Post on 23-Jan-2022
Overview
Rotations of thecoordinate systemEuler angles
Rotation of scalarfunctions
Invarianceprinciples —generalRepresentations
SphericalharmonicsRepresentation ofd
mm′ (α)
Spherical vectorwaves
Planar vectorwaves
Cylindrical vectorwaves
List of references
Assignment
Vector Waves and Probe CompensationLecture 5: Rotations of Vector Waves
Gerhard Kristensson
Department of Electrical and Information Technology
Lund University
December 1, 2011
Overview
Rotations of thecoordinate systemEuler angles
Rotation of scalarfunctions
Invarianceprinciples —generalRepresentations
SphericalharmonicsRepresentation ofd
mm′ (α)
Spherical vectorwaves
Planar vectorwaves
Cylindrical vectorwaves
List of references
Assignment
Three giants in the theory of rotations
Leonhard Euler (1707–1783),Swiss mathematician andphysicist
Sir William Rowan Hamilton(1805–1865), Irish physicist,astronomer, and mathemati-cian
Eugene Paul Wigner (1902–1995), Hungarian Americanphysicist and mathematician(Nobel Prize in Physics in1963)
Rotations of the coordinate system Lecture 5: Rotations of Vector Waves December 1, 2011 Page 1/(38)
Overview
Rotations of thecoordinate systemEuler angles
Rotation of scalarfunctions
Invarianceprinciples —generalRepresentations
SphericalharmonicsRepresentation ofd
mm′ (α)
Spherical vectorwaves
Planar vectorwaves
Cylindrical vectorwaves
List of references
Assignment
Two coordinate systems — rotated I
e1
e2
e3
e′1
e′2
e′3
A rotation is characterized by an axis of rotation andan angle of rotation (proved by Euler)
Rotations of the coordinate system Lecture 5: Rotations of Vector Waves December 1, 2011 Page 2/(38)
Overview
Rotations of thecoordinate systemEuler angles
Rotation of scalarfunctions
Invarianceprinciples —generalRepresentations
SphericalharmonicsRepresentation ofd
mm′ (α)
Spherical vectorwaves
Planar vectorwaves
Cylindrical vectorwaves
List of references
Assignment
Two coordinate systems — rotated II
The two sets of units vectors are linearly relatede′1 = e1a11 + e2a12 + e3a13
e′2 = e1a21 + e2a22 + e3a23
e′3 = e1a31 + e2a32 + e3a33
or
e′i =3∑
j=1
ejaij, i = 1, 2, 3
The coefficients aij are the direction cosines
The inverse is (aij is an orthogonal matrix) [8]
ei =3∑
j=1
e′jaji, i = 1, 2, 3
Rotations of the coordinate system Lecture 5: Rotations of Vector Waves December 1, 2011 Page 3/(38)
Overview
Rotations of thecoordinate systemEuler angles
Rotation of scalarfunctions
Invarianceprinciples —generalRepresentations
SphericalharmonicsRepresentation ofd
mm′ (α)
Spherical vectorwaves
Planar vectorwaves
Cylindrical vectorwaves
List of references
Assignment
Two coordinate systems — rotated III
The position vector transforms (two rectilinearsystems)
r =3∑
i=1
xiei =3∑
i,j=1
xie′jaji =3∑
j=1
x′je′j
where
x′j =3∑
j=1
ajixi, i = 1, 2, 3
Rotations of the coordinate system Lecture 5: Rotations of Vector Waves December 1, 2011 Page 4/(38)
Overview
Rotations of thecoordinate systemEuler angles
Rotation of scalarfunctions
Invarianceprinciples —generalRepresentations
SphericalharmonicsRepresentation ofd
mm′ (α)
Spherical vectorwaves
Planar vectorwaves
Cylindrical vectorwaves
List of references
Assignment
Two coordinate systems — rotated IV
