# Euler angles Three giants in the theory of rotations

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Vector Waves and Probe Compensation - Lecture 5: Rotations of Vector WavesRotation of scalar functions

Invariance principles — general Representations

mm′ (α)

Assignment

Vector Waves and Probe Compensation Lecture 5: Rotations of Vector Waves

Gerhard Kristensson

Lund University

Rotation of scalar functions

Invariance principles — general Representations

mm′ (α)

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)

Overview

Rotation of scalar functions

Invariance principles — general Representations

mm′ (α)

e1

e2

e3

e′1

e′2

e′3

A rotation is characterized by an axis of rotation and an angle of rotation (proved by Euler)

Rotations of the coordinate system Lecture 5: Rotations of Vector Waves December 1, 2011 Page 2/(38)

Overview

Rotation of scalar functions

Invariance principles — general Representations

mm′ (α)

Two coordinate systems — rotated II

The two sets of units vectors are linearly related e′1 = e1a11 + e2a12 + e3a13

e′2 = e1a21 + e2a22 + e3a23

e′3 = e1a31 + e2a32 + e3a33

or

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

Rotation of scalar functions

Invariance principles — general Representations

mm′ (α)

The position vector transforms (two rectilinear systems)

r = 3∑

i=1

xiei = 3∑

i,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

Rotation of scalar functions

Invariance principles — general Representations

mm′ (α)

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 to the components (u′1, u

′ 2, u ′ 3) in the system (e′1, e

′ 2, e ′ 3) in

u′i = 3∑

aijuj i = 1, 2, 3

or expressed as column vectors and standard matrix multiplicationu′1

u′2 u′3

u1 u2 u3

Rotations of the coordinate system Lecture 5: Rotations of Vector Waves December 1, 2011 Page 5/(38)

Overview

Rotation of scalar functions

Invariance principles — general Representations

mm′ (α)

A general rotation can be made in three steps (Euler)

e1 e2e ′ 1

′′′ 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 axis 2 A rotation with the angle β around the e′2 axis 3 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

Rotation of scalar functions

Invariance principles — general Representations

mm′ (α)

Euler angles II

=

0 0 1

x1 x2 x3

x′′2 x′′3

sinβ 0 cosβ

The third rotation is represented byx′′′1

x′′′2 x′′′3

cos γ sin γ 0 − sin γ cos γ 0

0 0 1

x′′1 x′′2 x′′3

Rotations of the coordinate system Lecture 5: Rotations of Vector Waves December 1, 2011 Page 7/(38)

Overview

Rotation of scalar functions

Invariance principles — general Representations

mm′ (α)

In total, the rotation is made by

r′′′ = Rz(γ)Ry(β)Rz(α)r

The matrices are all orthogonal, R−1 = Rt (corresponds to a change in sign of the angle, e.g., α→ −α)

x′′′1 x′′′2 x′′′3

=

cos γ sin γ 0 − sin γ cos γ 0

0 0 1

sinβ 0 cosβ

0 0 1

x1 x2 x3

Rotations of the coordinate system Lecture 5: Rotations of Vector Waves December 1, 2011 Page 8/(38)

Overview

Rotation of scalar functions

Invariance principles — general Representations

mm′ (α)

α

β

Rotations of the coordinate system Lecture 5: Rotations of Vector Waves December 1, 2011 Page 9/(38)

Overview

Rotation of scalar functions

Invariance principles — general Representations

mm′ (α)

Two coordinate frames are related by

r′ = Rr

A scalar function f (r) in the unprimed system is related to 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

Rotation of scalar functions

Invariance principles — general Representations

mm′ (α)

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′′)

We conclude PSR = PSPR

In particular

I = PI = PR−1R = PR−1PR ⇒ P−1 R = PR−1

Notice that the identity I is used both as the identity operator (for functions) and the identity matrix (for rotations)

Rotation of scalar functions Lecture 5: Rotations of Vector Waves December 1, 2011 Page 11/(38)

Overview

Rotation of scalar functions

Invariance principles — general Representations

mm′ (α)

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

Rotation of scalar functions

Invariance principles — general Representations

mm′ (α)

