,
’
we
WIGNER
FUNCTION
AND
LASER
RADIATION
279
PRESSURE
lAl=Ifl--&wm
this gives a = d = (~)EElp,,~*/(A* b zz c = -AElp,,1*/(A2
+ r2/4 + E21p,,)*/2) ) + T2/4 + E21p,,1*/2) ,
which generates the following expression for F(R, t): F(R, t) = (~~,/~)E*[pr~1~hkcos(R.R
- wt){(l/2)rcos(k.R
- A sin(k* R - wt)} /(A* + r2/4 + E*[p,,1*/2) .
- wt) (3.6)
The expression for the time-averaged force, resulting from (3.6), when close to resonance, does not differ much from the one well known in the literature. It reads rhk @) = E21P1212 4 A2 + r*/4 + E21p,212/2 ’
(3.7)
However, expression (3.6) itself differs by the factor w,,/w from which is known as the first order approximation of the radiation pressure force, as well as by the different space-time behaviour, which cannot be cancelled in any limit.
4. Conclusions In this paper we have presented an analysis of the radiation pressure force exerted on atoms within the framework of the gauge independent Wigner function formalism. Our analysis of the interaction between atoms and electromagnetic field is new in a sense that it remains mainfestly gauge independent even if atoms were allowed to carry a nett charge (camp e.g. ref. 5, as well as later papers by Atkins and Woolley”) and Babiker and coworkers”). The results show that to obtain the proper space-time behaviour of the force, even in the lowest order of multipole expansion in the nonrelativistic limit, one has to include an interaction between atoms and the magnetic field. Within the method presented in this paper this becomes natural, because when radiation pressure is concerned the relevant contributions due to both E and B appear at the same level of approximation, even if the orders of magnitude of their sources in the Hamiltonian are not the same.
280
P. BADZIAG
References 1) 2) 3) 4) 5) 6) 7) 8) 9)
P.N. Lebedev, Ann. de Phys. 32 (1910) 411. A. Einstein, Mitt. Phys. Ges. (Zurich) 18 (1916) 47. S. Stenholm, Phys. Rep. 43 (1978) 151. V.S. Letokhov and V.G. Minogin, Phys. Rep. 73 (1981) 1. E.A. Power and S. Zienau, Phil. Trans. Roy. Sot. 251 (1959) 427. J. Fiutak and Z. Engels, Acta Physica Polonica A44 (1973) 795. E.P. Wigner, Phys. Rev. 40 (1932) 749. P. Badziag, Physica 130A (1985) 565. S.R. de Groot and L.C. Suttorp, Foundations of Electrodynamics (North-Holland. Amsterdam, 1972). 10) P.W. Atkins and R.G. Woolley, Proc. Roy. Sot. A319 (1970) 549. 11) M. Babiker. A.E. Power and T. Thirunamachandran, Proc. Roy. Sec. A338 (1974) 235, M. Babiker and R. London, Proc. Roy. Sot. A385 (1983) 439.
GAUGE INDEPENDENT WIGNER FUNCTION AND THE LASER RADIATION PRESSURE Piotr
BADZIAG
Department of Physics, University of South Africa,
Received Revised
P. 0.
Box 392, Pretoria, 0001, South Africa
18 December
1984
25 September
1985
An analysis of atomic motion in electromagnetic field using gauge independent Wigner function formalism is presented and the expression for the radiation pressure force is rederived using multipole expansion of terms in equations of motion rather than in the interaction Hamiltonian. The lowest order result is then compared with the expression for the radiation pressure force based on E*d approximation of the interaction Hamiltonian. The comparison shows that our approach is more consistent than the standard approach. This is due to the fact that the equation governing the motion of the atomic mass centre does not depend on the Hamiltonian itself but on its derivatives.
1. Introduction There is a long, over 70-year’s history of the study of light pressure upon free atoms and molecules. It starts with pioneer researches by Lebedev’) and Einstein2) and is still continuing, especially in connection with possible applications. For references, see e.g. papers by Stenholml) and Letokhov and Minogin4). Authors usually describe coupling of particles and field by the d - E term obtainable from the minimal coupling by the Power and Zienau transformation”’ ). Because the motion of atoms and molecules is in many aspects classical, the Wigner distribution’) is often used as a convenient tool to describe them. In this paper we follow ideas presented recently’) and describe an atom interacting with an electromagnetic field in terms of the gauge independent Wigner function. As was shown in ref. 8, this approach enables us to avoid both electromagnetic potentials and infinite terms in the interaction Hamiltonian (like the polarization term in the Power and Zienau approach’). When dealing with the radiation pressure, equations we derive contain coupling of the atomic motion to both electric and magnetic fields in the leading term responsible for the pressure. This result, however not new (see e.g. ref. 9), 0378-4371/86/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
272
P. BADZIAG
has not been,
up to our knowledge,
the radiation
pressure
compare Hamiltonian
the result
with one based
and presented
The paper is organized equations
elaborated
In this paper
force.
