APPENDIX A

EQUIVALENCE OF ROTATION ABOUT AN ARBITRARY

AXIS TO EULER ANGLES OF ROTATION

Rotation of the coordinate system through an angle about an arbi-trary axis denoted by the unit vector (θ,φ) is equivalent to successive rotations through the Euler angles α, β, γ about the z-axis, the new y-axisand the new z-axis respectively. In what follows, we shall try to obtain a re-lation between the two sets of three parameters θ, φ, and α, β, γ describingthe rotation.

The procedure is outlined below. First, we consider the rotation about an arbitrary axis and obtain the transformation matrix M ( e0, e ) in terms of certain parameters e0, e, known as the Euler parameters which are related to the rotation parameters θ, φ, Then we consider unitary transforma-tions in a two-dimensional complex space, which is equivalent to a rotation in the three-dimensional real space. The unitary transformation matrix Qin complex space when expressed in terms of certain parameters yields the same transformation matrix for three-dimensional rotation obtained earlier in terms of Euler parameters for rotation about an arbitrary axis Thus, we identify the parameters used in the description of unitary matrix Q incomplex space with the Euler parameters. Since the parameters describ-ing the complex unitary matrix Q are related to Euler angles of rotation α, β, γ, we deduce the required relation between the two sets of rotation parameters θ, φ, and α, β, γ.

It is also possible to obtain the transformation matrix M ( α,β,γ ) due to Euler angles of rotation directly, and comparing this with the trans-formation matrix obtained by rotation through an angle about an axis denoted by the unit vector we deduce the required relations for their equivalence.

A.1. Rotation About an Arbitrary Axis

Rotation of the coordinate system through an angle about an axis in the anticlockwise direction is equivalent to a clockwise rotation of the object through the same angle about the same axis in the fixed coordinate system.

216

EQUIVALENCE TO EULER ANGLES OF ROTATION 217

Figure A.1. (a) Effect of rotation about an arbitrary axis through an angle The point P moves to point Q and the vector r r'. (b) The section

normal to the plane is shown separately.

Consider a vector r denoted by OP rotated in the clockwise direction through an angle about the axis The new vector r' is denoted by OQ.From Figure A.1, we obtain the following relations.

(A.1)

(A.2)

(A.3)

(A.4)

(A.5)

(A.6)

(A.7)

(A.8)

Hence

such that

or equivalently

It follows that

218 APPENDIX A

Let us now introduce the Euler parameters e0, e1, e2, e3 and express the vector r' in terms of r and the Euler parameters.

(A.9)

(A.10)

(A.11)

(A.12)

(A.13)

Using the above relations, r' can be written as a function of r and the Euler parameters e0 and e.

(A.14)

Using Eq. (A.14), the components x' y', z' can be explicitly written in terms of x, y, z.

(A.15)

(A.16)

(A.17)

Writing more elegantly in the matrix form

we obtain the transformation matrix M ( e0, e ),

(A.18)

(A.19)

with the Euler parameters expressed in terms of the rotation parameters

(A.20)

and perform unitary transformation Q on it.

The conditions (A.24) yields the relations

EQUIVALENCE TO EULER ANGLES OF ROTATION 219

A.2. The Euler Angles of Rotation

Rotation in a two dimensional complex space is equivalent to a rotation in the three-dimensional real space.

Choose a matrix operator P

(A.21)

(A.22)

Since Q is a unitary unimodular operator, one can obtain certain con-ditions between the elements of this unitary matrix in the two-dimensionalcomplex space.

(A.23)

(A.24)

(A.25)

(A.26)

and the matrix

has only four parameters, of which only three are independent because of the unimodular condition (det Q = 1).

The transformed operator P' is given by

From Eq. (A.27), one can obtain the following relations:

(A.27)

(A.28)

(A.29)

(A.30)

220 APPENDIX A

from which one can deduce the transformation matrix M(a,b ) in the three dimensional real space.

where,

(A.31)

(A.32)

The parameters a and b occurring in Eq. (A.32) are complex quantities and let us define them in terms of the real parameters e0, e1, e2 and e3.

(A.33)

The transformation matrix M ( a,b ) can be rewritten in terms of e0, e1, e2

and e3 using the definition (A.33) of a and b. The transformation matrix M ( e0,e1,e2,e3 ) so obtained is identical with the transformation matrix given in Eq. (A.19) describing the rotation about an arbitrary axis and the quantities e0, e1, e2, e3 defined in Eq. (A.33) are identical with the Euler parameters introduced earlier in Eq. (A.9) or Eq. (A.20).