Invariance principles — general I

If A(r) is an operator acting on f (r), giving a new function 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

f )(r) = (PRA(r)P−1 R f )(R−1r)

If the operator A satisfies

A(r) = PRA(r)P−1 R ⇔ A(r)PR = PRA(r) (Commutes)

it is called symmetric or invariant under the transformation R — A commutes with PR (in our case symmetric 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

Rotation of scalar functions

Invariance principles — general Representations

mm′ (α)

Invariance principles — general II

We now study what happens to the eigenfunctions of an 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

Rotation of scalar functions

Invariance principles — general Representations

mm′ (α)

Invariance principles — general III

If the operator A is symmetric under the transformation R, the rotated functions, PRψn, are also eigenfunctions 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 {ψn}N n=1 (linearly

independent set of functions)

(notice the same argument r)

Invariance principles — general Lecture 5: Rotations of Vector Waves December 1, 2011 Page 15/(38)

Overview

Rotation of scalar functions

Invariance principles — general Representations

mm′ (α)

PSPRψn = PS

n′′=1

The matrices D(R) represents the group of rotations Invariance principles — general Lecture 5: Rotations of Vector Waves December 1, 2011 Page 16/(38)

Overview

Rotation of scalar functions

Invariance principles — general Representations

mm′ (α)

or D(R)−1 = D(R−1)

Notice that the identity I is used both as the identity operator for the matrices D(R) and as the identity matrix for rotations in R3

Invariance principles — general Lecture 5: Rotations of Vector Waves December 1, 2011 Page 17/(38)

Overview

Rotation of scalar functions

Invariance principles — general Representations

mm′ (α)

(PRψn)(Rr) def = ψn(r), (definition of PR)

and

= (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

Rotation of scalar functions

Invariance principles — general Representations

mm′ (α)

Representations IV

If the operator A is invariant under the transformation R, then PR are unitary operators, i.e., P†R = P−1

R (the adjoint operator, denoted by a dagger †, is the inverse operator)

f , g = PRf ,PRg = ⟨

P†RPRf , g ⟩

All “observations” in the original and the rotated system must be identical (otherwise there is a special coordinate system, which violated the invariance)

Invariance principles — general Lecture 5: Rotations of Vector Waves December 1, 2011 Page 19/(38)

Overview

Rotation of scalar functions

Invariance principles — general Representations

mm′ (α)

If ψn are orthogonal, the PRψn are also orthogonal (PR

unitary operators), i.e.,

δnn′ = PRψn,PRψn′

n′′=1

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

Rotation of scalar functions

Invariance principles — general Representations

mm′ (α)

and

n′=1

Dn′n(R)ψn′(Rr)

Invariance principles — general Lecture 5: Rotations of Vector Waves December 1, 2011 Page 21/(38)

Overview

Rotation of scalar functions

Invariance principles — general Representations

mm′ (α)

Spherical harmonics I

Let the operator A be the Laplace operator on the unit sphere ∇2

, and we then have N = 2l + 1 eigenfunctions ψn, i.e., {Ylm(θ, φ)}l

m=−l

where

∇2f (r, θ, φ) = 1 r ∂2(rf (r, θ, φ))

∂r2 + 1 r2∇

2 f (r, θ, φ)

The rotation is made with the Euler angles (α, β, γ) [4]

Ylm(θ, φ) = ∑ m′

′, φ′)

Spherical harmonics Lecture 5: Rotations of Vector Waves December 1, 2011 Page 22/(38)

Overview

Rotation of scalar functions

Invariance principles — general Representations

mm′ (α)

Assignment

Spherical harmonics II 1 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∑

′′, φ′′)

3 A rotation with the angle γ around the e′′3 axis

Ylm(θ′′, φ′′) = eimγYlm(θ′′, φ′′′)

m′m(β)eimα

Spherical harmonics Lecture 5: Rotations of Vector Waves December 1, 2011 Page 23/(38)

Overview

Rotation of scalar functions

Invariance principles — general Representations

mm′ (α)

d(l) m′m(β) =

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-negative Spherical harmonics Lecture 5: Rotations of Vector Waves December 1, 2011 Page 24/(38)

Overview

Rotation of scalar functions

Invariance principles — general Representations

mm′ (α)

dn

dxn

) P(α,β)

