Table Of ContentJournal of Nonlinear MathematicalPhysics 1999, V.6, N 1, 51–65. Article
Contact Symmetry of Time-Dependent
Schr¨odinger Equation for a Two-Particle System:
Symmetry Classification of Two-Body Central
Potentials
9
9
9 P. RUDRA
1
n Department of Physics, University of Kalyani, Kalyani, WB, 741-235, India
a
E-mail: [email protected] and [email protected]
J
1
Received July 31, 1998; Accepted September 8, 1998
1
v
Abstract
3
2 Symmetry classification of two-body central potentials in a two-particle Schr¨odinger
0
equation in terms of contact transformations of the equation has been investigated.
1
Explicit calculation has shown that they are of the same four different classes as
0
9 for the point transformations. Thus in this problem contact transformations are not
9 essentially different from point transformations. We have also obtained the detailed
/ algebraic structures of the corresponding Lie algebras and the functional bases of
h
p invariants for the transformationgroups in all the four classes.
-
h
t
a 1 Introduction
m
: Thepositionofcontact transformations [1,2,3,4]liesinbetween thepointtransforma-
v
i tions and theLie-Ba¨cklund transformations [2,5]. However, in studyingthe dynamics of a
X
systemthegroupofcontact transformationshasavery importantposition. Becauseofthe
r
a continuity conditions in quantum mechanics of the wavefunction and its space derivatives,
groups of contact transformations play such important roles in the dynamics of physical
systems.
Pointtransformationgroupsforanysetofdifferentialequationsinvolvetransformations
among the independent space-time variables and the dependent variables. The generators
of the transformation group thus involve only these variables. The groups of contact
transformations, on the other hand, involve these variables as well as the gradients of
the dependent variables. The contact relations connecting the gradients with the original
variables are thus included in the set of differential equations. If the generators do not
involve the gradients in an essential manner, then the contact transformation group is not
essentially different from the point transformation group [2].
Copyright c 1999 by P. Rudra
(cid:13)
52 P. Rudra
In a previous work [6] we have studied symmetry classification of two-body central po-
tentials for the point transformation groups of time-dependent Schro¨dinger equation for
a two-body system. Here, we have done a similar study of the groups of contact trans-
formations of the same system. What we have found here, and what could not be known
withoutthisdetailedanalysis, isthatthegroupsofcontact transformationsforthissystem
arenot essentially differentfromthecorrespondinggroupsof pointtransformations. Thus
we have four classes of two-body central potentials:
1. a constant potential with a 31-parameter Lie group,
2. a harmonic oscillator potential with a 20-parameter Lie group,
3. an inverse square potential with a 16-parameter Lie group, and
4. all other potentials with a 14-parameter Lie group.
Using Lie-Jacobi’s method [4, 5] we have calculated the functional bases of invariants
for all these cases. Any invariant of these groups can be functionally expressed in terms
of these base invariants. For the constant potential case there are 5 base invariants. The
corresponding number for all the other cases is 4.
2 Contact transformations and Schr¨odinger equation
The method [2] for obtaining the group of contact transformation is a generalization of
the method of extended group [1, 2] for obtaining that of point transformations. We give
below the essential points as it appears in our system of 2 particles of masses m and m
1 2
at positions ~r and ~r .
1 2
We use the relative space coordinate ~r = ~r ~r , the centre-of-mass space coordinate
1 2
−
R~ = (m ~r +m ~r )/(m +m ), the reduced mass m = m m /(m +m ) and the total
1 1 2 2 1 2 1 2 1 2
mass M = m +m . It is to be noted that 0 < m/M 1/4. The limit m/M = 1/4 occurs
1 2
≤
when the two particles have equal masses. Positronium atom and a homonuclear diatomic
molecule are two important physical systems which have this limiting value of m/M. The
other limiting value m/M = 0 occurs when one of the masses and hence M is . This is
∞
sameas ignoringthemotion of thecentre-of-mass. Thisdoesnot describeatruetwo-body
system and cannot thus throw any light on the classification of inter-particle potentials.