For Euler angles of rotation, the unitary matrix Q can be written as a product of three unitary matrices

(A.34)

The unitary matrices Q α, Q β, and Q γ can be deduced from the known prop-erties of coordinate transformation under rotation. It is found on inspection that the matrices Q α, Q β and Q γ can be written in a compact form using the Pauli matrices. The matrices so obtained are

(A.35)

(A.36)

(A.37)

EQUIVALENCE TO EULER ANGLES OF ROTATION 221

The product of these matrices is denoted by Q.

(A.38)

Comparing the unitary matrix (A.38) with (A.26) and expressing the com-plex elements in terms of the real parameters e0, e1, e2, e3 defined in Eq. (A.33), we obtain

(A.39)

From Eqs. (A.39) and (A.20), the required relations between the two sets of rotation parameters, θ, φ, and α, β, γ are deduced.

(A.40)

(A.41)

(A.42)

(A.43)

Squaring Eqs. (A.41) and (A.42) and adding, we obtain

the square root of which yields the relation

Substituting this in Eq. (A.42), we get

Thus

(A.44)

(A.45)

(A.46)

(A.47)

Equations (A.40), (A.45) and (A.47) relate the two sets of rotation param-eters.

222 APPENDIX A

A.3. Direct method of obtaining the transformation matrix

The vector r in the original coordinate system becomes vector r' in the rotated coordinate system obtained by successive Euler angles of rotation α, β, γ.

The transformation matrix for these rotations are discussed in Chap.4.

(A.48)

where

By direct matrix multiplication, the matrix elements of M (α,β,γ) are ob-tained as given below:

(A.50)

(A.51)

(A.52)

(A.53)

(A.54)

(A.55)

(A.56)

(A.57)

(A.58)

(A.59)

Using the trignometric relations,

EQUIVALENCE TO EULER ANGLES OF ROTATION 223

(A.60)

(A.61)

(A.62)

(A.63)

(A.64)

and defining the parameters as given in Eq. (A.39), it can be shown that the transformation matrix M (α,β,γ) is identical with the transformation matrix M (e0,e ) defined in Eq. (A.19). For the purpose of illustration, let us choose the matrix element M11 (α,β,γ).

In a similar way, all the other matrix elements can be expressed in terms of the parameters e0, e1, e2, e3 and the resulting transformation matrix M (α,β,γ) is identical with the matrix (A.19).

APPENDIX B

TABLES OF CLEBSCH-GORDAN COEFFICIENTS

224

APPENDIX C

TABLES OF RACAH COEFFICIENTS

225

APPENDIX C

226

APPENDIX D

THE SPHERICAL HARMONICS

The spherical harmonics are the solutions of the differential equation

(D.1)

and they can be expressed in terms of the associated Legendre functions

(D.2)

where x = cos θ and the associated Legendre functions are the derivatives of the Legendre function Pl (x )

(D.3)

Note that = Pl(x) and the Legendre functions are defined by

(D.4)

Substituting the expression for the associated Legendre functions, a general expression for the spherical harmonics is obtained.

The spherical harmonics satisfy the relation

227

(D.5)

(D.6)

228 APPENDIX D

and they are normalized such that

(D.7)

The completeness relation for the spherical harmonics is given by

(D.8)

Let us list some of the relations involving sums over m but with fixed lfor spherical harmonics.

(D.9)

(D.10)

(D.11)

The spherical harmonics have an inversion symmetry property of great importance. The direction opposite to (θ,φ) is (π - θ,φ + π). From an examination of Eq. (D.5), we obtain a relation

(D.12)

which means that the spherical harmonics have positive parity for even land negative parity for odd l.

It can be easily verified that

(D.13)

The first four Legendre polynomials Pl(x) are given in Table D1 and the explicit forms of the spherical harmonics for l = 0, 1, 2, 3, 4 are presented in Table D2.

THE SPHERICAL HARMONICS 229

APPENDIX E

THE SPHERICAL BESSEL AND NEUMANN FUNCTIONS

The spherical Bessel function jl (x ) and the spherical Neumann function nl(x) are defined in terms of the ordinary Bessel functions of odd-half-integer order.