(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

Rotation of scalar functions

Invariance principles — general Representations

mm′ (α)

mm′(β)

m0(β)eimγ = (−1)m

P(m,m) l−m (x) =

(1− x2)−m/2Pm l (x)

Spherical harmonics Lecture 5: Rotations of Vector Waves December 1, 2011 Page 26/(38)

Overview

Rotation of scalar functions

Invariance principles — general Representations

mm′ (α)

jl(kr)

vlm′(kr′) ulm′(kr′) wlm′(kr′)

Note that the distance to the origin r is the same in both the primed and the unprimed system

Spherical vector waves Lecture 5: Rotations of Vector Waves December 1, 2011 Page 27/(38)

Overview

Rotation of scalar functions

Invariance principles — general Representations

mm′ (α)

Spherical vector waves I

The position vector transforms as (between two rectilinear systems)x′1

x′2 x′3

x1 x2 x3

∂

aij ∂

∂xj

Spherical vector waves Lecture 5: Rotations of Vector Waves December 1, 2011 Page 28/(38)

Overview

Rotation of scalar functions

Invariance principles — general Representations

mm′ (α)

Spherical vector waves II

Of particular interest is (somewhat harder to prove, involves the Levi-Civita symbol εijk, permutation symbol [2])

∇′ × (r′f (r)) = ∇× (rf (r))

Spherical vector waves Lecture 5: Rotations of Vector Waves December 1, 2011 Page 29/(38)

Overview

Rotation of scalar functions

Invariance principles — general Representations

mm′ (α)

=

1√ l(l + 1)

= ∑

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. the primed spherical unit vectors r′, θ

′ , φ ′

Spherical vector waves Lecture 5: Rotations of Vector Waves December 1, 2011 Page 30/(38)

Overview

Rotation of scalar functions

Invariance principles — general Representations

mm′ (α)

φ1(γ; kr) =

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

Rotation of scalar functions

Invariance principles — general Representations

mm′ (α)

where

Rττ ′ =

cos −i sin 0 −i sin cos 0

0 0 1

sinα′

sinα′

cosα′ = cosα cos η + sinα sin η cos(β − ψ)

Note vector identity Planar vector waves Lecture 5: Rotations of Vector Waves December 1, 2011 Page 32/(38)

Overview

Rotation of scalar functions

Invariance principles — general Representations

mm′ (α)

ψ

η

Planar vector waves Lecture 5: Rotations of Vector Waves December 1, 2011 Page 33/(38)

Overview

Rotation of scalar functions

Invariance principles — general Representations

mm′ (α)

Cylindrical vector waves

Rotation of cylindrical vector wave is less often employed due to an extended singularity along the e3 axis (radiating waves)

The only one used is rotation of the regular cylindrical vector waves

The rotation of the regular cylindrical vector waves is done by first transforming into planar spherical vector waves (Lecture 7), then a rotation followed by a transformation back

Cylindrical vector waves Lecture 5: Rotations of Vector Waves December 1, 2011 Page 34/(38)

Overview

Rotation of scalar functions

Invariance principles — general Representations

mm′ (α)

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 Electromagnetic Waves.

List of references Lecture 5: Rotations of Vector Waves December 1, 2011 Page 35/(38)

Overview

Rotation of scalar functions

Invariance principles — general Representations

mm′ (α)

List of references II

[9] J. Kuipers. Quaternions and rotation sequences: a primer with applications to orbits, aerospace, and virtual 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

Rotation of scalar functions

Invariance principles — general Representations

mm′ (α)

Assignment 5

1 Write explicitly down the transformation of the coordinate system in terms of the Euler angles α, β, and γ = 0, and identify the transformed spherical angles θ′ and φ′ in terms of the unprimed 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 spherical harmonics

Y00(r) =

√ 1

4π

Y10(r) =

√ 3

for α, β, and γ = 0 — compare with Item 1

Assignment Lecture 5: Rotations of Vector Waves December 1, 2011 Page 37/(38)

Overview

Rotation of scalar functions

Invariance principles — general Representations

mm′ (α)

Assignment

Hint

Aim at expressing the spherical angles for the primed coordinates 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)