In terms of these variables, the 2-particle Schro¨dinger equation for the wavefunction
Ψ0 becomes
~ ~
∆0 iΨ0 + Ψcα + Ψrλ v(r)Ψ0 = 0, (1)
≡ q0 2M qcα 2m qrλ −
α λ
X X
with the contact conditions
∆cα Ψ0 Ψcα = 0, (2)
qcα
≡ −
∆rλ Ψ0 Ψrλ = 0. (3)
≡ qrλ −
We have written the physicist’s version of the equation, keeping the Planck’s constant,
the relevant masses and the imaginary number ı. We use a compact notation qa, a = 0,
Contact Symmetry of Time-Dependent Schro¨dinger Equation 53
cα, rλ, where q0 = t, qcα = R , qrλ = r . Here and later on α, λ etc. will mean the
α λ
cartesian components, subscripted variables (other than the cartesian components) will
mean derivatives with respect to those subscripts, the letters c and r will mean center-of-
mass and relative coordinates, and t will mean time.
Equations (2), (3) are actually the definitions of the gradient variables. The inter-
particle potential v(r) has been taken to be of central nature. In terms of the gradient
variables the Schro¨dinger equation is now of the first order.
The generators of the Lie group transformations in the space of qa, Ψa are of the form
∂ ∂
X = ξa(q,Ψ) +χa(q,Ψ) , (4)
∂qa ∂Ψa
a (cid:20) (cid:21)
X
with a= 0, cα, rλ. The arguments of ξa and χa contain the collection of all qa and Ψa. In
theRacahnomenclaturetheξsandχsarecalled thevelocity vectors ofthegenerator X. If
χ0 explicitlycontainsthegradientsΨcα,Ψrλ,thenthecontacttransformationisessentially
different from a point transformation [2].
As in the case of point transformations, the first extention of X is written as
∂
X(1) = X + χa;b . (5)
∂Ψa
a,b qb
X
Here,
χa;b = χaqb − Ψaqb′ξqb′b − Ψaqb′Ψaqb′ξΨb′a′ + Ψaqb′χaΨa′. (6)
b′ a′,b′ a′
X X X
The effect of X(1) on the different ∆s of equations (1), (2), (3) are
~ ~ v(r)′
X(1)∆0 iχ0;t+ χcα,cα+ χrλ,rλ v(r)χ0 ξrλr Ψ = 0,(7)
λ
≡ 2M 2m − − r
α λ λ
X X X
X(1)∆cα χ0;cα χcα = 0, (8)
≡ −
X(1)∆rλ χ0;rλ χrλ = 0. (9)
≡ −
3 Defining equations of group generators
Thegroupofcontacttransformationsfortheequation(1)willbeuniquelyknownwhenthe
velocity vectors ξa and χa willbeobtained. Thesevelocity vectors satisfy an overcomplete
set of differential equations known as the defining equations. These defining equations
are obtained by separately equating to zero the coefficients of the different monomials in
Ψa appearing in equations (7), (8), (9).
qb
In our case we get the defining relations:
ξ0 ξ0 = 0, (10)
Ψcα ≡ Ψrλ
ξ0 +Ψcαξ0 ξ0 +Ψrλξ0 = 0, (11)
qcα Ψ0 ≡ qrλ Ψ0
ξcα ξrλ = 0, (12)
Ψrλ ≡ Ψcα
54 P. Rudra
1 1
δ ξrλ + δ ξcα = 0, (13)
M αβ Ψrµ m λµ Ψcβ
δ ξrσ +δ ξrν = 0, (14)
νλ Ψrµ σµ Ψrλ
δ ξcσ +δ ξcγ = 0, (15)
γα Ψcβ σβ Ψcα
χ0 Ψcβξcβ Ψrµξrµ = 0, (16)
Ψcα − Ψcα − Ψcα
β µ
X X
χ0 Ψcβξcβ Ψrµξrµ = 0, (17)
Ψrλ − Ψrλ − Ψrλ
β µ
X X
χ0 Ψcβξcβ Ψrµξrµ
qcα − qcα − qcα
β µ
X X
(18)
+Ψcα χ0 Ψcβξcβ Ψrµξrµ χcα = 0,
Ψ0 − Ψ0 − Ψ0−
β µ
X X
χ0 Ψcβξcβ Ψrµξrµ
qrλ − qrλ − qrλ
β µ
X X
(19)
+Ψrλ χ0 Ψcβξcβ Ψrµξrµ χrλ = 0,
Ψ0 − Ψ0 − Ψ0−
β µ
X X
δ χ0 Ψcβξcβ Ψrµξrµ ξ0 +ξcγ χcγ +Ψcαξcγ = 0, (20)
αγ Ψ0 − Ψ0 − Ψ0 − q0 qcα − Ψcα Ψ0
β µ
X X
δ χ0 Ψcβξcβ Ψrµξrµ ξ0 +ξrν χrν +Ψrλξrν = 0, (21)
λν Ψ0 − Ψ0 − Ψ0 − q0 qrλ − Ψrλ Ψ0
β µ
X X
~ ~
χcα ξcα +Ψrλξcα = 0, (22)
2M Ψrλ − 2m qrλ Ψ0
h i
~ ~
χrλ ξrλ +Ψcαξrλ = 0, (23)
2m Ψcα − 2M qcα Ψ0
h i
i χ0 Ψcβξcβ Ψrµξrµ +v(r)Ψ0 χ0 Ψcβξcβ Ψrµξrµ
q0 − q0 − q0 Ψ0 − Ψ0 − Ψ0
β µ β µ
X X X X
~ ~
+ χcα +Ψcαχcα + χrλ +Ψrλχrλ (24)
2M qcα Ψ0 2m qrλ Ψ0
Xα (cid:2) (cid:3) Xλ h i
′
v(r)
v(r) χ0+Ψ0ξ0 + Ψ0 r ξrλ = 0.
