Interaction of "rigid" quantum systems

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by  A. A. Kolpakov

Interaction of “rigid” quantum systems
A.A. Kolpakov (Universit´e de Fribourg, Suisse)
A.G. Kolpakov (Marie Curie Fellow, Novosibirsk, Russia)
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1
0 We introduce the notion of a “rigid” quantum system as a system with
2
constant relative positions of its nuclei and constant relative distribution of

n
the electrons with respect to the nuclei.
a
J In accordance with this definition, a molecule which does not interact

3 with other objects, is a “rigid” quantum system. Molecule is also “rigid” if it

interacts with other objects, but the interaction does not change the intrinsic
]
h
structure of the molecule (or this change can be neglected).
p
- Several “rigid” quantum systems interact one with another in the quan-
h
tummanner. The interaction is ruled by theSchro¨dinger equation [1] written
t
a
for all the particles of the systems under consideration. We consider the case
m
when the external potential is zero.
[

Since the Schro¨dinger equation for particle systems is invariant with re-
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v spect to translations and rotations [1], every “rigid” quantum system can
2 move as a “rigid body” without any change of its relative characteristics, in
0
particular, without any change of its total energy. However, the total energy
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3 of several interacting “rigid” quantum systems changes when one of them
.
2 moves even as a rigid body, because in quantum mechanics systems interact
1
always.
2
1 Inreality, theaboveassumptionsmeanthattherelativepositionsofnuclei
:
v and the distribution of electrons in every “rigid” quantum system changes so
Xi slightly, that the changes may be neglected.
r Our aim is to demonstrate that the interaction of two “rigid” quantum
a
system(i.e., systemsdescribed bytheSchro¨dinger equation)canbeexpressed
as an electrostatic interaction of two systems of charges.
Although we demonstrate that the systems under consideration, finally,
interact in the electrostatic manner, we cannot use the electrostatic model as
a starting point, because a system of charges is electrostatic unstable [2]. We
have to use the quantum model as a starting point, because only this model
is in agreement with the fact of stable charged particles systems existence.
Accepting the quantum model as a starting point, we have to derive the
electrostatic principle for the interaction of charged systems from it.
1
In other words, we demonstrate that the following decomposition is pos-
sible:
1. the individual (microscopic) integrity and stability of every quantum
system is a result of quantum forces (electrostatic forces cannot form stable
objects);
2. theinteraction of“rigid”quantum systems asindividual (macroscopic)
objects is powered by electrostatic forces corresponding to the (microscopic)
distribution of charges in every quantum system.
This decomposition is inspirited by the behaviour of some kinds of mo-
lecule, which may interact as individual objects without forming a new sub-
stance.
Let one system D be formed by n electrons numbered 1,...,n and m
1 1 1 1
nuclei numbered 1,...,m . Let another system D be formed by n electrons
1 2 2
numbered n +1,...,n +n and m nuclei numbered m +1,...,m +m .
1 1 2 2 1 1 2
We assume that the electric charge of each electron is 1, and denote the
electric charge of the n-th nucleus by Z (in the electron units).
n
We use the notation
−
x= (x ,...,x ),
1 n1
=
x= (x ,...,x ).
n1+1 n1+n2
− =
With this notations x = (x,x).
The indices of the electrons are split into two sets: I – the indices of the
1
electrons that belong to the first system and I – the indices of the electrons
2
that belong to the second system. The indices of the nuclei are also split into
two sets: N – the indices of the electrons that belong to the first system and
1
N – the indices of the electrons that belong to the second system.
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The problem could be written as [1]
E(φ) → min, (1)
|φ(x)|2dx = 1. (2)
ZR3n1×R3n2
Arguments of the wave function φ(x) is x = (x ,...,x ,x ,...,x ).