A vector u is a geometric quantity with components(u1, u2, u3) in the system (e1, e2, e3), which are related tothe components (u′1, u
′2, u′3) in the system (e′1, e
′2, e′3) in
the following way [3, 8, 9]:
u′i =3∑
j=1
aijuj i = 1, 2, 3
or expressed as column vectors and standard matrixmultiplicationu′1
u′2u′3
=
a11 a12 a13a21 a22 a23a31 a32 a33
u1u2u3
Rotations of the coordinate system Lecture 5: Rotations of Vector Waves December 1, 2011 Page 5/(38)
Overview
Rotations of thecoordinate systemEuler angles
Rotation of scalarfunctions
Invarianceprinciples —generalRepresentations
SphericalharmonicsRepresentation ofd
mm′ (α)
Spherical vectorwaves
Planar vectorwaves
Cylindrical vectorwaves
List of references
Assignment
Euler angles I
A general rotation can be made in three steps (Euler)
e1 e2e′1
e′2
e′3 = e3
α α
e′1
e′3
e′′1
e′′2 = e
′2
e′′3
β
β
e′′1
e′′2
e′′′1
e′′′2e
′′3 = e
′′′3
γ
γ
(e1, e2, e3) and (e′′′1 , e′′′2 , e
′′′3 ), are related to each other by
the three Euler angles (α, β, γ) [4, 8]1 A rotation with the angle α around the e3 axis2 A rotation with the angle β around the e′2 axis3 A rotation with the angle γ around the e′′3 axis
Rotations of the coordinate system Lecture 5: Rotations of Vector Waves December 1, 2011 Page 6/(38)
Overview
Rotations of thecoordinate systemEuler angles
Rotation of scalarfunctions
Invarianceprinciples —generalRepresentations
SphericalharmonicsRepresentation ofd
mm′ (α)
Spherical vectorwaves
Planar vectorwaves
Cylindrical vectorwaves
List of references
Assignment
Euler angles II
The first rotation is represented byx′1x′2x′3
=
cosα sinα 0− sinα cosα 0
0 0 1
x1x2x3
The second rotation is represented byx′′1
x′′2x′′3
=
cosβ 0 − sinβ0 1 0
sinβ 0 cosβ
x′1x′2x′3
The third rotation is represented byx′′′1
x′′′2x′′′3
=
cos γ sin γ 0− sin γ cos γ 0
0 0 1
x′′1x′′2x′′3
Rotations of the coordinate system Lecture 5: Rotations of Vector Waves December 1, 2011 Page 7/(38)
Overview
Rotations of thecoordinate systemEuler angles
Rotation of scalarfunctions
Invarianceprinciples —generalRepresentations
SphericalharmonicsRepresentation ofd
mm′ (α)
Spherical vectorwaves
Planar vectorwaves
Cylindrical vectorwaves
List of references
Assignment
Euler angles III
In a compact notation
r′ = Rz(α)r, r′′ = Ry(β)r′, r′′′ = Rz(γ)r′′
In total, the rotation is made by
r′′′ = Rz(γ)Ry(β)Rz(α)r
The matrices are all orthogonal, R−1 = Rt (correspondsto a change in sign of the angle, e.g., α→ −α)
x′′′1x′′′2x′′′3
=
cos γ sin γ 0− sin γ cos γ 0
0 0 1
cosβ 0 − sinβ0 1 0
sinβ 0 cosβ
cosα sinα 0− sinα cosα 0
0 0 1
x1x2x3
Rotations of the coordinate system Lecture 5: Rotations of Vector Waves December 1, 2011 Page 8/(38)
Overview
Rotations of thecoordinate systemEuler angles
Rotation of scalarfunctions
Invarianceprinciples —generalRepresentations
SphericalharmonicsRepresentation ofd
mm′ (α)
Spherical vectorwaves
Planar vectorwaves
Cylindrical vectorwaves
List of references
Assignment
Euler angles IV
Simplified picture γ = 0
e1e2
e3
e′3r
φ
φ′θθ′
α
β
Rotations of the coordinate system Lecture 5: Rotations of Vector Waves December 1, 2011 Page 9/(38)
Overview
Rotations of thecoordinate systemEuler angles
Rotation of scalarfunctions
Invarianceprinciples —generalRepresentations
SphericalharmonicsRepresentation ofd
mm′ (α)
Spherical vectorwaves
Planar vectorwaves