(E.1)

(E.2)

where l is an integer. The spherical Bessel and Neumann functions are the solutions of the differential equation

(E.3)

Explicit expressions for the fist few spherical Bessel and Neumann functions are given below:

with

The spherical Bessel and Neumann functions take simple forms in the lim-iting cases. As x 0,

(E.5)

(E.6)

(E.7)

In the asymptotic limit i.e., as

230

THE SPHERICAL BESSEL FUNCTIONS 231

We give below some useful recurrence relations for jl(x):

(E.8)

(E.9)

(E.10)

(E.11)

(E.12)

The same recurrence relations are obeyed by the spherical Neumann func-tions nl(x) also. The reader may find the following integarl formulas involv-ing spherical Bessel functions useful.

(E.13)

(E.14)

(E.15)

(E.16)

APPENDIX F

THE BERNOULLI POLYNOMIALS

The Bernoulli polynomial Bs(x) is defined by

Bs (0) are called Bernoulli numbers Bs.

Equating coefficients of equal powers oft, we get

(F.1)

(F.2)

With the exception of B1, all odd Bernoulli numbers vanish. From Eqs. (A.1) and (A.2), we get

(F.3)

(F.4)

The first few Bernoulli numbers and Bernoulli polynomials (Miller, 1960; Subramanian, 1974) are listed below:

Bernoulli numbers

(F.5)

232

THE BERNOULLI POLYNOMIALS 233

Bernoulli polynomials

APPENDIX G

LIST OF SYMBOLS AND NOTATION

Angular Momentum Coupling Coefficients

Clebsch-Gordan coefficient or C.G. coefficient

3-j symbol

U-coefficient (Unitary transformation coefficient for the coupling of three angular momenta)

Racah coefficient

6-j symbol

LS-jj coupling coefficient

9-j symbol

Angular Momentum Eigenstates and Operators

Angular momentum eigenstate

Eigenstate of two angular momenta in uncoupled representation

Eigenstate of two angular momenta in coupled representation

234

LIST OF SYMBOLS AND NOTATION 235

Eigenstate of three angular momenta in one cou-pled representation

Eigenstate of three angular momenta in another coupled representation

Eigenstate of four angular momenta in L-S cou-pled representation

Eigenstate of four angular momenta in j-j coupled represent at ion

Spin-up state for the Dirac particle

Spin-down state for the Dirac particle

Helicity state of a particle moving with momen-tum p (= p, θ, φ) and helicity λ

Two-particle helicity state moving with relative momentum p and helicities λ1 and λ2 as described in centre of momentum frame

Square of the angular momentum operator

Components of angular momentum operator in Cartesian basis

Raising and lowering angular momentum opera-tors (Ladder operators)

Components of angular momentum operator in spherical basis

Pauli spin operators

Iso-spin operators for nucleons

Rotation Operator and Rotation Matrices

Euler angles of rotation

Rotation operators

Rotation matrix

Rotation matrix for rotation about the y-axis

Transformation matrix

236 APPENDIX G

Special Functions

Bs Bernoulli number

Bs(x) Bernoulli polynomial

Jv ( x )

Nv ( x )

jl (x ) Spherical Bessel function

Bessel function of order v

Neumann function of order v

nl (x ) Spherical Neumann function

Pl(x) Legendre function

Associated Legendre function

Vectors, Tensors and Tensor Operators

ex, ey, ez

Ax, Ay, Az

Spherical harmonic

Vector spherical harmonic

Unit vectors in Cartesian basis

Unit vectors in spherical basis

Polar basis vectors

Unit vector with polar angles θ,φ

Cartesian components of a vector A

Spherical components of a vector A.

Spherical tensor operator of rank k and projec-tion µ

Gk ( j ) Fano’s statistical tensor

Spherical tensor parameter

ρ Density matrix

LIST OF SYMBOLS AND NOTATION 237

Miscellaneous Symbols

δ ij Kronecker δ -symbol

dΩ Element of solid angle

dΩ = sin θ dθ dφ

[ j]

Binomial coefficient

α x, α y, α z, β

γ x, γ y, γ z, γ0

Dirac matrices

Gamma matrices

α 0 γ 0 - α xγ x - α yγ y - α zγ zMatrix element of a tensor operator

Reduced matrix element of a tensor operator of rank k

Coefficient of fractional parentage (c.f.p.)