1 n1 n1+1 n1+n2
The total energy E(φ)) of the integral system (the system formed by
both, the first and the second systems) is written as in [1] (we assume that
there is no external potential):
E(φ) = |∇ φ(x)|2 + (3)
k
ZR3n1×R3n2 k∈XI1∪I2(cid:20)
2
φ2(x) φ2(x)Z
n
+ + dx+
|x −x | |x −x |(cid:21)
j∈XI1∪I2 k j n∈XN1∪N2 k n
Z Z
n m
+ .
|x −x |
n<mX∈N1∪N2 n m
The first three terms in (3) corresponds to the electrons: the first is the
kinetic energy of the electrons, the second one is the potential energy of
the mutual electrons interaction, the third one is the potential energy of the
interaction between the electrons and the kernels. The last term in (3) is the
potential energy of the mutual kernel interaction, the kinetic energy of the
kernels is neglected (the notation n < m ∈ M means that n,m ∈ M and
n < m).
In each of the “rigid” quantum systems, the electron density distribution
is its own specific property, that does not depend on its counterpart. Thus,
the electron density distributions over two “rigid” quantum systems areinde-
pendent functions. So we can construct a solution to the problem (1), (2) for
two “rigid” quantum systems in the form (which is often used in the analysis
of the interaction of a pair of quantum objects [1]):
− − = =
φ(x) =Φ (x) Φ (x). (4)
The advantage of the presentation used in (4) is that if
− − − = = =
| Φ (x)|2d x= 1, | Φ (x)|2d x= 1
ZR3n1 ZR3n2
than the equality
|φ(x)|2dx = 1
ZR3n1×R3n2
is satisfied automatically.
By substituting (4) into (3), we obtain
− − − = = =
E = |∇ Φ (x)|2d x | Φ (x)|2d x + (5)
k
kX∈I1ZR3n1 ZR3n2
− − − = = =
+ | Φ (x)|2d x |∇ Φ (x)|2d x +
k
kX∈I2ZR3n1 ZR3n2
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−2 −
Φ (x) − =2 = =
+ d x Φ (x)d x +
j∈XI1k∈I1ZR3n1 |xk −xj| ZR3n2
−2 − =2 =
Φ (x) Φ (x) − =
+ d x d x +
j∈XI1k∈I2ZR3n1 ZR3n2 |xk −xj|
−2 −
+ Φ (x)Znd x− Φ=2 (x=)d x= +
j∈IX1n∈N1ZR3n1 |xk −xn| ZR3n2
−2 − =2 =
+ Φ (x) Φ (x)Znd x− d x= +
k∈IX1n∈N2ZR3n1 ZR3n2 |xk −xn|
Z Z
n m
+ .
|x −x |
n<mX∈N1∪N2 n m
We investigate the terms forming the right hand part of (5).
We have that
−2 − − =2 = =
Φ (x)d x= Φ (x)d x= 1. (6)
ZR3n1 ZR3n2
We integrate in
−2 −
Φ (x) −
d x (7)
ZR3n1 |xk −xj|
−
with respect to all variables x except for x . As a result, we obtain
k
−2 − −
−2 − Φ (x)d(x \x )
Φ (x) d x−= ZR3n1−3 k dx . (8)
k
ZR3n1 |xk −xj| ZR3 |xk −xj|
− −
The symbol f(x)d(x \x ) means the integration of the function
k
ZR3n1−3
− − −
f(x) of the variables x= (x ,...,x ) with respect to all variables x except
1 n1
for x (k ∈ 1,...,n ).
k 1
4
= =
The symbol f(x)d(x \x ) means the integration of the function
k
ZR3n1−3
= = =
f(x) of the variables x with respect to all variables x except for x (k ∈
k
n +1,...,n +n ).
1 1 2
The integral
− − −
| Φ (x)|2d(x \x ) (9)
k
ZR3n1−3
has the meaning of the probability density of the event “the k-th electron is
placed in the vicinity of the point x” and can be regarded as the contribution
of the density of the k-th electron to the density of the total electron field in
the system D . We denote it by ρ (x). By analogy, we denote the density
1 1k
of the electron field in the system D by ρ (x).