Cylindrical vectorwaves
List of references
Assignment
Rotation of scalar functions I
Two coordinate frames are related by
r′ = Rr
A scalar function f (r) in the unprimed system is relatedto the function f ′(r′) in the primed system
f ′(r′) def= f (r)
“Same values at the same point in space”
Define an operation PR on scalar functions as f ′ = PRf
(PRf )(Rr) def= f (r)
R rotates coordinates r to r′
PR “rotates” functions f to f ′
Rotation of scalar functions Lecture 5: Rotations of Vector Waves December 1, 2011 Page 10/(38)
Overview
Rotations of thecoordinate systemEuler angles
Rotation of scalarfunctions
Invarianceprinciples —generalRepresentations
SphericalharmonicsRepresentation ofd
mm′ (α)
Spherical vectorwaves
Planar vectorwaves
Cylindrical vectorwaves
List of references
Assignment
Rotation of scalar functions II
Successive rotations, first by R, then by S, i.e.,r′′ = Sr′ = SRr leads to
(PSf ′)(Sr′) def= f ′(r′) = f (r) def
= (PSRf )(r′′)
On the other hand
(PSf ′)(Sr′) def= (PSPRf )(Sr′) = (PSPRf )(r′′)
We concludePSR = PSPR
In particular
I = PI = PR−1R = PR−1PR ⇒ P−1R = PR−1
Notice that the identity I is used both as the identityoperator (for functions) and the identity matrix (forrotations)
Rotation of scalar functions Lecture 5: Rotations of Vector Waves December 1, 2011 Page 11/(38)
Overview
Rotations of thecoordinate systemEuler angles
Rotation of scalarfunctions
Invarianceprinciples —generalRepresentations
SphericalharmonicsRepresentation ofd
mm′ (α)
Spherical vectorwaves
Planar vectorwaves
Cylindrical vectorwaves
List of references
Assignment
Rotation of scalar functions III
The operators PR are linear
(PR(af + bg))(r) = a(PRf )(r) + b(PRg)(r)
and
(PR(fg))(r) = f (R−1r)g(R−1r) = (PRf )(r)(PRg)(r)
Rotation of scalar functions Lecture 5: Rotations of Vector Waves December 1, 2011 Page 12/(38)
Overview
Rotations of thecoordinate systemEuler angles
Rotation of scalarfunctions
Invarianceprinciples —generalRepresentations
SphericalharmonicsRepresentation ofd
mm′ (α)
Spherical vectorwaves
Planar vectorwaves
Cylindrical vectorwaves
List of references
Assignment
Invariance principles — general I
If A(r) is an operator acting on f (r), giving a newfunction g(r) = A(r)f (r), then
(PR(Af ))(r) = (PRg)(r) = g(R−1r) = A(R−1r)f (R−1r)
On the other hand
(PRA P−1R PR︸ ︷︷ ︸=I
f )(r) = (PRA(r)P−1R f )(R−1r)
If the operator A satisfies
A(r) = PRA(r)P−1R ⇔ A(r)PR = PRA(r) (Commutes)
it is called symmetric or invariant under thetransformation R — A commutes with PR (in our casesymmetric under rotations, e.g., A(r) = ∇2) [5, 10]
Invariance principles — general Lecture 5: Rotations of Vector Waves December 1, 2011 Page 13/(38)
Overview
Rotations of thecoordinate systemEuler angles
Rotation of scalarfunctions
Invarianceprinciples —generalRepresentations
SphericalharmonicsRepresentation ofd
mm′ (α)
Spherical vectorwaves
Planar vectorwaves
Cylindrical vectorwaves
List of references
Assignment
Invariance principles — general II
We now study what happens to the eigenfunctions ofan operator A
Let ψn(r) be eigenfunctions to A(r) with N (distinct)eigenvalues λn, i.e.,
A(r)ψn(r) = λnψn(r), n = 1, 2, . . . ,N
Invariance principles — general Lecture 5: Rotations of Vector Waves December 1, 2011 Page 14/(38)
Overview