Rotations of the coordinate system

Rotation of scalar functions

Invariance principles — general Representations

mm′ (α)

Assignment

Vector Waves and Probe Compensation Lecture 5: Rotations of Vector Waves

Gerhard Kristensson

Lund University

Rotation of scalar functions

Invariance principles — general Representations

mm′ (α)

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)

Overview

Rotation of scalar functions

Invariance principles — general Representations

mm′ (α)

e1

e2

e3

e′1

e′2

e′3

A rotation is characterized by an axis of rotation and an angle of rotation (proved by Euler)

Rotations of the coordinate system Lecture 5: Rotations of Vector Waves December 1, 2011 Page 2/(38)

Overview

Rotation of scalar functions

Invariance principles — general Representations

mm′ (α)

Two coordinate systems — rotated II

The two sets of units vectors are linearly related e′1 = e1a11 + e2a12 + e3a13

e′2 = e1a21 + e2a22 + e3a23

e′3 = e1a31 + e2a32 + e3a33

or

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

Rotation of scalar functions

Invariance principles — general Representations

mm′ (α)

The position vector transforms (two rectilinear systems)

r = 3∑

i=1

xiei = 3∑

i,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

Rotation of scalar functions

Invariance principles — general Representations

mm′ (α)

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 to the components (u′1, u

′ 2, u ′ 3) in the system (e′1, e

′ 2, e ′ 3) in

u′i = 3∑

aijuj i = 1, 2, 3

or expressed as column vectors and standard matrix multiplicationu′1

u′2 u′3

u1 u2 u3

Rotations of the coordinate system Lecture 5: Rotations of Vector Waves December 1, 2011 Page 5/(38)

Overview

Rotation of scalar functions

Invariance principles — general Representations

mm′ (α)

A general rotation can be made in three steps (Euler)

e1 e2e ′ 1

′′′ 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 axis 2 A rotation with the angle β around the e′2 axis 3 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

Rotation of scalar functions

Invariance principles — general Representations

mm′ (α)

Euler angles II

=

0 0 1

x1 x2 x3

x′′2 x′′3

sinβ 0 cosβ

The third rotation is represented byx′′′1

x′′′2 x′′′3

cos γ sin γ 0 − sin γ cos γ 0

0 0 1

x′′1 x′′2 x′′3

Rotations of the coordinate system Lecture 5: Rotations of Vector Waves December 1, 2011 Page 7/(38)

Overview

Rotation of scalar functions

Invariance principles — general Representations

mm′ (α)

In total, the rotation is made by

r′′′ = Rz(γ)Ry(β)Rz(α)r

The matrices are all orthogonal, R−1 = Rt (corresponds to a change in sign of the angle, e.g., α→ −α)

x′′′1 x′′′2 x′′′3

=

cos γ sin γ 0 − sin γ cos γ 0

0 0 1

sinβ 0 cosβ

0 0 1

x1 x2 x3

Rotations of the coordinate system Lecture 5: Rotations of Vector Waves December 1, 2011 Page 8/(38)

Overview

Rotation of scalar functions

Invariance principles — general Representations

mm′ (α)

α

β

Rotations of the coordinate system Lecture 5: Rotations of Vector Waves December 1, 2011 Page 9/(38)

Overview

Rotation of scalar functions

Invariance principles — general Representations

mm′ (α)

Two coordinate frames are related by

r′ = Rr

A scalar function f (r) in the unprimed system is related to 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

Rotation of scalar functions

Invariance principles — general Representations

mm′ (α)

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′′)

We conclude PSR = PSPR

In particular

I = PI = PR−1R = PR−1PR ⇒ P−1 R = PR−1

Notice that the identity I is used both as the identity operator (for functions) and the identity matrix (for rotations)

Rotation of scalar functions Lecture 5: Rotations of Vector Waves December 1, 2011 Page 11/(38)

Overview

Rotation of scalar functions

Invariance principles — general Representations

mm′ (α)

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

Rotation of scalar functions

Invariance principles — general Representations

mm′ (α)

Invariance principles — general I

If A(r) is an operator acting on f (r), giving a new function 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

f )(r) = (PRA(r)P−1 R f )(R−1r)