2 2k
The sum over the indices of all electrons in the system D
1
ρ (x) = ρ (x) (10)
1 1j
Xj∈I1
is the density of the total electron field in the system D at the point x. A
1
similar equality is valid for the system D : ρ (x) = ρ (x).
2 2 2j
Xj∈I2
Note that
− − −
ρ (x)dx = | Φ (x)|2d(x \x )dx =
1 k k
ZR3 jX∈I1ZR3n1−3ZR3n1−3
− − −
= | Φ (x)|2d x= n
1
Xj∈I1ZR3n1
and
ρ (x)dx = n .
2 2
ZR3
We compute the integral
−2 − =2 =
Φ (x) Φ (x) − =
d x d x= (11)
j∈XI1k∈I2ZR3n1 ZR3n2 |xk −xj|
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−2 − − =2 = =
Φ (x)d(x \x ) Φ (x)d(x \x )
k k j
= ZR3n1−3 ZR3n2−3 dx dx =
k j
j∈XI1k∈I2ZR3×R3 |xk −xj|
ρ (x )ρ (x )
= 1j k 2k j dx dx =
k j
j∈XI1k∈I2ZR3×R3 |xk −xj|
ρ (x)ρ (y)
= 1j 2k dxdy =
j∈XI1k∈I2ZR3×R3 |x−y|
ρ (x) ρ (y)
1j 2k
= Xj∈I1 kX∈I2 dxdy =
ZR3×R3 |x−y|
ρ (x)ρ (y)
= 1 2 dxdy.
ZR3×R3 |x−y|
We also compute the integral
−2 − =2 =
Φ (x) Φ (x)Znd x− d x== (12)
k∈IX1n∈N2ZR3n1 ZR3n2 |xk −xn|
−2 −
= Φ (x)Znd x− Φ=2 (x=)d x==
k∈IX1n∈N2ZR3n2 |xk −xn| ZR3n1
−2 −
= Φ (x)Znd x−=
k∈IX1n∈N2ZR3n2 |xk −xn|
−2 − −
Φ (x)Z d(x \x )
n k
= ZR3n1−3 dx =
k
k∈IX1n∈N2ZR3 |xk −xn|
ρ (x )Z
= 1k k ndx =
k
k∈IX1n∈N2ZR3 |xk −xn|
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ρ (x)Z
= 1k ndx =
k∈IX1n∈N2ZR3 |x−xn|
ρ (x)Z
1k n
= kX∈I1 dx =
nX∈N2ZR3 |x−xn|
ρ (x)Z
= 1 ndx.
nX∈N2ZR3 |x−xn|
We split the sum into three terms
Z Z
n m
= (13)
|x −x |
n<mX∈N1∪N2 n m
Z Z
n m
= +
|x −x |
n∈NX1m∈N1 n m
Z Z
n m
+ +
|x −x |
n∈NX2m∪N2 n m
Z Z
n m
+ .
|x −x |
n∈N1∪NX2m∈N1∪N2 n m
By using (11), (12) and (13), we can write (5) in the form
− − −
E = |∇ Φ (x)|2d x + (I) (14)
k
kX∈I1ZR3n1
= = =
+ |∇ Φ (x)|2d x + (II)
k
kX∈I2ZR3n2
−2 −
Φ (x) −
+ d x + (I)
j∈XI1k∈I1ZR3n1 |xk −xj|
=2 =
Φ (x) =
+ d x + (II)
j∈XI2k∈I2ZR3n1 |xk −xj|
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ρ (x)ρ (y)
+ 1 2 dxdy+ (e)
ZR3×R3 |x−y|
−2 −
+ Φ (x)Znd x− + (I)
j∈IX1n∈N1ZR3n1 |xk −xn|
ρ (x)Z
+ 1 ndx+ (e)
nX∈N2ZR3 |x−xn|
=2 =
+ Φ (x)Znd x= + (II)
j∈IX2n∈N2ZR3n1 |xk −xn|
ρ (x)Z
+ 2 ndx+ (e)
nX∈I1ZR3 |x−xn|
Z Z
n m
+ + (I)
|x −x |
n∈NX1m∈N1 n m
Z Z
n m
+ + (II)
|x −x |
n∈NX2m∪N2 n m
Z Z
n m
+ . (e)
|x −x |
n∈NX1m∈∪N2 n m
In(14), bycollecting thetermscorrespondingtothesystem D (theterms
1
marked by I), the terms corresponding to the system D (the terms marked
2
by II) and the remaining terms (those marked by e) that correspond to the
interaction of the elements of different systems, we can write (14) in the form
− =
E = E (Φ)+E (Φ)+E (ρ ,ρ ), (15)