Rotations of thecoordinate systemEuler angles
Rotation of scalarfunctions
Invarianceprinciples —generalRepresentations
SphericalharmonicsRepresentation ofd
mm′ (α)
Spherical vectorwaves
Planar vectorwaves
Cylindrical vectorwaves
List of references
Assignment
Invariance principles — general III
If the operator A is symmetric under thetransformation R, the rotated functions, PRψn, are alsoeigenfunctions of A with the same eigenvalue
A(r)(PRψn)(r) = (PRA(r)ψn)(r) = λn(PRψn)(r), n = 1, 2, . . . ,N
PRψn can be expanded in the set ψnNn=1 (linearly
independent set of functions)
(PRψn)(r) =N∑
n′=1
Dn′n(R)ψn′(r), (definition of Dn′n(R))
(notice the same argument r)
Invariance principles — general Lecture 5: Rotations of Vector Waves December 1, 2011 Page 15/(38)
Overview
Rotations of thecoordinate systemEuler angles
Rotation of scalarfunctions
Invarianceprinciples —generalRepresentations
SphericalharmonicsRepresentation ofd
mm′ (α)
Spherical vectorwaves
Planar vectorwaves
Cylindrical vectorwaves
List of references
Assignment
Representations I
For consecutive rotations we have
PSPRψn = PS
N∑n′=1
Dn′n(R)ψn′ =N∑
n′=1
Dn′n(R)PSψn′
=N∑
n′,n′′=1
Dn′n(R)Dn′′n′(S)ψn′′ =N∑
n′′=1
[D(S)D(R)]n′′n ψn′′
But also
PSPRψn = PSRψn =N∑
n′=1
Dn′n(SR)ψn′
D(SR) = D(S)D(R), (homeomorphism)
The matrices D(R) represents the group of rotationsInvariance principles — general Lecture 5: Rotations of Vector Waves December 1, 2011 Page 16/(38)
Overview
Rotations of thecoordinate systemEuler angles
Rotation of scalarfunctions
Invarianceprinciples —generalRepresentations
SphericalharmonicsRepresentation ofd
mm′ (α)
Spherical vectorwaves
Planar vectorwaves
Cylindrical vectorwaves
List of references
Assignment
Representations II
In particular
I = D(I) = D(R−1R) = D(R−1)D(R)
orD(R)−1 = D(R−1)
Notice that the identity I is used both as the identityoperator for the matrices D(R) and as the identitymatrix for rotations in R3
Invariance principles — general Lecture 5: Rotations of Vector Waves December 1, 2011 Page 17/(38)
Overview
Rotations of thecoordinate systemEuler angles
Rotation of scalarfunctions
Invarianceprinciples —generalRepresentations
SphericalharmonicsRepresentation ofd
mm′ (α)
Spherical vectorwaves
Planar vectorwaves
Cylindrical vectorwaves
List of references
Assignment
Representations III
We can now combine the results above
(PRψn)(Rr) def= ψn(r), (definition of PR)
and
ψn(Rr) = ψn(r′) def= (PR−1ψn)(R−1r′)
= (PR−1ψn)(r) def=
N∑n′=1
Dn′n(R−1)ψn′(r)
Invariance principles — general Lecture 5: Rotations of Vector Waves December 1, 2011 Page 18/(38)
Overview
Rotations of thecoordinate systemEuler angles
Rotation of scalarfunctions
Invarianceprinciples —generalRepresentations
SphericalharmonicsRepresentation ofd
mm′ (α)
Spherical vectorwaves
Planar vectorwaves
Cylindrical vectorwaves
List of references
Assignment
Representations IV
If the operator A is invariant under the transformationR, then PR are unitary operators, i.e., P†R = P−1
R (theadjoint operator, denoted by a dagger †, is the inverseoperator)
〈f , g〉 = 〈PRf ,PRg〉 =⟨
P†RPRf , g⟩
All “observations” in the original and the rotatedsystem must be identical (otherwise there is a specialcoordinate system, which violated the invariance)
Invariance principles — general Lecture 5: Rotations of Vector Waves December 1, 2011 Page 19/(38)
