144
O. Ragnisco
=
ρ + − ρ −
2
cos(2(ωt + α)) +
ρ + + ρ −
2
(9)
with 2ω = |a|
1
2 , and α an arbitrary phase.
So, we have a simple harmonic motion with frequency given by |a|
1
2 . We notice
that both the frequency and the amplitude are (algebraic) functions of the constants
of motion ε and l 2 , as well as of the coupling constant k.
Remark 1 One could ask about the possibility of manufacturing superintegrable
examples leading to a radial time evolution such that (˙ r) 2 be given by a higher
degree polynomial. The simplest, and possibly most interesting case, would be a
third degree polynomial, entailing its solvability in terms of Weierstrass P function.
Work is in progress in that direction.
2 The Quantum Model
Hereafter, we will use the standard definitions for the quantum position ˆ
q and
momentum ˆ
p operators:
ˆ
q i ψ(q) = q i , ˆ
p i ψ(q) = −i ¯
h
∂ ψ(q)
∂q i
, [ ˆ
q i , ˆ
p j ] = i ¯
hδ ij ,
i, j = 1, . . . , N,
together with the conventions
∇ =
∂
∂q 1
, . . . ,
∂
∂q N
, Δ = ∇
2
=
∂ 2
∂ 2 q 1
+ · · · +
∂ 2
∂ 2 q N
, q · ∇ =
N
i=1
q i
∂
∂q i
.
Note that the operator | ˆ
q| is defined as | ˆ
q| ψ(q) = |q| ψ(q).
We will apply the so-called direct Schrödinger quantization prescription, and
will take the hyperspherical coordinates (2) together with the usual definition of the
linear momentum operators, namely
ˆ
p r = −i ¯
h
∂
∂r
,
ˆ
p θ j = −i ¯
h
∂
∂θ j
,
j = 1, . . . , N − 1,
(10)
so that the quantum radial Hamiltonian ˆ
H r reads
ˆ
H r =
1
2
1 −
r
ξ
r
ξ
1
ˆ
r N −1 ˆ
p r ˆ
r
N −1
ˆ
p r +
ˆ
L 2
ˆ
r 2
+
k
ξ − ˆ
r
(11)
where ˆ
L 2 is the square of the total quantum angular momentum operator, i.e.
O. Ragnisco
=
ρ + − ρ −
2
cos(2(ωt + α)) +
ρ + + ρ −
2
(9)
with 2ω = |a|
1
2 , and α an arbitrary phase.
So, we have a simple harmonic motion with frequency given by |a|
1
2 . We notice
that both the frequency and the amplitude are (algebraic) functions of the constants
of motion ε and l 2 , as well as of the coupling constant k.
Remark 1 One could ask about the possibility of manufacturing superintegrable
examples leading to a radial time evolution such that (˙ r) 2 be given by a higher
degree polynomial. The simplest, and possibly most interesting case, would be a
third degree polynomial, entailing its solvability in terms of Weierstrass P function.
Work is in progress in that direction.
2 The Quantum Model
Hereafter, we will use the standard definitions for the quantum position ˆ
q and
momentum ˆ
p operators:
ˆ
q i ψ(q) = q i , ˆ
p i ψ(q) = −i ¯
h
∂ ψ(q)
∂q i
, [ ˆ
q i , ˆ
p j ] = i ¯
hδ ij ,
i, j = 1, . . . , N,
together with the conventions
∇ =
∂
∂q 1
, . . . ,
∂
∂q N
, Δ = ∇
2
=
∂ 2
∂ 2 q 1
+ · · · +
∂ 2
∂ 2 q N
, q · ∇ =
N
i=1
q i
∂
∂q i
.
Note that the operator | ˆ
q| is defined as | ˆ
q| ψ(q) = |q| ψ(q).
We will apply the so-called direct Schrödinger quantization prescription, and
will take the hyperspherical coordinates (2) together with the usual definition of the
linear momentum operators, namely
ˆ
p r = −i ¯
h
∂
∂r
,
ˆ
p θ j = −i ¯
h
∂
∂θ j
,
j = 1, . . . , N − 1,
(10)
so that the quantum radial Hamiltonian ˆ
H r reads
ˆ
H r =
1
2
1 −
r
ξ
r
ξ
1
ˆ
r N −1 ˆ
p r ˆ
r
N −1
ˆ
p r +
ˆ
L 2
ˆ
r 2
+
k
ξ − ˆ
r
(11)
where ˆ
L 2 is the square of the total quantum angular momentum operator, i.e.