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.)
REFERENCES
Ambler, E., Eisenstein, J.C. and Schooley, J.F. (1962a) Traces of products of angular momentum matrices, J. Math. Phys. 3, 118-130.
Ambler, E., Eisenstein, J.C. and Schooley, J.F. (1962b) Traces of products of angular momentum matrices II - Spherical basis, J. Math. Phys. 3, 760-771.
Berman, S.M. and Jacob, M. (1965) Systematics of angular and polarization distributions in three-body decays, Phys. Rev. B139, 1023-1038.
Bernabeu, J. (1975) Restrictions for asymmetry and polarizations of recoil in muon capture, Phys. Lett. B55, 313-317.
Biedenharn, L.C. (1970) Interpretation of the Symmetry of the Clebsch-GordanCoefficients Discovered by Regge, J. Math. Phys. 11, 1835-1844.
Biedenharn, L.C., Blatt, J.M. and Rose, M.E. (1952) Some Properties of the Racah and Associated Coefficients, Rev. Mod. Phys. 24, 249-257.
Biedenharn, L.C. and Louck, J.D. (1981) Angular Momentum in Quantum Physics, Theory and Application, Encyclopedia of Mathematics and Its Applications, Addison-Wesley, Reading, Massachusetts.
Biedenharn, L.C. and Van Dam, H. (eds.), (1965) Quantum Theory of Angular Momentum, Academic Press, New York.
Blatt, J.M. and Weisskopf, V.F. (1952) Theoretical Nuclear Physics, John Wiley & Sons, New York.
Blin-Stoyle, R.J. and Grace, M.A. (1957) Oriented nuclei, in Handbuch der Physik, Springer-Verlag, Berlin, pp. 557-610.
Bohr, A and Mottelson, B.R. (1969) Nuclear Structure, Vol. 1, W.A. Benjamin, New York.
Brink, D.M. and Satchler, G.R. (1962) Angular Momentum, Clarendon Press, Oxford.
Byers, N and Fenster, S. (1963) Determination of spin and decay parameters of fermion states, Phys. Rev. Letters 11, 52-55.
Condon, E.U. and Shortley, Q.W. (1935) The Theory of Atomic Spectra, Cam-bridge University Press, Cambridge.
De Meyer, H and Vanden Berghe, G. (1978a) Traces of products of angular mo-mentum operators in the spherical basis, J. Phys. A11, 485-500.
De Meyer, H and Vanden Berghe, G. (1978b) Recursion relations for traces of products of angular momentum operators in the spherical basis, J. Phys. A11,
Devanathan, V. and Girija, V. (1985) The gradient formula revisited, J. Madras University, B48, 93-98.
Devanathan, V., Parthasarathy, R. and Subramanian, P.R. (1972) Recoil nuclear polarization in muon capture, Annals of Physics (N.Y.) 73, 291-302.
Devanathan, V. and Subramanian, P.R. (1975) Cascade process in muon capture and gamma-neutrino angular correlations, Annals of Physics (N.Y.) 92, 25-43.
Devons, S. and Goldfarb, J.B. (1957) Angular correlation, in Handbuch der Physik, Vol. 62, Springer Verlag, Berlin, pp. 362-554.
1295-1301.
238
REFERENCES 239
Edmonds, A.R. (1957) Angular Momentum in Quantum Mechanics, Princeton
Eisenberg, J.M. and Greiner, W. (1976) Nuclear Theory, Vol. 2: Excitation Mech-
Feynman, R.P. (1962) Quantum Electrodynamics, W.A. Benjamin, Inc., New
Fano, U and Racah, G. (1959) Irreducible Tensorial Sets, Academic Press, New
Goldstein, H. (1950) Classical Mechanics, Addison-Wesley, U.S.A. Jackson, J.D. (1965) Particle and polarization angular distributions for two and
three body decays, in C. de Witt and M. Jacob (eds.), High Energy Physics, Les Houches 1965, Gordon and Breach, p.327.
Jacob, M and Wick, G.C. (1959) On the general theory of collisions for particles with spin, Annals of Physics (N.Y.) 7, 404-428.
Jacob, M. (1964) in G.F. Chew and M. Jacob (eds.), Strong Interaction Physics, Benjamin, New York.
Kaplan, D.M. and Zia, R.K.P. (1979) The nature of the S dependence of spin traces, J, Phys. A12, 167-170.
Lee, T.D. and Yang, C.N. (1957) Possible determination of the spin of Λ0 fromits large decay angular asymmetry, Phys. Rev. 109, 1755-1758.
Martin, A.D. and Spearman, T.D. (1970) Elementary Particle Theory, NorthHolland, Amsterdam.
Miller, K.S. (1960) An Introduction to the Calculus of Finite Differences and Difference Equations, Henry Holt, New York.
Pal, M.K. (1982) Theory of Nuclear Structure, Affiliated East-West Press, New Delhi.