− q0 rv(r) λ
" #
λ
X
Contact Symmetry of Time-Dependent Schro¨dinger Equation 55
We have obtained the general solution of these defining equations and have indicated the
method in the Appendix 1. The velocity vectors are expressed in terms of the auxilliary
functions
iM im
F0(t,R~,~r) = A (t) fcα(t)′R frλ(t)′r
0 − ~ 0 α− ~ 0 λ
α λ
X X (25)
iM im
+ b(t)′′ R2 + b(t)′′ r2,
4~ α 4~ λ
α λ
X X
1
Fcα(t,R~,~r)= fcα(t) b(t)′R + e fcγR +m fcα;rλr , (26)
0 − 2 α αβγ 1 β 0 λ
βγ λ
X X
1
Frλ(t,R~,~r)= frλ(t) b(t)′r + e frνr M fcα;rνR , (27)
0 − 2 λ λµν 1 µ− 0 α
µν α
X X
and are of the form
ξ0(t)= b(t), (28)
ξcα(t,R~,~r)= Fcα(t.R~,~r), (29)
−
ξrλ(t,R~,~r)= Frλ(t,R~,~r), (30)
−
χ0(t,R~,~r,Ψ0) = f0(t,R~,~r)+Ψ0F0(t,R~,~r), (31)
∂χ0(t,R~,~r,Ψ0)
χcα(t,R~,~r,Ψa) = M Ψrλfcα;rλ
∂R − 0
α
λ
X
(32)
1
Ψcβ δ b(t)′ δ F0(t,R~,~r)+ e fcγ ,
− 2 αβ − αβ αβγ 1
" #
β γ
X X
∂χ0(t,R~,~r,Ψ0)
χrλ(t,R~,~r,Ψa) = +m Ψcαfcα;rλ
∂r 0
λ
α
X (33)
1
Ψrµ δ b(t)′ δ F0(t,R~,~r)+ e frν ,
− 2 λµ − λµ λµν 1
" #
µ ν
X X
where F0(t,R~,~r) and f0(t,R~,~r) satisfy
∂F0(t,R~,~r) ~ ∂2F0(t,R~,~r) ~ ∂2F0(t,R~,~r)
i + +
∂t 2M (∂R )2 2m (∂r )2
α λ
α λ
X X (34)
′
v(r)
v(r)b(t)′+ r Frλ(t,R~,~r)= 0,
λ
− r
λ
X
∂f0(t,R~,~r) ~ ∂2f0(t,R~,~r) ~ ∂2f0(t,R~,~r)
i + +
∂t 2M (∂R )2 2m (∂r )2
α α λ λ (35)
X X
v(r)f0(t,R~,~r) = 0.