If the operator A satisfies

A(r) = PRA(r)P−1 R ⇔ A(r)PR = PRA(r) (Commutes)

it is called symmetric or invariant under the transformation R — A commutes with PR (in our case symmetric 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

Rotation of scalar functions

Invariance principles — general Representations

mm′ (α)

Invariance principles — general II

We now study what happens to the eigenfunctions of an 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

Rotation of scalar functions

Invariance principles — general Representations

mm′ (α)

Invariance principles — general III

If the operator A is symmetric under the transformation R, the rotated functions, PRψn, are also eigenfunctions 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 {ψn}N n=1 (linearly

independent set of functions)

(notice the same argument r)

Invariance principles — general Lecture 5: Rotations of Vector Waves December 1, 2011 Page 15/(38)

Overview

Rotation of scalar functions

Invariance principles — general Representations

mm′ (α)

PSPRψn = PS

n′′=1

The matrices D(R) represents the group of rotations Invariance principles — general Lecture 5: Rotations of Vector Waves December 1, 2011 Page 16/(38)

Overview

Rotation of scalar functions

Invariance principles — general Representations

mm′ (α)

or D(R)−1 = D(R−1)

Notice that the identity I is used both as the identity operator for the matrices D(R) and as the identity matrix for rotations in R3

Invariance principles — general Lecture 5: Rotations of Vector Waves December 1, 2011 Page 17/(38)

Overview

Rotation of scalar functions

Invariance principles — general Representations

mm′ (α)

(PRψn)(Rr) def = ψn(r), (definition of PR)

and

= (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

Rotation of scalar functions

Invariance principles — general Representations

mm′ (α)

Representations IV

If the operator A is invariant under the transformation R, then PR are unitary operators, i.e., P†R = P−1

R (the adjoint operator, denoted by a dagger †, is the inverse operator)

f , g = PRf ,PRg = ⟨

P†RPRf , g ⟩

All “observations” in the original and the rotated system must be identical (otherwise there is a special coordinate system, which violated the invariance)

Invariance principles — general Lecture 5: Rotations of Vector Waves December 1, 2011 Page 19/(38)

Overview

Rotation of scalar functions

Invariance principles — general Representations

mm′ (α)

If ψn are orthogonal, the PRψn are also orthogonal (PR

unitary operators), i.e.,

δnn′ = PRψn,PRψn′

n′′=1

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

Rotation of scalar functions

Invariance principles — general Representations

mm′ (α)

and

n′=1

Dn′n(R)ψn′(Rr)

Invariance principles — general Lecture 5: Rotations of Vector Waves December 1, 2011 Page 21/(38)

Overview

Rotation of scalar functions

Invariance principles — general Representations

mm′ (α)

Spherical harmonics I

Let the operator A be the Laplace operator on the unit sphere ∇2

, and we then have N = 2l + 1 eigenfunctions ψn, i.e., {Ylm(θ, φ)}l

m=−l

where

∇2f (r, θ, φ) = 1 r ∂2(rf (r, θ, φ))

∂r2 + 1 r2∇

2 f (r, θ, φ)

The rotation is made with the Euler angles (α, β, γ) [4]

Ylm(θ, φ) = ∑ m′

′, φ′)

Spherical harmonics Lecture 5: Rotations of Vector Waves December 1, 2011 Page 22/(38)

Overview

Rotation of scalar functions

Invariance principles — general Representations

mm′ (α)

Assignment

Spherical harmonics II 1 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∑

′′, φ′′)

3 A rotation with the angle γ around the e′′3 axis

Ylm(θ′′, φ′′) = eimγYlm(θ′′, φ′′′)

m′m(β)eimα

Spherical harmonics Lecture 5: Rotations of Vector Waves December 1, 2011 Page 23/(38)

Overview

Rotation of scalar functions

Invariance principles — general Representations

mm′ (α)

d(l) m′m(β) =

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-negative Spherical harmonics Lecture 5: Rotations of Vector Waves December 1, 2011 Page 24/(38)

Overview

Rotation of scalar functions

Invariance principles — general Representations

mm′ (α)

dn

dxn

) P(α,β)