enough
to include
we perform
calculations
of
the calculations
on E *d approximation
and
of the interaction
e.g. by Stenholm’). as follows:
which in section
in section
3 are applied
2 we present
to the calculation
basic definitions of radiation
and
pressure
force exerted on the two-level atom in a field of a plane monochromatic electromagnetic wave. Comparison of the result with those analyzed in literature together with some concluding remarks are given in section 4. 2. Gauge independent
Wigner
The gauge independent particle in electromagnetic
function
Wigner function operator describing field can be defined as follows:
a charged
f(R,p;i):=&$+:@+(R-r12.i)
da a(R
+ ar, t)
-1.1 Here,
as usual,
:( ): denotes
normal
(2.1)
ordering.
is electromagnetic potential operator. The Wigner function describing a particle expectation value off in this state. The density matrix evolution equation in a electromagnetic Wigner function: t:{$
in a given
equation
t)] + k it, X k[i(R,
field operator,
quantum
for the nonrelativistic
field gives the following
+ i, * V + e{k[k(R,
@is the particle
state
charged
A
is the particle
for the gauge independent
t)]} *
$ jf(R,
p; I):) = 0, (2.2)
where
WIGNER
FUNCTION
AND LASER RADIATION
PRESSURE
273
We have written the integral operations in the form of formal series of differential operators to show their semiclassical limit. In this limit (2.2) gives the classical kinetic equation for the position-kinetic momentum space density of the charged particles in the electromagnetic field. Many particle generalizations of (2.1) and (2.2) are straightforward. In the case of an atom with one optically sensitive electron, one has to consider two particles: a positively charged core and an electron. The Hamiltonian for such a system in a electromagnetic field reads
1 2
A=& ,
+
[
ti(R, -
Ri) +
L @,+
:+&
j+i(R,J)
e
1 2
f$R,,l)
:
e[@(Ri,t>- +(R,, t)l .
(2.3)
Quantities with index “i” describe here a positively charged core and those with index “e” describe an electron. The density matrix evolution equation gives the following equation for the gauge independent Wigner function:
Eq. (2.4) is a direct two particle generalization
of (2.2). The new term:
is responsible for the mutual electron-ion interaction. In the long wave (semiclassical) limit we can put
G[F(Ri, ‘>I= ‘CR,, ‘13 ‘1 i, e,
e,
e =
Pi,
elMi,
e
.
Leaving only the terms of the first order in pi, J(Mi,e.c) and in spatial derivatives of fields (dipole approximation) and neglecting MelM, in comparison
274
P. BADZIAG
with 1 we obtain
:($ +u -7 x
from +
(2.4):
ti[-o,ti(r)] * ; -
eji(R, t)(l
+
PR*r)
+ f (V + u)
&R, t)
xf((r,p;R,P;t):=O. Here
lower
case
letters
(2.5) describe
relative
motion
of the
electron
and
the
positively charged core whereas the capital ones describe the center of mass motion. If we now describe relative motion in the representation given by the atom’s stationary states, we get the following equation instead of (2.5):
(2.6) We have neglected functions
&
here (1 /c)V
are defined
X Up(rl)*ucy(r2)f( where
uct(r) is the
x B and (r,t/j - V)E
in comparison
with E. The
as follows:
electron
it-, + wave
ir,. p;R, P; t) , function
in the
stationary
state
ICY) and
r
WIGNER
FUNCTION
AND
LASER
RADIATION
PRESSURE
27.5
radiation pressure, because, in principle, it is of the same order as the one arising from the electric field and responsible for the radiation pressure (it follows from the Maxwell equations that (1 /c) a B,l at is of the same order as VjEk). To see how it influences the final results, let us examine the two-level atoms with the dipole moments induced along electric field, in the field of the electromagnetic wave given by: E(R, t) = E cos(k.R B(R, t) = kl;r
-
wt) ,
(2.7a)
-
(2.7b)
cos(k.R
wt) ,
where k=(O,O,k),
E=(E,O,O),
w=ck.
3. Two-level atom in a plane monochromatic
electromagnetic
wave
In this case eq. (2.6) with E(R, t) and B(R, t) given by (2.7a) and (2.7b) respectively, together with the phenomenological terms responsible for spontaneous emission gives the following equations for functions fij:
- iih,,Ecos(k.R-wt)
{p2,fi2-p12f21}=0,
(3.la)
f2?+iEcos(k.R--Ot)(P21fi2-P1Zf21) -fib%
Ecos(k*R-ot)
-$ +iw,, + ir+ -iih%
~J*V
Ecos(k*R-wt)
{p21fi2-p12f21}=0,
f2, +iEcos(k*R-ot){p,,f,,
(3.lb) -p2,fz2}
{pzlf,l+p21f22}=0.
These equations give the following set of equations for the functions:
(3.lc)
P. BADZIAG
276
i
l
f+E%hcos(k.R-wt)
-$+d’
$
+ v - V + l-
(3.2~1)
k C
(3.2b)
g + 2E cos(k - R - wt)< - rf=o,
I
(3.2~)
~-w~,~~-~E(P~,~~cos(~.R-
wt)g=O.