−
56 P. Rudra
Name/
Form of the generator
Symbol of the generator
Scaling: ∂ ∂ ∂
Ψ0 + Ψcα + Ψrλ
XS ∂Ψ0 ∂Ψcα ∂Ψrλ
α λ
X X
time translation: ∂
i
Xt ∂t
centre of mass coordinate
∂
space translations: i
− ∂R
Xcα α
T
relative coordinate
∂
space translations: i
− ∂r
Xrλ λ
T
centre of mass coordinate
M ∂
Galilean transformations: tXcα+ R X +Ψ0
T ~ α S ∂Ψcα
Xcα (cid:20) (cid:21)
G
relative coordinate
m ∂
Galilean transformations: tXrλ+ r X +Ψ0
T ~ λ S ∂Ψrλ
Xrλ (cid:20) (cid:21)
G
centre of mass coordinate
∂ ∂
space rotations: i e R +Ψcβ
− αβγ β∂R ∂Ψcγ
XRcα Xβγ (cid:20) γ (cid:21)
relative coordinate
∂ ∂
space rotations: i e r +Ψrµ
− λµν µ∂r ∂Ψrν
Xrλ µν (cid:20) ν (cid:21)
R X
cross-rotations: m ∂ ∂ M ∂ ∂
i r Ψcα +i R Ψrλ
Xcα;rλ − M λ∂R − ∂Ψrλ m α∂r − ∂Ψcα
r (cid:20) α (cid:21) r (cid:20) λ (cid:21)
∂
X 2tXt R Xcα r Xrλ+2v tX +iΨ0
1 − α T − λ T 0 S ∂Ψ0
α λ
X X
X t2Xt t R Xcα t r Xrλ
2 − α T − λ T
α λ
X X
M m
R2 + r2 +4it v t2 X
−"2~ α 2~ λ − 0 # S
α λ
X X
∂ M ∂ m ∂
Ψ0 it + R + r
− "− ∂Ψ0 ~ α∂Ψcα ~ λ∂Ψrλ#
α λ
X X
relative vibrations: ∂ imω imω ∂
e±iωt i r X Ψ0
XVrλ,(±) (cid:20)− ∂rλ ± ~ λ S ± ~ ∂Ψrλ(cid:21)
Table 1. Generators that describe the different Lie Algebras of the four classes of inter-particle
potential.
Here, a prime on the function of a single variable denotes derivative with respect to that
variable, e and e are the permutation symbols and fcα, frλ, fcα;rλ are constants.
αβγ λµν 1 1 0
Since χ0(t,R~,~r,Ψ0) does not contain Ψcα and Ψrλ, the group of contact transformations
for this system is not essentially different from that of point transformations. However,
the generators now contain derivatives with respect to the gradient variables, because the
Contact Symmetry of Time-Dependent Schro¨dinger Equation 57
group of contact transformations is the first extension of the group of point transforma-
tions.
Equation (35) is nothing but the original Schro¨dinger equation and f0(t,R~,~r) is a
solution of equation (1). This is the symmetry corresponding to linear superposition
principleoftheSchro¨dingerequationandformsaninfinitedimensionalinvariantsubgroup
of thetotal group. Thefactor groupmodulothissubgroupis thephysical groupof interest
and will be referred to as the group of contact transformations.
Substituting F0(t,R~,~r) from equation (25) in equation (34) and equating coefficients
of different monomials in the space coordinates to zero, we get
′′′
b(t) = 0, (36)
fcα(t)′′ = 0, (37)
0
′
m v(r)
frλ(t)′′ + frλ(t) =0, (38)
~ 0 r 0
3i 1
′ ′′ ′ ′
iA (t) + b(t) v(r)+ rv(r) b(t) = 0, (39)
0
2 − 2
(cid:20) (cid:21)
and either
′
v(r) = 0 (40)
or
cα;rλ
f = 0. (41)
0
4 Symmetry classification of interparticle potentials
¿From the solution of equations (36)–(41) we get four different classes of interparticle
potentials. This complete symmetry analysis of the 2-particle Schro¨dinger equation as
far as the dynamics of the system is concerned shows that contact transformations do not
enforceanymorerestrictionthanthatalreadyrequiredonthebasisofpointtransformation
symmetry of the system. Their group algebras are the first extensions of the algebras of
point transformation symmetries of the corresponding potentials [6].
These group algebras are described in terms of the generators given in Table 1.
The letters T, G, R and V in the symbols of the generators denote space translational,
galilean, space rotational and vibrational modes described by these generators. We have
kept the imaginary number ı in the forms of the generators so that these can be identified
with the usual quantum mechanical operators for energy, linear and angular momenta.
In actual calculations with the group algebras, the structure constants (commutation
relations in physicists’ parlance) are of the greatest help. They are given in Appendix 2.