1 2 e 1 2
where
− − − −
E (Φ) = |∇ Φ (x)|2d x + (16)
1 k
kX∈I1ZR3n1
−2 −
Φ (x) −
+ d x +
j∈XI1k∈I1ZR3n1 |xk −xj|
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−2 −
+ Φ (x)Znd x− +
j∈IX1n∈N1ZR3n1 |xk −xn|
Z Z
n m
+
|x −x |
n<mX∈∪N1 n m
is the total energy of the first quantum system neglecting the interaction
with the second quantum system,
= = = =
E (Φ) = |∇ Φ (x)|2d x + (17)
2 k
kX∈I2ZR3n2
=2 =
Φ (x) =
+ d x +
j∈XI2k∈I2ZR3n2 |xk −xj|
=2 =
+ Φ (x)Znd x= +
j∈IX2n∈N2ZR3n2 |xk −xn|
Z Z
n m
+
|x −x |
n<mX∈∪N2 n m
is the total energy of the second quantum system neglecting the interaction
with the first quantum system, and
ρ (x)ρ (y)
E (ρ ,ρ ) = 1 2 dxdy+ (18)
e 1 2 ZR3×R3 |x−y|
ρ (x)Z
+ 1 ndx+
nX∈N2ZR3 |x−xn|
ρ (x)Z
+ 2 ndx+
nX∈N1ZR3 |x−xn|
Z Z
n m
+
|x −x |
n∈NX1m∈N2 n m
describes the energy of the mutual interaction of two quantum systems under
consideration. It is seen that the energy E is an electrostatic-type energy
e
[2].
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We see that in (18) the electrons’ charges are “smeared” over the cor-
responding systems in accordance with the prediction by the Schro¨dinger
equation for the corresponding system. The electron densities ρ (y) and
1
ρ (y) may be deduced in different ways: either computed or determined by
2
an experiment.
The energies E (16) and E (17) of the first and second “rigid” quantum
1 2
systems are constants (do not depend on the translation and/or rotation of
the systems). They do not change when the systems move as rigid bodies.
The energy E (18) depends on the positions of the systems considered as
e
rigid bodies: the positions of the fixed points x and x in the systems and
1 2
the Euler angles [3] ω and ω : E = E (x ,x ,ω ,ω ).
1 2 e e 1 2 1 2
Finally, the original problem (1), (2) takes the form
E = E +E +E (x ,x ,ω ,ω ) → min, (19)
1 2 e 1 2 1 2
or, which is the same,
E (x ,x ,ω ,ω ) → min, (20)
e 1 2 1 2
where
ρ (x)ρ (y)
E = 1 2 dxdy+ (21)
e
ZG(R3)×G(R3) |x−y|
ρ (x)Z
+ 1 n dx+
nX∈N2ZG(R3) |x−G(xn)|
ρ (x)Z
+ 2 n dx
nX∈N1ZG(R3) |x−G(xn)|
Z Z
n m
+
|G(x )−G(x )|
n∈N1∪NX2m∈N1∪N2 n m
and the mapping G acts as follows:
− −
G = G(x ,ω ) :x∈ R3 → S(ω ) x +x ∈ R3, (22)
1 1 1 1
= =
G = G(x ,ω ) :x∈ R3 → S(ω ) x +x ∈ R3.
2 2 2 2
It describes the movement of the systems as rigid bodies. Here S(ω) means
therotationwiththeEuleranglesω. Inotherwords, wearriveattheproblem
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