174
K. B. Wolf et al.
The functions Ξ J,◦,◦ in (13) and (14) of angles in the 2-sphere S
2 can be isolated
through integrating, respectively, over the circles of φ 1 and of φ; this will also set to
zero the corresponding labels m 1 and m in those equations.
We perform this integration for the solutions in the coordinate System I of
cylindrical coordinates (13), without regard for normalization at this stage,
Ψ
I
n,m (x, y) :=
1
2π
π
−π
dφ 1 Φ
I
J,m 1 ,m 2
(γ , φ 1 , φ 2 )
= e
−i
1
2 πm (x
2
+y
2 )
1
2 |m| P
(|m|,0)
n r
1−2(x
2
+y
2 )
e imφ .
(19)
The index ranges are: n = J ∈ Z
+
0 is the principal quantum number, we have now
fixed m 1 = 0 so we write m := −m 2 ; and we have the radial quantum number
n r :=
1
2 (n−|m|) ∈ Z
+
0 . For φ =
1
2 π −φ 2 and the ranges γ |
π/2
0
and φ| π
−π , the
Cartesian coordinates on the disk (x, y) ∈ D and positive half -sphere S
2
+ are
x = ξ 1 = sin γ sin φ 2 , y = ξ 2 = sin γ cos φ 2 , ξ 3 = cos γ 0.
(20)
These are polar coordinates with radius sin γ 1 and angle φ 2 over a circle.
The integration for the System II solutions in polar coordinates on the sphere (14)
yields
Ψ
II
m 1 ,m 2
(x, y) :=
1
2π
π
−π
dφ Φ
II
J,,,μ (χ, θ, φ)
= (1 − x
2 )
1
2 m 1 C
m 1 +1
m 2
(x) P m 1
y
√
1−x 2
.
(21)
Here the principal quantum number is also n = m 1 +m 2 = J ∈ Z
+
0 , while we can
set := m 1 . The coordinates on the disk and half-sphere for the ranges χ | π
0 and
θ | π
0 , are
x = ξ 1 = cos χ, y = ξ 2 = sin χ cos θ, ξ 3 = sin χ sin θ 0.
(22)
This coordinate system can be visualized as polar coordinates on a sphere, projected
on a plane that contains its poles.
We identify this construction to pertain the Zernike system because the defining
Hamiltonian, the first quadratic invariant of the Lie algebra so(4), is the Casimir
operator that serves also to classify hyper-spherical harmonics and generate free
evolution for ideal quantum systems given in (18), surprisingly contains the Zernike
system Hamiltonian (6). For functions Φ J,ν,μ (α, β, φ) such as (13) and (14) with a
factor exp(iμφ), the Laplace–Beltrami operator on S
3 is
K. B. Wolf et al.
The functions Ξ J,◦,◦ in (13) and (14) of angles in the 2-sphere S
2 can be isolated
through integrating, respectively, over the circles of φ 1 and of φ; this will also set to
zero the corresponding labels m 1 and m in those equations.
We perform this integration for the solutions in the coordinate System I of
cylindrical coordinates (13), without regard for normalization at this stage,
Ψ
I
n,m (x, y) :=
1
2π
π
−π
dφ 1 Φ
I
J,m 1 ,m 2
(γ , φ 1 , φ 2 )
= e
−i
1
2 πm (x
2
+y
2 )
1
2 |m| P
(|m|,0)
n r
1−2(x
2
+y
2 )
e imφ .
(19)
The index ranges are: n = J ∈ Z
+
0 is the principal quantum number, we have now
fixed m 1 = 0 so we write m := −m 2 ; and we have the radial quantum number
n r :=
1
2 (n−|m|) ∈ Z
+
0 . For φ =
1
2 π −φ 2 and the ranges γ |
π/2
0
and φ| π
−π , the
Cartesian coordinates on the disk (x, y) ∈ D and positive half -sphere S
2
+ are
x = ξ 1 = sin γ sin φ 2 , y = ξ 2 = sin γ cos φ 2 , ξ 3 = cos γ 0.
(20)
These are polar coordinates with radius sin γ 1 and angle φ 2 over a circle.
The integration for the System II solutions in polar coordinates on the sphere (14)
yields
Ψ
II
m 1 ,m 2
(x, y) :=
1
2π
π
−π
dφ Φ
II
J,,,μ (χ, θ, φ)
= (1 − x
2 )
1
2 m 1 C
m 1 +1
m 2
(x) P m 1
y
√
1−x 2
.
(21)
Here the principal quantum number is also n = m 1 +m 2 = J ∈ Z
+
0 , while we can
set := m 1 . The coordinates on the disk and half-sphere for the ranges χ | π
0 and
θ | π
0 , are
x = ξ 1 = cos χ, y = ξ 2 = sin χ cos θ, ξ 3 = sin χ sin θ 0.
(22)
This coordinate system can be visualized as polar coordinates on a sphere, projected
on a plane that contains its poles.
We identify this construction to pertain the Zernike system because the defining
Hamiltonian, the first quadratic invariant of the Lie algebra so(4), is the Casimir
operator that serves also to classify hyper-spherical harmonics and generate free
evolution for ideal quantum systems given in (18), surprisingly contains the Zernike
system Hamiltonian (6). For functions Φ J,ν,μ (α, β, φ) such as (13) and (14) with a
factor exp(iμφ), the Laplace–Beltrami operator on S
3 is