AnyLieAlgebraLisasemi-directproductofitsRadicalR andasemisimplepartL/R.
The semisimple part L/R is again a direct sum of ideals which, as subalgebras, are simple
[7, 8]. These characteristics of the Lie Algebras for the different classes of inter-particle
potentials are given in Table 2.
It is to be noted that the generators for the space translational and Galilean as well as
vibrational transformations always belong to the Radical of the Lie Algebra.
58 P. Rudra
Inter-particle Constant Harmonic Inverse square Arbitrary
potential: oscillator
mω2r2 v
1
v(r) =v =v + =v =v ,
0 0 2~ 0− r2 6 0
mω2r2
v + ,
0 2~
v
1
v
0− r2
Generators of X ,Xt,Xcα , X ,Xt, Xcα , X ,Xt,Xcα , X , Xt,
{ S G,T.R { S G,T,R { S G,T,R { S
Lie Algebra, Xrλ ,Xcα;rλ, Xrλ,Xrλ Xrλ, X ,X Xcα ,Xrλ
G,T,R R V,(±)} R 1 2} G,T,R R }
L X ,X
1 2
}
Dimension 31 20 16 14
Solvability, None None None None
Nilpotency,
Simplicity,
Semisimplicity
Centre, Z(L) X X X X
S S S S
{ } { } { } { }
Radical, R X ,Xcα ,Xrλ X ,Xcα ,Xrλ X ,Xcα X ,Xcα
{ S G,T G,T} { S G,T V,(±)} { S G,T} { S G,T}
Semisimple I = Xt,X ,X , I = Xt , I = Xt,X ,X , I = Xt ,
1 1 2 1 1 1 2 1
{ } { } { } { }
part as I = Xcα,Xrλ, I = Xcα , I = Xcα , I = Xcα ,
2 { R R 2 { R } 2 { R } 2 { R }
direct sum Xcα;rλ I = Xrλ I = Xrλ I = Xrλ
⊕ } 3 { R } 3 { R } 3 { R }
L/R= I
i i
of simple
P
ideals
Cartan X ,Xc3,Xr3, X ,Xt,Xc3 , X ,Xc3,Xr3, X ,Xt,
{ S R R { S G,T,R { S R { S
subalgebra Xc3;r3,X Xr3 X , Xc3 ,Xr3
1} R } 1} G,T,R R }
H
Table 2. Characteristics of the Lie Algebras of the Symmetry groups of the different classes of
inter-particle potential.
WenotethattheRadical, beingsolvable,hasonly1-dimensionalirreduciblerepresenta-
tions (irreps)[7]. Thusthe irrepsofL are obtained if theirrepsof L/R areknown. Tothis
end we give in Table 3 the algebraic characteristics of the simple subalgebras appearing
in the direct sum of L/R for the different classes of inter-particle potential.
5 Functional bases of invariants
In order to investigate integrability of a dynamical system we require the functional bases
of invariants in terms of which all invariants of the dynamical system can be expressed
functionally. These base invariants of a Lie algebra L generate the Centre of the Universal
Enveloping algebra of L. Since they commute with all the generators of L, they appear as
conserved quantities of the system. Lie’s method [9, 10, 11, 12] has been utilized to obtain
the base invariants for the four symmetry groups of Section 4. If the functional base has
s invariants I ,I ,...,I , each a function of the r generators X of the symmetry group,
1 2 s a
then
[X ,I ] = 0, a = 1,...,r, b= 1,...,s.