(3.2d)
Here, in (3.2d), the term proportional to hk - df/dP is neglected, because in the final equation for f (f describes the atom’s probability density) this term gives a contribution of the order (hkia,,)’ and such quantities are neglected in (2.6). One can point out appropriate equations Instead
here that, although eqs. (3.2b-d) are identical in, for example, Stenholm’s pape?), eq. (3.2a)
as the is not.
of our
(w,,/w)cos(k.R Stenholm
- wt).
has
sin(k - R - wt) . The difference between our equations and those of Stenholm is due to the fact that our equations do not neglect terms with the magnetic field. which are usually disregarded in the first order approximation. These terms should have already
been included
in the first order
the semiclassical
level, where
of the interaction
Lagrangian
approximation.
the second includes
order
One can see that even at
term in the multipole
expansion
the expression
fr.[VxB(R,r)]. In the equations ; Because
of motion
$ [r x B(R, of Maxwell
er*E(R,
this gives the following
for R:
t)] equations
this term is of the same order as those due to the
t)
part of the interaction
term in the equation
Lagrangian
-
WIGNER
FUNCTION
AND
LASER
RADIATION
PRESSURE
277
To show how the presence of the magnetic field in the equations of motion influences the final formula for the radiation pressure force, we hereafter calculate c[ f] from (3.2b-d) and then substitute the resulting expression for 5 in (3.2a). Thus we arrive at the equation for f(R, P; t) in the form:
[
I
i+“.v+F.$
f=O.
(3.3)
The resulting expression for F is given in (3.6). As usual we perform our calculations in the limit:
(3.4)
We also assume that f is a slowly varying function of R and t. To simplify the equation we rewrite them in the variables R,=R-vt,
t,=t.
In these variables we obtain
(& 1 +r
g = rf-
2E cos(kz, - LZt)l ,
(3Sb) (3Sc)
(3.5d) where O=w-k.v.
Zero order approximation g
in (3Sb) gives
(0) -f,
so that in the first order solution we can put 5, = a cos(kz, - Ot) + c sin(kz, - L?t) , q = b cos(kz,
- Ot) + d sin(kz, - L?t) .
P. BADZIAG
278
This and (3.5b)
gives
g=f-(Ely)aS where
y = 1 r and g2 is a rapidly
oscillating
term of the order:
(aE/r)(rln) less than (&Z/T),
This is one order can write:
so that in the first order
(Ely)a
g =f-
This and (3.5c-d)
gives the following
Ru + yc - q,d
set of equations
for a, b, c, d:
= 0,
fJb+wz,c+yd=O, w?,u + yb - f..d = 0, (y + E’~p,,~*/~)u Simple
algebra
- q,b
- fjc = 2E(p,,l’f.
gives now
a =2yElp,,12
;
,
z b=
w,I
“f,
R -
Y &[(I-
where
In the limit
-
fl’+
y*
2 cd*, L
I_ i
L
Y
c=
approximation
w,,
2 -
o2
-
y2
R’
U
w;’ +
-0’
1+ “5’ i w,, +a’
” ) ;Rwzl lu, w;, + R2 w2, + 0’
R’
il
f12
+
wt,
’
U
Jl fl*+w;,
’
we
WIGNER
FUNCTION
AND
LASER
RADIATION
279
PRESSURE
lAl=Ifl--&wm
this gives a = d = (~)EElp,,~*/(A* b zz c = -AElp,,1*/(A2
+ r2/4 + E21p,,)*/2) ) + T2/4 + E21p,,1*/2) ,
which generates the following expression for F(R, t): F(R, t) = (~~,/~)E*[pr~1~hkcos(R.R
- wt){(l/2)rcos(k.R
- A sin(k* R - wt)} /(A* + r2/4 + E*[p,,1*/2) .
- wt) (3.6)
The expression for the time-averaged force, resulting from (3.6), when close to resonance, does not differ much from the one well known in the literature. It reads rhk @) = E21P1212 4 A2 + r*/4 + E21p,212/2 ’
(3.7)
However, expression (3.6) itself differs by the factor w,,/w from which is known as the first order approximation of the radiation pressure force, as well as by the different space-time behaviour, which cannot be cancelled in any limit.
4. Conclusions In this paper we have presented an analysis of the radiation pressure force exerted on atoms within the framework of the gauge independent Wigner function formalism. Our analysis of the interaction between atoms and electromagnetic field is new in a sense that it remains mainfestly gauge independent even if atoms were allowed to carry a nett charge (camp e.g. ref. 5, as well as later papers by Atkins and Woolley”) and Babiker and coworkers”). The results show that to obtain the proper space-time behaviour of the force, even in the lowest order of multipole expansion in the nonrelativistic limit, one has to include an interaction between atoms and the magnetic field. Within the method presented in this paper this becomes natural, because when radiation pressure is concerned the relevant contributions due to both E and B appear at the same level of approximation, even if the orders of magnitude of their sources in the Hamiltonian are not the same.
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