Overview
Rotations of thecoordinate systemEuler angles
Rotation of scalarfunctions
Invarianceprinciples —generalRepresentations
SphericalharmonicsRepresentation ofd
mm′ (α)
Spherical vectorwaves
Planar vectorwaves
Cylindrical vectorwaves
List of references
Assignment
Representations V
If ψn are orthogonal, the PRψn are also orthogonal (PR
unitary operators), i.e.,
〈PRψn,PRψn′〉 =⟨
P†RPRψn, ψn′⟩
= 〈ψn, ψn′〉 = δnn′
Insert the relation from above
δnn′ = 〈PRψn,PRψn′〉
=
⟨N∑
n′′=1
Dn′′n(R)ψn′′ ,N∑
n′′′=1
Dn′′′n′(R)ψn′′′
⟩
=N∑
n′′=1
D∗n′′n(R)Dn′′n′(R) =N∑
n′′=1
D†nn′′(R)Dn′′n′(R)
and the matrices are unitary matrices, D(R)−1 = D(R)†
Invariance principles — general Lecture 5: Rotations of Vector Waves December 1, 2011 Page 20/(38)
Overview
Rotations of thecoordinate systemEuler angles
Rotation of scalarfunctions
Invarianceprinciples —generalRepresentations
SphericalharmonicsRepresentation ofd
mm′ (α)
Spherical vectorwaves
Planar vectorwaves
Cylindrical vectorwaves
List of references
Assignment
Representations VI
Specifically,
D(R−1) = D(R)−1 = D†(R)
and
ψn(Rr) =N∑
n′=1
Dn′n(R−1)ψn′(r) =N∑
n′=1
D∗nn′(R)ψn′(r)
or the inverse
ψn(r) =N∑
n′=1
Dn′n(R)ψn′(Rr)
Invariance principles — general Lecture 5: Rotations of Vector Waves December 1, 2011 Page 21/(38)
Overview
Rotations of thecoordinate systemEuler angles
Rotation of scalarfunctions
Invarianceprinciples —generalRepresentations
SphericalharmonicsRepresentation ofd
mm′ (α)
Spherical vectorwaves
Planar vectorwaves
Cylindrical vectorwaves
List of references
Assignment
Spherical harmonics I
Let the operator A be the Laplace operator on the unitsphere ∇2
Ω, and we then have N = 2l + 1eigenfunctions ψn, i.e., Ylm(θ, φ)l
m=−l
∇2ΩYlm(θ, φ) = −l(l + 1)Ylm(θ, φ)
where
∇2Ω =
1sin θ
∂
∂θ
(sin θ
∂
∂θ
)+
1sin2 θ
∂2
∂φ2
Notice
∇2f (r, θ, φ) =1r∂2(rf (r, θ, φ))
∂r2 +1r2∇
2Ωf (r, θ, φ)
The rotation is made with the Euler angles (α, β, γ) [4]
Ylm(θ, φ) =∑m′
D(l)m′m(α, β, γ)Ylm′(θ
′, φ′)
Spherical harmonics Lecture 5: Rotations of Vector Waves December 1, 2011 Page 22/(38)
Overview
Rotations of thecoordinate systemEuler angles
Rotation of scalarfunctions
Invarianceprinciples —generalRepresentations
SphericalharmonicsRepresentation ofd
mm′ (α)
Spherical vectorwaves
Planar vectorwaves
Cylindrical vectorwaves
List of references
Assignment
Spherical harmonics II1 A rotation with the angle α around the e3 axis
Ylm(θ, φ) = eimαYlm(θ′, φ′)
2 A rotation with the angle β around the e′2 axis
Ylm(θ′, φ′) =l∑
m′=−l
d(l)m′m(β)Ylm′(θ
′′, φ′′)
3 A rotation with the angle γ around the e′′3 axis
Ylm(θ′′, φ′′) = eimγYlm(θ′′, φ′′′)
In total
D(l)m′m(α, β, γ) = eim′γd(l)
m′m(β)eimα
Spherical harmonics Lecture 5: Rotations of Vector Waves December 1, 2011 Page 23/(38)
Overview
Rotations of thecoordinate systemEuler angles
Rotation of scalarfunctions
Invarianceprinciples —generalRepresentations
SphericalharmonicsRepresentation ofd
mm′ (α)
Spherical vectorwaves
Planar vectorwaves
Cylindrical vectorwaves
List of references
Assignment
Wigner’s functions d(l)mm′(β)
d(l)m′m(β) =
√(l + m′)!(l + m)!
(l− m′)!(l− m)!
∑j
(−1)l−m′−j
(l + m
l− m′ − j
)(l− m
j
)(cos
β
2
)2j+m+m′ (sin
β
2
)2l−2j−m−m′
Another representation1
d(l)m′m(β) =
√(l + m′)!(l + m)!
(l− m′)!(l− m)!