a b
Contact Symmetry of Time-Dependent Schro¨dinger Equation 59
Algebra: L L L L L
1 2 3 4
Generators: Xt,X ,X Xcα, Xrλ, Xcα;rλ Xα Xt
{ 1 2} { R R } { R} { }
X
i
{ }
Dimension 3 15 3 1
Cartan
i
subalgebra: X Xc3,Xr3, Xc3;r3 X3 Xt
−2 1 { R R } { R} { }
(cid:26) (cid:27)
H
Rank 1 3 1 1
Base of
1 1
simple − −
α=1 α = 0 , α = 0 , α = 1 void
roots: 1 2
1 1
∆ −
1
α = 1
3
0
α , α , α ,
Root ± 1 ± 2 ± 3
α (α +α ), (α +α ), α void
system ± ± 1 3 ± 2 3 ±
(α +α +α )
1 2 3
±
Isomorphy
O(3) O(6) O(3) U(1)
to
E±α1 = XRc1∓iXRc2
+ Xc1;r3 iXc2;r3 ,
(cid:0) (cid:1)
∓
E±(cid:0)α2 = XRc1∓iXRc(cid:1)2
Xc1;r3 iXc2;r3 ,
(cid:0) (cid:1)
− ∓
E±α3(cid:0)= Xc1;r1−Xc2(cid:1);r2
Stasentdoafrd Eα =X2 ±i Xc(cid:0)1;r2+Xc2;r1 , (cid:1) E±α = void
generators E−α =Xt E±(α1(cid:0)+α3) = XRr1±iX(cid:1)Rr2 XR1 ±iXR2
Xc3;r1 iXc3;r2 ,
(cid:0) (cid:1)
− ±
E = Xr1 iXr2
±(α(cid:0)2+α3) R ± (cid:1)R
+ Xc3;r1 iXc3;r2 ,
(cid:0) (cid:1)
±
E = Xc1;r1+Xc2;r2
±(α1+α2(cid:0)+α3) (cid:1)
i Xc1;r2 Xc2;r1
(cid:0) (cid:1)
± −
(cid:0) (cid:1)
Table 3. Characteristics of the simple subalgebras forming the semisimple parts in Table 2.
If I is any other invariant so that
[X ,I] = 0, a= 1,...,r,
a
then I can be functionally expressed as
I I(I ,...,I ).
1 s
≡
If the base has the constant as its only member, then the system is completely chaotic. If
on the other hand s = r and all the generators give mutually commuting invariants, then
the system is fully integrable. Actual dynamical systems are almost always in between
60 P. Rudra
these two extreme cases. In the four symmetry classes obtained in Section 4, the same
thing happens.
In Table 4 we define auxilliary operators in terms of which the functional bases of
invariants for the four classes of distinct inter-particle potentials are given in Table 5.
Inter particle
− Auxilliary operators
potential
~
Ycα =X Xcα+ e XcβXcγ,
R S R M αβγ T G
βγ
~ X
Yrλ =X Xrλ+ e XrµXrν,
R S R m λµν T G
µν
X ~
Ycα;rλ =X Xcα;rλ+ XcαXrλ XcαXrλ ,
Constant S √mM G T − T G
potential Yt =X Xt+v (X )2+ ~ (cid:2) (Xcα)2+ ~ (cid:3) Xrλ 2,
S 0 S 2M T 2m T
α λ
~ X ~ X(cid:0) (cid:1)
Y =X X 4i(X )2+ XcαXcα+ XrλXrλ,
1 S 1− S M T G m T G
α λ
~ X ~ X
Y =X X + (Xcα)2+ Xrλ 2
2 S 2 2M G 2m G
α λ
~X X(cid:0) (cid:1)
Ycα =X Xcα+ e XcβXcγ,
R S R M αβγ T G
βγ
Harmonic i~X
oscillator YRrλ =XSXRrλ+ 2mω eλµνXVrµ,(+)XVrν,(−),
potential µν
~ X ~
Yt =X Xt+ (Xcα)2+ Xrλ Xrλ
S 2M T 2m V,(+) V,(−)
α λ
X X
~
Ycα =X Xcα+ e XcβXcγ,
R S R M αβγ T G
βγ
X ~
Inverse Yt =XSXt+v0(XS)2+ 2M (XTcα)2,
square α
~ X
potential Y =X X 4i(X )2+ XcαXcα,
1 S 1− S M T G
α
~ X
Y =X X + (Xcα)2
2 S 2 2M G
α
X
~
Ycα =X Xcα+ e XcβXcγ,
Arbitrary R S R M αβγ T G
βγ
potential ~ X
Yt =X Xt+ (Xcα)2
S 2M T
α
X
Table 4. Auxilliary operators for the four distinct classes of inter-particle potentials.
In all these cases I is the scaling operator and It is essentially the energy operator, Ic
S R
and Ir are the c.m.coordinate and the relative coordinate angular momentum operators.
R
Except for the case of constant potential, these are theonly invariants. Theonly potential
c;r
that has more invariants is the constant potential and the two extra invariants I and
3
c;r
I of degrees 3 and 4 in the generators make this apparently simplesystem actually more
4
complicated yet more systematic.
