Euler angles Three giants in the theory of rotations

20
Overview Rotations of the coordinate system Euler angles Rotation of scalar functions Invariance principles — general Representations Spherical harmonics Representation of d mm 0 (α) Spherical vector waves Planar vector waves Cylindrical vector waves List of references Assignment Vector Waves and Probe Compensation Lecture 5: Rotations of Vector Waves Gerhard Kristensson Department of Electrical and Information Technology Lund University December 1, 2011 Overview Rotations of the coordinate system Euler angles Rotation of scalar functions Invariance principles — general Representations Spherical harmonics Representation of d mm 0 (α) Spherical vector waves Planar vector waves Cylindrical vector waves List of references Assignment Three giants in the theory of rotations Leonhard Euler (1707–1783), Swiss mathematician and physicist Sir William Rowan Hamilton (1805–1865), Irish physicist, astronomer, and mathemati- cian Eugene Paul Wigner (1902– 1995), Hungarian American physicist and mathematician (Nobel Prize in Physics in 1963) Rotations of the coordinate system Lecture 5: Rotations of Vector Waves December 1, 2011 Page 1/(38)

Transcript of Euler angles Three giants in the theory of rotations

Page 1: Euler angles Three giants in the theory of rotations

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)

Page 2: Euler angles Three giants in the theory of rotations

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)

Page 3: Euler angles Three giants in the theory of rotations

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)

Page 4: Euler angles Three giants in the theory of rotations

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)

Page 5: Euler angles Three giants in the theory of rotations

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)

Page 6: Euler angles Three giants in the theory of rotations

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)

Page 7: Euler angles Three giants in the theory of rotations

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)

Page 8: Euler angles Three giants in the theory of rotations

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)

Page 9: Euler angles Three giants in the theory of rotations

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)

Page 10: Euler angles Three giants in the theory of rotations

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)

Page 11: Euler angles Three giants in the theory of rotations

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)

Page 12: Euler angles Three giants in the theory of rotations

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)

Page 13: Euler angles Three giants in the theory of rotations

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)

Page 14: Euler angles Three giants in the theory of rotations

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)

Page 15: Euler angles Three giants in the theory of rotations

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)

Page 16: Euler angles Three giants in the theory of rotations

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∇×

rvlm(kr)rulm(kr)rwlm(kr)

=

1√l(l + 1)

(1k∇′×

r′vlm(kr)r′ulm(kr)r′wlm(kr)

=

l∑m′=−l

D(l)m′m(α, β, γ)

1√l(l + 1)

(1k∇′×

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)

Page 17: Euler angles Three giants in the theory of rotations

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)

Page 18: Euler angles Three giants in the theory of rotations

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)

Page 19: Euler angles Three giants in the theory of rotations

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

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)

Page 20: Euler angles Three giants in the theory of rotations

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)