(cos
β
2
)m′+m(sin
β
2
)m′−m
P(m′−m,m′+m)l−m′ (cosβ)
where P(a,b)l (x) are the Jabobi polynomials [1, 4, 7]
1Strictly speaking, m′ − m and m′ + m has to be non-negativeSpherical harmonics Lecture 5: Rotations of Vector Waves December 1, 2011 Page 24/(38)
Overview
Rotations of thecoordinate systemEuler angles
Rotation of scalarfunctions
Invarianceprinciples —generalRepresentations
SphericalharmonicsRepresentation ofd
mm′ (α)
Spherical vectorwaves
Planar vectorwaves
Cylindrical vectorwaves
List of references
Assignment
Jacobi polynomials I
Definition [1, 4, 7]
P(α,β)n (x) =
(−1)n
2nn!(1−x)−α(1+x)−β
dn
dxn
((1− x)α+n(1 + x)β+n
)P(α,β)
0 (x) = 1
P(α,β)1 (x) =
12
(α+ β + 2)x +12
(α− β)
All authors agree on the normalization!!
Special case: If α = β = 0 then P(α,β)n (x) are the
Legendre polynomials Pn(x)
Spherical harmonics Lecture 5: Rotations of Vector Waves December 1, 2011 Page 25/(38)
Overview
Rotations of thecoordinate systemEuler angles
Rotation of scalarfunctions
Invarianceprinciples —generalRepresentations
SphericalharmonicsRepresentation ofd
mm′ (α)
Spherical vectorwaves
Planar vectorwaves
Cylindrical vectorwaves
List of references
Assignment
Jacobi polynomials II
From the definition, we have
d(l)m′m(β) = (−1)m′+md(l)
mm′(β)
and
D(l)m0(α, β, γ) = d(l)
m0(β)eimγ = (−1)m
√4π
2l + 1Ylm(β, γ)
since
d(l)m0(β) =
√(l + m)!(l− m)!
2ml!sinm βP(m,m)
l−m (cosβ)
and (Rodrigues’ generating function [7])
P(m,m)l−m (x) =
2ml!(l + m)!
(1− x2)−m/2Pml (x)
Spherical harmonics Lecture 5: Rotations of Vector Waves December 1, 2011 Page 26/(38)
Overview
Rotations of thecoordinate systemEuler angles
Rotation of scalarfunctions
Invarianceprinciples —generalRepresentations
SphericalharmonicsRepresentation ofd
mm′ (α)
Spherical vectorwaves
Planar vectorwaves
Cylindrical vectorwaves
List of references
Assignment
Spherical scalar waves
vlm(kr)ulm(kr)wlm(kr)
=
jl(kr)
h(1)l (kr)
h(2)l (kr)
Ylm(θ, φ)
=l∑
m′=−l
D(l)m′m(α, β, γ)
jl(kr)
h(1)l (kr)
h(2)l (kr)
Ylm′(θ′, φ′)
=l∑
m′=−l
D(l)m′m(α, β, γ)
vlm′(kr′)ulm′(kr′)wlm′(kr′)
Note that the distance to the origin r is the same inboth the primed and the unprimed system
Spherical vector waves Lecture 5: Rotations of Vector Waves December 1, 2011 Page 27/(38)
Overview
Rotations of thecoordinate systemEuler angles
Rotation of scalarfunctions
Invarianceprinciples —generalRepresentations
SphericalharmonicsRepresentation ofd
mm′ (α)
Spherical vectorwaves
Planar vectorwaves
Cylindrical vectorwaves
List of references
Assignment
Spherical vector waves I
The position vector transforms as (between tworectilinear systems)x′1
x′2x′3
=
a11 a12 a13a21 a22 a23a31 a32 a33
x1x2x3
where
aij =∂x′i∂xj
and aij =∂xj
∂x′iand the gradient transforms as
∂
∂x′i=
3∑j=1
∂xj
∂x′i
∂
∂xj=
3∑j=1
aij∂
∂xj
Spherical vector waves Lecture 5: Rotations of Vector Waves December 1, 2011 Page 28/(38)
Overview
Rotations of thecoordinate systemEuler angles
Rotation of scalarfunctions
Invarianceprinciples —generalRepresentations
SphericalharmonicsRepresentation ofd
mm′ (α)
Spherical vectorwaves
Planar vectorwaves
Cylindrical vectorwaves
List of references
Assignment
Spherical vector waves II
Of particular interest is (somewhat harder to prove,involves the Levi-Civita symbol εijk, permutationsymbol [2])
∇′ × (r′f (r)) = ∇× (rf (r))
Spherical vector waves Lecture 5: Rotations of Vector Waves December 1, 2011 Page 29/(38)
Overview
Rotations of thecoordinate systemEuler angles
Rotation of scalarfunctions
Invarianceprinciples —generalRepresentations
SphericalharmonicsRepresentation ofd
mm′ (α)
Spherical vectorwaves