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SUBJECT INDEX

Addition of two angular momenta, 10 Algebra of γ matrices, 210-212Angular momentum eigenfunctions, 2,

Angular momentum matrices, 6, 7, 150-

Density of final states, 206 Dirac equation, 199-203

negative energy solutions, 201

orthogonal and closure properties of

positive energy solutions, 200 positive energy states, 200

34 negative energy states, 201

152, 162 solutions, 201-203products, 150, 162

Angular momentum operator, 1, 2, 8-10, 157, 163, 164

Angular momentum vector, 1, 2

Bernoulli numbers, 232 Bernoulli polynomials, 153-154, 232,

Bosons, 119, 126

Dirac Hamiltonian, 199 Dirac hole theory, 207 Dirac matrices, 199, 209, 214 Distorted waves, 102, 103

Euler angles of rotation, 34, 216-221Euler parameters, 218 Evaluation of matrix elements, 82-84,

Fano’s statistical tensors, 134-138

Feynman’s notation, 206-209Fractional parentage coefficient, 121,

Gell-Mann-Nishijima relation, 16 Gradient formula, 103-105

233

Cartesian basis, 24 95-96, 108-112unit vectors, 24

general expression, 13 Fermions, 119 symmetry properties, 14 tables, 224

Clebsch-Gordan series, 48, 49 inverse, 49, 50

Commutation relations, 1, 3, 10 Coulomb scattering of electrons, 203,

Clebsch-Gordan coefficients, 10-23

123

derivation, 105-108

212 Half-vectors, 47 scattering cross section, 205 sum over spin states, 203-206

Helicity amplitude, 175, 176, 180, 189, 192, 197, 198

Coupling of four angular momenta, 88- invariance under parity and time re-99 versal, 175-176

87 two-body scattering, 172-176

23 polarization studies, 176-180

Coupling of three angular momenta, 77-

Coupling of two angular momenta, 10-

Coupled representation, 11, 12

Density matrix, 130-149

Helicity formalism, 165-198

two-body decay, 181-191

muon capture, 191-194Helicity quantum number, 165, 180 Helicity states, 165-172

144 two-particle, 171, 172

Identical particles, 119-129144 three identical fermions, 120-123

application to cascade process. 143,

application to nuclear reactions, 140-

construction, 133, 134 two identical fermions, 119, 120

241

242 SUBJECT INDEX

Inverse Clebsch-Gordan series, 49 Irreducible tensor, 34 Irreducible tensor operators, 61

Racah’s definition, 61-63 Iso-spin, 15, 125 159 Isotopic spin, 15

closed expression, 81, 82 symmetry properties, 81 tables, 225, 226

Reduced matrix element, 95, 112, 158,

Rotation matrices, 34-60 orthogonality and normalization, 55-

unitarity and symmetry properties, Legendre function, 227 57

associated, 227 Legendre polynomial, 228, 229 LS-jj coupling coefficient, 88-92

50, 51 Rotation operator, 42-44

Scattering amplitude, 173, 174, 178, 197 Scattering cross section, 172-175 Scattering of particles with spin, 172-

Spherical basis, 24 unit vectors, 24

Spherical Bessel function, 230, 231 Spherical harmonic addition theorem,

Spherical harmonics, 25, 52, 227-229

simple properties, 92

m-scheme, 126-128 Muon capture, 191-194

Non-oriented system, 139

Oriented system, 139

Parity and time reversal, 174, 175

Partial waves, 100-118

180

Parity C.G. coefficient, 14 52-54

Raleigh’s expansion of plane wave, addition theorem, 52, 53 coupling rule, 54, 55 100-1 02

Pauli spin matrices, 7, 15 Polarization studies, 176-180

longitudinal, 176, 187, 191 transverse, 176-1 80, 188-190

Spherical Neumann function, 230, 231 Spherical tensor parameter, 134 Spherical tensors, 27 Statistical tensors, 130-149, 158, 159 Sum over spin states, 203-206, 208

Tensors, 72-74 Tensor operators, 72-74 Tensor product, 72-74 Transformation matrix, 36-41 Two-body decay, 181-191

parity conserving, 186-190 parity non-conserving weak decay,

Polarization vector, 152 Position vector, 24, 74, 105

Products of angular momentum matri- spherical components, 24, 25

ces, 150-164 in Cartesian basis, 151-155 in spherical basis, 152, 155-157 their traces, 150-164 evaluation of traces, 152-157, 160-

recurrence relations for traces, 157,

161 190, 191

U-coefficients, 77-81 158 orthonormality, 80

214 Unoriented system, 139

208

208

Projection operators, 203-205, 208, 209, Uncoupled representation, 11, 12

for Dirac’s negative eigenvalue states,

for Dirac’s positive eigenvalue states,

for negative energy states, 203, 214 for positive energy states, 203, 214

Vector spherical harmonics, 104, 106

Wigner 3-j symbol, 16 Wigner 6-j symbol, 81 Wigner 9-j symbol, 92-95 Wigner-Eckart theorem, 63-65

Racah coefficients, 81, 82 proofs, 65-72