Pearce, P.A. (1976) Monotonicity of correlation function, J. Math. Phys. 17, 744-746.
Racah, G. (1942a) Theory of Complex Spectra I, Phys. Rev. 61, 186-197.Racah, G. (1942b) Theory of Complex Spectra II, Phys. Rev. 62, 438-462.Racah, G. (1943) Theory of Complex Spectra III, Phys. Rev. 63, 367-382.Racah, G. (1951) Directional Correlation of Successive Nuclear Radiations, Phys.
Ramachandran, G. (1962) Matscience Report on Angular Momentum, The Insti-
Ramachandran, G. (1987) Non-oriented spin systems, Journal of the Madras Uni-
Ramachandran, G., Mallesh, K.S. and Ravishankar, V. (1984) On polarized spin-
Ramachandran, G. and Ravishankar, V. (1986) On polarized spin-j assemblies,
Ramachandran, G., Ravishankar, V., Sandhya, S.N. and Sirsi, S. (1987) Non-oriented spin-1 systems, J. Phys. G13, L271-L273.
Ramakrishnan, Alladi (1962) Elementary Particles and Cosmic Rays, PergamonPress, Oxford.
Ramakrishnan, Alladi (1967a) The Dirac Hamiltonian as a member of the hier-archy of matrices, J. of Mathematical Analysis and applications 20, 9-16.
Ramakrishnan, Alladi (1967b) Helicity and energy as members of hierarchy of eigenvalues, J. of Mathematical Analysis and applications 20, 397-401.
University Press, Princeton.
anisms of the Nucleus, North-Holland, Amsterdam.
York.
York.
Rev. 84, 910-912.
tute of Mathematical Sciences, Madras.
versity B50, 383-409.
1 systems, J. Phys. G10, L163-L166.
J. Phys. G12, L143-L145.
240 REFERENCES
Ramakrishnan, Alladi (1972) L-matrix Theory or the Grammar of Dirac matrices,
Rashid, M.A. (1979) Traces of products of angular momentum operators J. Phys.
Rose, M.E. (1955) Multipole Fields, John Wiley, New York. Rose, M.E. (1957a) Elementary Theory of Angular Momentum, John Wiley, New
Rose, M.E. (1957b) Statistical tensors for oriented nuclei, Phys. Rev. 108, 362-
Rose, M.E. (1961) Relativistic Electron Theory, John Wiley, New York. Rose, M.E. (1962) Properties of the irreducible angular momentum tensors, J.
Schiff, L.I. (1968) Quantum Mechanics, McGraw-Hill Kogakusha, Ltd., Tokyo. Schweber, S.S., Bethe, H.A. and de Hoffmann, F. (1956) Mesons and Fields,
Srinivasa Rao, K and Rajeswari, V. (1993) Quantum Theory of Angular Momen-
Subramanian, P.R. (1974) A short note on the Bernoulli polynomial of the first
Subramanian, P.R. (1986a) Evaluation of Tr using the Brillouin function, J.
Subramanian, P.R. (1986b) Generating functions for angular momentum traces,
Subramanian, P.R. and Devanathan, V. (1974) Trace techniques for angular mo- mentum operators, J. Phys. A7, 1995-2007.
Subramanian, P.R. and Devanathan, V. (1979) Higher order tensor orientations of recoil nuclei in muon capture, Phys. Rev. C20, 1615.
Subramanian, P.R. and Devanathan, V. (1980) Recurrence relations for angular momentum traces, J. Phys. A13, 2689-2693.
Subramanian P.R. and Devanathan, V. (1985) Generation of angular momentum traces and Euler polynomials by recurrence relations, J. Phys. A18, 2909-
Subramanian, P.R., Parthasarathy, R and Devanathan, V. (1976) Nucl. Phys. A262, 433.
Thakur, J. (1975) General expressions for the trace of the product of any arbitrary number of components of angular momentum, Ind. J. Pure and Appl. Phys.
Ullah, M. (1980a) Note on the traces of the angular momentum operators, Phys.
Ullah, M. (1980b) On the derivation of the traces of products of angular momen-
Varshalovich, D.A., Moskalev, A.N. and Khersonskii, V.K. (1988) Quantum The-
Witschel, W. (1971) Traces of products of angular momentum operators, Molec.
Witschel, W. (1975) Ordered operator expansions by comparison, J. Phys. A8,
Tata McGraw-Hill, New Delhi.
A12, L199-201.
York.
365.
Math. Phys. 3, 409-413.
Vol. 1, Row, Peterson and Co., New York.
tum, Narosa Publishers, New Delhi.
kind, The Mathematics Student (India) 42, 57-59.
Phys. A19, 1179-1187.
J. Phys. A19, 2667-2670.
2915.
13, 792-793.
Rev. A21 1069-1070.
tum operators using operator identities J. Phys. A13 1603-1606.
ory of Angular Momentum, World Scientific, Singapore.
Phys. 22, 869-879.
143-155.
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
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