6.4.1 Spinor Space and Its Matrix Representation . 127 6.4.2 Angular Momentum Operators in Spinor Representation . 128 6.4.3 IncludingSpinSpace ..........................128 6.5 Single Particle States with Spin . 131 6.5.1 Coordinate-Space-Spin Representation . 131 6.5.2 Helicity Representation . 131 6.6 Isospin......................................133 6.6.1 Nucleon-Nucleon (NN) Scattering . 135 7 Combining Angular Momentum Eigenstates 137 7.1 AdditionofTwoAngularMomenta . .137 7.2 ConstructionoftheEigenstates . 141 7.2.1 Symmetry Properties of Clebsch-Gordan Coefficients . 143 7.3 SpecialCases ..................................144 7.3.1 Spin-OrbitInteraction . .144 1 7.3.2 Coupling of Two Spin- 2 Particles ...................146 7.4 PropertiesofClebsch-GordanCoefficients . 147 7.5 Clebsch-GordanSeries .............................149 7.5.1 Addition theorem for spherical Harmonics . 150 7.5.2 Coupling Rule for spherical Harmonics . 151 7.6 Wigner’s 3 j Coefficients...........................152 − 7.7 CartesianandSphericalTensors . 155 v 7.8 TensorOperatorsinQuantumMechanics . 159 7.9 Wigner-EckartTheorem . .. .. .162 7.9.1 Qualitative . 162 7.9.2 ProofoftheWigner-EckartTheorem . 163 7.10Applications...................................164 8 Symmetries III: Time Reversal 170 8.1 Anti-UnitaryOperators. .. .. .170 8.2 Time Reversal for Spinless Particles . 171 8.3 Time Reversal for Particles with Spin . 173 8.4 InvarianceUnderTimeReversal . 174 8.5 AddendumtoParityTransformations . 176 8.6 Application: Construction of the Nucleon Nucleon Potential FromInvarianceRequirements. .177 9 Quantum Mechanics of Identical Particles 183 9.1 General Rules for Describing Several Identical Particles . ......183 9.2 SystemofTwoIdenticalParticles . 185 9.3 PermutationGroupforThreeParticles . 188 9.4 MatrixRepresentationofaGroup. 189 9.5 Important Results About Representations of the Permutation Group . 190 9.6 Fermi-,Bose-andPara-Particles. 201 9.7 TheHeliumAtom................................206 vi Chapter 7 Combining Angular Momentum Eigenstates 7.1 Addition of Two Angular Momenta (1) (2) Let Jk and Jk be two sets of angular momentum operators with (1) (1) (1) Jk ,Jℓ = i εkℓm Jm m h i X (2) (2) (2) Jk ,Jℓ = i εkℓmJm m h i X (1) (2) Jk ,Jℓ = 0 h i (7.1) for k,ℓ,m =1, 2, 3. Furthermore, R(j , j ) is defined as a (2j +1) (2j + 1)-dimensional 1 2 1 · 2 space, which is spanned by the common eigenvectors j , m ; j , m = j m j m (7.2) | 1 1 2 2i | 1 1i | 2 2i ~ (1) 2 ~ (2) 2 (1) (2) of the set of operators (J ) , (J ) ,J3 ,J3 . The following relations shall hold for ν =1, 2: (J~(ν))2 j , m ; j , m = j (j + 1) j , m ; j , m | 1 1 2 2i ν ν | 1 1 2 2i J (ν) j , m ; j , m = m j , m ; j , m 3 | 1 1 2 2i ν | 1 1 2 2i (7.3) 137 where j1, j2 are fixed and mν can take the values jν , jν 1, , jν. The states in (7.2) − shall be normalized according to ··· − j , m′ ; j , m′ j , m ; j , m = δ ′ δ ′ . (7.4) h 1 1 2 2 | 1 1 2 2i m1m1 m2m2 In this space we want to consider the operators (1) (2) Jk = Jk + Jk . (7.5) Considering the commutation relations (7.1), it follows that [Jk,Jℓ] = i εkℓmJm . (7.6) m X The space in which J~ acts in a direct product Hilbert space, and we want to determine in this space R(j1, j2) the eigenvalues and common eigenvectors of J~ and J3. This means we want to find states j j JM = j j m m j j m m j j JM . (7.7) | 1 2 i | 1 2 1 2ih 1 2 1 2 | 1 2 i m1m2 X The coefficients j j m m j j JM that give the amplitude for each product state in h 1 2 1 2 | 1 2 i the combined state are called Clebsch-Gordan or vector-coupling coefficients: j j m m j j JM C(j j J, m m M) Cj1 j2 J . (7.8) h 1 2 1 2 | 1 2 i ≡ 1 2 1 2 ≡ m1m2M As first step we can immediately determine the eigenvalues of J3 and their degeneracy. From (1) (2) J3 = J3 + J3 (7.9) follows J j j JM = M j j JM 3 | 1 2 i | 1 2 i = (m + m ) j j m m C(j j J, m m M) 1 2 | 1 2 1 2i 1 2 1 2 m1m2 X = M j j m m C(j j J, m m M) . | 1 2 1 2i 1 2 1 2 m1m2 X (7.10) Since the basis elements are orthogonal, one has (m1 + m2) C(j1j2J, m1m2M) = M C(j1j2J, m1m2M) . (7.11) 138 From this follows that C(j j J, m m M) = 0 if M = m + m . (7.12) 1 2 1 2 6 1 2 Furthermore C(j j J; j ,J j J) = 0, unless 1 2 1 − 1 j J j j or j j J j + j − 2 ≤ − 1 ≤ 2 1 − 2 ≤ ≤ 1 2 (7.13) and C(j j J; J j , j J) = 0, unless 1 2 − 2 2 j J j j or j j J j + j . − 1 ≤ − 2 ≤ 1 2 − 1 ≤ ≤ 1 2 (7.14) From (7.13) and (7.14) follows that there are j j J j + j (7.15) | 1 − 2 | ≤ ≤ 1 2 allowed values for J. In general we can say that the eigenvalues of J~ 2 can be numbers J(J + 1) with J = 0, 1 , 1, 3 , . If there is a quantum number J, then there has to be a (2J +1) dimensional 2 2 ··· space R(J) of eigenvectors JM belonging to the operators J~ 2 and J~ with M = | i 3 J, J 1, , J. From (7.12) followed that M = m1 + m2, where j1 m1 j1 and j − m··· − j . Let us consider now the number of states g(M)− for different≤ ≤ values of − 2 ≤ 2 ≤ 2 M. Without loss of generality we assume j j . 1 ≥ 2 m 2 j 2 m -j j 1 1 1 -j 2 139 The largest value of M is j1 + j2, thus one has g(j1 + j2) = 1 g(j + j 1) = 2 1 2 − g(j + j 2) = 3 1 2 − (7.16) The number increases by 1 until g(j j ) = 2j + 1 (7.17) 1 − 2 2 is reached. Then it will stay the same unit M = (j j ) is reached and will decrease − 1 − 2 again by 1 from (j j ) 1 or until g( j j ) = 1. − 1 − 2 − − 1 − 2 In principle, one can have different states belonging to the eigenvalues J and M. We want to determine this number of states p(J). From the number of states belonging to a given M, g(M) = p(J = M ) + p(J = M +1) + p(J = M +2) + (7.18) | | | | | | ··· we can conclude by reversing (7.18) g(M = J) = p(J) + p(J +1)+ ··· g(M = J +1) = p(J +1) + p(J +2)+ ··· (7.19) Subtracting both relations leads to p(J) = g(M = J) g(M = J + 1) . (7.20) − Now we can count the states p(J > j + j ) = 0 , since g(M)=0 for M > j + j 1 2 | | 1 2 p(J = j1 + j2) = g(M = j1 + j2)=1 p(J = j + j 1) = g(M = j + j 1) g(M = j + j )=1 1 2 − 1 2 − − 1 2 and continue up to p(J = j j ) = g(M = j j ) g(M = j j +1) = 1 , 1 − 2 1 − 2 − 1 − 2 and finally p(J < j j ) = 0 . 1 − 2 140 Thus the eigenvalues J within the allowed interval are given by J = j + j , j + j 1, , j j , (7.21) 1 2 1 2 − ··· 1 − 2 and they occur exactly only once. We verify that the number of states j1m1 j2m2 is identical with the number of states j j JM , where j j J |j + ji|and iM J M: | 1 2 i | 1 − 2 | ≤ ≤ 1 2 − ≤ ≤ j1+j2 (2J +1) = (2j + 1)(2j2 + 1) . (7.22) J= j1 j2 X| − | 7.2 Construction of the Eigenstates Since we know which vectors j1j2JM exist, we are left with their explicit construction and with the determination of| the Clebsch-Gordani coefficients. In analogy to the determination of single angular momentum states, we start from a state J = j + j , M = j + j j j ; j j (7.23) | 1 2 1 2i ≡| 1 1 2 2i and apply the ladder operator (1) (2) J = (J1 iJ2) = (J + J ) . (7.24) − − − − In general we have J j1j2JM = J(J + 1) M(M 1) j1j2JM 1 (7.25) ± | i − ± | ± i p (1) (2) and the corresponding relations for J and J . Applying the ladder operators on a state as given in (7.7) leads to ± ± J(J + 1) M(M 1) j j ; JM 1 − ± | 1 2 ± i = C(j j J, m m M) j (j + 1) m (m 1) j j m 1 m p 1 2 1 2 1 1 − 1 1 ± | 1 2 1 ± 2i m1m2 X hp + j (j + 1) m (m 1) j j m m 1 . (7.26) 2 2 − 2 2 ± | 1 2 1 2 ± i p i If we project this result on states j j m m , we obtain the following relation for the | 1 2 1 2i coefficients: 141 J(J + 1) M(M 1) C(j j J, m m M 1) − ± 1 2 1 2 ± = j (j + 1) m (m 1) C(j j J, m 1, m M) p 1 1 − 1 1 ∓ 1 2 1 ∓ 2 + j (j + 1) m (m 1) C(j j J, m m 1, M) .