Planar vectorwaves
Cylindrical vectorwaves
List of references
Assignment
Spherical vector waves III
vn(kr)un(kr)wn(kr)
=1√
l(l + 1)
(1k∇×
)τ
rvlm(kr)rulm(kr)rwlm(kr)
=
1√l(l + 1)
(1k∇′×
)τ
r′vlm(kr)r′ulm(kr)r′wlm(kr)
=
l∑m′=−l
D(l)m′m(α, β, γ)
1√l(l + 1)
(1k∇′×
)τ
r′vlm′(kr′)r′ulm′(kr′)r′wlm′(kr′)
=∑
n′
Dn′n(α, β, γ)
v′n′(kr′)u′n′(kr′)w′n′(kr′)
Dn′n(α, β, γ) = δττ ′δll′D
(l)m′m(α, β, γ) = δττ ′δll′eim′γd(l)
m′m(β)eimα
Note vector identity — primed vectors w.r.t. theprimed spherical unit vectors r′, θ
′, φ′
Spherical vector waves Lecture 5: Rotations of Vector Waves December 1, 2011 Page 30/(38)
Overview
Rotations of thecoordinate systemEuler angles
Rotation of scalarfunctions
Invarianceprinciples —generalRepresentations
SphericalharmonicsRepresentation ofd
mm′ (α)
Spherical vectorwaves
Planar vectorwaves
Cylindrical vectorwaves
List of references
Assignment
Recapitulation — Planar vector waves I
φ1(γ; kr) =
14πk sinα
∇×(
zeikγ·r)
= −β i4π
eikγ·r
φ2(γ; kr) =1
4πk2 sinα∇×
(∇×
(zeikγ·r
))= −α 1
4πeikγ·r
φ3(γ; kr) =1
4πk∇(
zeikγ·r)
= γi
4πeikγ·r
α = x cosα cosβ + y cosα sinβ − z sinα
β = −x sinβ + y cosβ
γ = x sinα cosβ + y sinα sinβ + z cosα
γ transforms as a vector, but α and β do not
Planar vector waves Lecture 5: Rotations of Vector Waves December 1, 2011 Page 31/(38)
Overview
Rotations of thecoordinate systemEuler angles
Rotation of scalarfunctions
Invarianceprinciples —generalRepresentations
SphericalharmonicsRepresentation ofd
mm′ (α)
Spherical vectorwaves
Planar vectorwaves
Cylindrical vectorwaves
List of references
Assignment
Planar vector waves I
φ′τ (γ ′; kr′) =
3∑τ ′=1
Rττ ′φτ ′(γ; kr)
where
Rττ ′ =
cos Ω −i sin Ω 0−i sin Ω cos Ω 0
0 0 1
cos Ω =
sinα cos η − cosα sin η cos(β − ψ)
sinα′
sin Ω =sin η sin(β − ψ)
sinα′
and sinα′ is obtained from
cosα′ = cosα cos η + sinα sin η cos(β − ψ)
Note vector identityPlanar vector waves Lecture 5: Rotations of Vector Waves December 1, 2011 Page 32/(38)
Overview
Rotations of thecoordinate systemEuler angles
Rotation of scalarfunctions
Invarianceprinciples —generalRepresentations
SphericalharmonicsRepresentation ofd
mm′ (α)
Spherical vectorwaves
Planar vectorwaves
Cylindrical vectorwaves
List of references
Assignment
Planar vector waves II
e1e2
e3
e′3r
φ
φ′θθ′
ψ
η
Planar vector waves Lecture 5: Rotations of Vector Waves December 1, 2011 Page 33/(38)
Overview
Rotations of thecoordinate systemEuler angles
Rotation of scalarfunctions
Invarianceprinciples —generalRepresentations
SphericalharmonicsRepresentation ofd
mm′ (α)
Spherical vectorwaves
Planar vectorwaves
Cylindrical vectorwaves
List of references
Assignment
Cylindrical vector waves
Rotation of cylindrical vector wave is less oftenemployed due to an extended singularity alongthe e3 axis (radiating waves)
The only one used is rotation of the regularcylindrical vector waves
The rotation of the regular cylindrical vectorwaves is done by first transforming into planarspherical vector waves (Lecture 7), then a rotationfollowed by a transformation back
Cylindrical vector waves Lecture 5: Rotations of Vector Waves December 1, 2011 Page 34/(38)
Overview
Rotations of thecoordinate systemEuler angles
Rotation of scalarfunctions
Invarianceprinciples —generalRepresentations
SphericalharmonicsRepresentation ofd
mm′ (α)
Spherical vectorwaves
Planar vectorwaves
Cylindrical vectorwaves
List of references
Assignment
List of references I
[1] M. Abramowitz and I. A. Stegun, editors.Handbook of Mathematical Functions.Applied Mathematics Series No. 55. National Bureau of Standards, Washington D.C., 1970.