(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

Rotation of scalar functions

Invariance principles — general Representations

mm′ (α)

mm′(β)

m0(β)eimγ = (−1)m

P(m,m) l−m (x) =

(1− x2)−m/2Pm l (x)

Spherical harmonics Lecture 5: Rotations of Vector Waves December 1, 2011 Page 26/(38)

Overview

Rotation of scalar functions

Invariance principles — general Representations

mm′ (α)

jl(kr)

vlm′(kr′) ulm′(kr′) wlm′(kr′)

Note that the distance to the origin r is the same in both the primed and the unprimed system

Spherical vector waves Lecture 5: Rotations of Vector Waves December 1, 2011 Page 27/(38)

Overview

Rotation of scalar functions

Invariance principles — general Representations

mm′ (α)

Spherical vector waves I

The position vector transforms as (between two rectilinear systems)x′1

x′2 x′3

x1 x2 x3

∂

aij ∂

∂xj

Spherical vector waves Lecture 5: Rotations of Vector Waves December 1, 2011 Page 28/(38)

Overview

Rotation of scalar functions

Invariance principles — general Representations

mm′ (α)

Spherical vector waves II

Of particular interest is (somewhat harder to prove, involves the Levi-Civita symbol εijk, permutation symbol [2])

∇′ × (r′f (r)) = ∇× (rf (r))

Spherical vector waves Lecture 5: Rotations of Vector Waves December 1, 2011 Page 29/(38)

Overview

Rotation of scalar functions

Invariance principles — general Representations

mm′ (α)

=

1√ l(l + 1)

= ∑

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. the primed spherical unit vectors r′, θ

′ , φ ′

Spherical vector waves Lecture 5: Rotations of Vector Waves December 1, 2011 Page 30/(38)

Overview

Rotation of scalar functions

Invariance principles — general Representations

mm′ (α)

φ1(γ; kr) =

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

Rotation of scalar functions

Invariance principles — general Representations

mm′ (α)

where

Rττ ′ =

cos −i sin 0 −i sin cos 0

0 0 1

sinα′

sinα′

cosα′ = cosα cos η + sinα sin η cos(β − ψ)

Note vector identity Planar vector waves Lecture 5: Rotations of Vector Waves December 1, 2011 Page 32/(38)

Overview

Rotation of scalar functions

Invariance principles — general Representations

mm′ (α)

ψ

η

Planar vector waves Lecture 5: Rotations of Vector Waves December 1, 2011 Page 33/(38)

Overview

Rotation of scalar functions

Invariance principles — general Representations

mm′ (α)

Cylindrical vector waves

Rotation of cylindrical vector wave is less often employed due to an extended singularity along the e3 axis (radiating waves)

The only one used is rotation of the regular cylindrical vector waves

The rotation of the regular cylindrical vector waves is done by first transforming into planar spherical vector waves (Lecture 7), then a rotation followed by a transformation back

Cylindrical vector waves Lecture 5: Rotations of Vector Waves December 1, 2011 Page 34/(38)

Overview

Rotation of scalar functions

Invariance principles — general Representations

mm′ (α)

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 Electromagnetic Waves.

List of references Lecture 5: Rotations of Vector Waves December 1, 2011 Page 35/(38)

Overview

Rotation of scalar functions

Invariance principles — general Representations

mm′ (α)

List of references II

[9] J. Kuipers. Quaternions and rotation sequences: a primer with applications to orbits, aerospace, and virtual 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

Rotation of scalar functions

Invariance principles — general Representations

mm′ (α)

Assignment 5

1 Write explicitly down the transformation of the coordinate system in terms of the Euler angles α, β, and γ = 0, and identify the transformed spherical angles θ′ and φ′ in terms of the unprimed 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 spherical harmonics

Y00(r) =

√ 1

4π

Y10(r) =

√ 3

for α, β, and γ = 0 — compare with Item 1

Assignment Lecture 5: Rotations of Vector Waves December 1, 2011 Page 37/(38)

Overview

Rotation of scalar functions

Invariance principles — general Representations

mm′ (α)

Assignment

Hint

Aim at expressing the spherical angles for the primed coordinates 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)

Rotations of the coordinate system

Rotation of scalar functions