[2] G. B. Arfken and H. J. Weber.Mathematical methods for physicists.Academic Press, New York, sixth edition, 2005.
[3] L. Brillouin.Tensors in mechanics and elasticity.Academic Press, New York, 1964.
[4] A. R. Edmonds.Angular Momentum in Quantum Mechanics.Princeton University Press, Princeton, New Jersey, second edition, 1960.
[5] M. Hammermesh.Group Theory.Addison-Wesley, Reading, MA, USA, second edition, 1964.
[6] J. E. Hansen, editor.Spherical Near-Field Antenna Measurements.Number 26 in IEE electromagnetic waves series. Peter Peregrinus Ltd., Stevenage, UK, 1988.ISBN: 0-86341-110-X.
[7] G. Kristensson.Second order differential equations — Special functions and their classification.Springer-Verlag, London, 2010.
[8] G. Kristensson.Spherical vector waves, 2011.This booklet is an excerpt of the textbook Gerhard Kristensson, Scattering of ElectromagneticWaves.
List of references Lecture 5: Rotations of Vector Waves December 1, 2011 Page 35/(38)
Overview
Rotations of thecoordinate systemEuler angles
Rotation of scalarfunctions
Invarianceprinciples —generalRepresentations
SphericalharmonicsRepresentation ofd
mm′ (α)
Spherical vectorwaves
Planar vectorwaves
Cylindrical vectorwaves
List of references
Assignment
List of references II
[9] J. Kuipers.Quaternions and rotation sequences: a primer with applications to orbits, aerospace, andvirtual reality.John Wiley & Sons, New Jersey, 2002.
[10] E. Wigner and J. Griffin.Group theory and its application to the quantum mechanics of atomic spectra, volume 4.Academic Press, New York, 1959.
List of references Lecture 5: Rotations of Vector Waves December 1, 2011 Page 36/(38)
Overview
Rotations of thecoordinate systemEuler angles
Rotation of scalarfunctions
Invarianceprinciples —generalRepresentations
SphericalharmonicsRepresentation ofd
mm′ (α)
Spherical vectorwaves
Planar vectorwaves
Cylindrical vectorwaves
List of references
Assignment
Assignment 5
1 Write explicitly down the transformation of thecoordinate system in terms of the Euler angles α,β, and γ = 0, and identify the transformedspherical angles θ′ and φ′ in terms of theunprimed θ and φ
2 Write down the Wigner’s functions, d(l)m′m(β) and
D(l)m′m(α, β, 0), for l = 0 and l = 1
3 Verify the transformation of the sphericalharmonics
Y00(r) =
√1
4π
Y10(r) =
√3
4πcos θ
Y1±1(r) = ∓√
38π
sin θe±iφ
for α, β, and γ = 0 — compare with Item 1
Assignment Lecture 5: Rotations of Vector Waves December 1, 2011 Page 37/(38)
Overview
Rotations of thecoordinate systemEuler angles
Rotation of scalarfunctions
Invarianceprinciples —generalRepresentations
SphericalharmonicsRepresentation ofd
mm′ (α)
Spherical vectorwaves
Planar vectorwaves
Cylindrical vectorwaves
List of references
Assignment
Hint
Aim at expressing the spherical angles for the primedcoordinates in the following form
x′3 = r cos θ′, x′1 ± ix′2 = r sin θ′e±iφ′
Assignment Lecture 5: Rotations of Vector Waves December 1, 2011 Page 38/(38)