Clebsch-Gordan Coefficients
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1 Phy489 Lecture 7
2 Clebsch-Gordan Coefficients 2 j j j2 m m m 2 j= j j2 j + j j m > j m > = C jm > m = m + m Two systems with spin j and j 2 and z components m and m 2 can combine to give a system which (quantum-mechanically) is a linear combination of states having spin j from j -j 2 to j +j 2 (in integer steps) each having a z component of m = m + m 2. So if we make a measurement of the total angular momentum of a state made up of the two spin states as defined above, the square of the Clebsch-Gordan (CG) coefficient j j j2 C m m m 2 represents the probability of obtaining a measurement of J 2 = j(j+) ħ 2
3 Tables of Clebsch-Gordan Coefficients j X j 2 Table for combining spin j with spin j 2 2 j j j2 m m m 2 j= j j2 j + j j m > j m > = C jm > m = m + m J J... m m... m m 2 m m j j j2 C m m m 2
4 Problem from Last Time An electron in a hydrogen atom is in a state with orbital angular momentum number =. If the total angular momentum quantum number is j=3/2, and the z component of total angular momentum is / 2 what is the probability of finding the electron with? m s = + / 2 Need x Clebsch-Gordan Table (spin + = )
5 Clesch Gordan Coefficients from PDG Listings
6 Combine 2 spin particles as in Griffiths example 4.4 Four possible outcomes for J,M> X + + /2 + /2 + /2 - /2 - /2 + /2 /2 /2 /2 - /2 - /2 - /2 -
7 Combine 2 spin particles as in Griffiths example 4.4 X + + /2 + /2 + /2 - /2 - /2 + /2 /2 /2 /2 - /2 - /2 - / =
8 Combine 2 spin particles as in Griffiths example 4.4 X + + /2 + /2 + /2 - /2 - /2 + /2 /2 /2 /2 - /2 - /2 - / =
9 Combine 2 spin particles as in Griffiths example 4.4 X + + /2 + /2 + /2 - /2 - /2 + /2 /2 /2 /2 - /2 - /2 - / =
10 Combine 2 spin particles as in Griffiths example 4.4 X + + /2 + /2 + /2 - /2 - /2 + /2 /2 /2 /2 - /2 - /2 - / =
11 Combine 2 spin particles as in Griffiths example 4.4 X + + /2 + /2 + /2 - /2 - /2 + /2 /2 /2 /2 - /2 - /2 - /2 - =
12 Combine 2 spin particles as in Griffiths example 4.4 X + + /2 + /2 + /2 - /2 - /2 + /2 /2 /2 /2 - /2 - /2 - /2 - =
13 Combine 2 spin particles as in Griffiths example 4.4 X + + /2 + /2 + /2 - /2 - /2 + /2 /2 /2 /2 - /2 - /2 - /2 - =
14 = + /2 + /2 + /2 - /2 - /2 + /2 /2 /2 /2 - /2 - = /2 - /2 = = Can re-write this in terms of states of definite J,M
15 Spin triplet Symmetric under 2 = /2 + /2 + /2 - /2 - /2 + /2 /2 /2 /2 - /2 - /2 - /2 - = = = Spin singlet Anti -symmetric under 2
16 Spin Formalism Write spin up and spin down states as two-component spinors spin up 2 2 = spin down 2 2 = Most general state of a spin system is then (a superposition of the two): α = α + β α, β β complex Probability that measurement of S z yields ħ/2 (e.g. spin up) is α 2 Probability that measurement of S z yields -ħ/2 (e.g. spin down) is β 2
17 Spin Matrices What about measurements of S x and S y? Must also be either ± ħ/2, but with what probability? To each component of S there is associated a 2x2 matrix Pauli spin matrix] ˆ ˆ i ˆ Sx = S y = Sz = 2 2 i 2 Each has eigenvalues of ± ħ/2 [e.g. from det( S ˆ x λi) = ] Normalized eigenvectors for S x are α = a + b a,b β 2 2 complex } χ = ± 2 ± a = 2 + b = 2 ( α β ) ( α β ) Probability of measuring S x = 2 is a 2
18 Flavour Symmetries (internal symmetries) Note that the proton (p) and neutron (n) have very similar masses: m p = MeV/c 2, m n = MeV/c 2 Heisenberg: try viewing this as two different states of the same particle (the nucleon, N). α N = = α + β p = n = β Reminiscent of spin formalism. p and n are related by a rotation in an abstract space referred to as isospin space (in analogy to spin). Adopt spin formalism to deal with systems of particles with different isospin (first for isospin in analogy to spin ). For any state we can define the isospin I and the third component I 3 (as for J). We label the components as I, I 2 and I 3 to make clear that we are not talking about components in coordinate space, but in some abstract space.
19 Warning. Isospin has NOTHING to do with real spin (e.g. angular momentum). The (confusing, so perhaps inappropriate) name is historical and comes from the similarity of the mathematical formalism, to that for handling spin. A rotation in isopin space does NOT have anything to do with rotations in physical (coordinate) space.
20 p = 2 2 Isospin We have seen that strong and electromagnetic interactions conserve strangeness, charm etc. Only the charged weak interaction can change quark flavour.there are no flavour changing neutral currents (FCNC) in the Standard Model. Isospin is just a version of conservation of quark number for u, d quarks, arising from the fact that their masses are so similar. Consider proton and neutron as members of an isospin doublet (the nucleon) n = 2 2 Heisenberg s proposal: the strong interaction is invariant under rotations in isospin space. Noether s theorem then implies that isospin is conserved in all strong interactions. Other particles fall into different isospin multiplets (each with 2I + elements) For example, pions have isospin I = (so three charge states): π π π + = = = e.g. total isospin I = /2
21 Isospin Rules for isopin assignment are a little obscure, having to do with the positions particles occupy in the eightfold way multiplets discussed in Chapter. Isospin relates states with different electric charges but very similar masses. (e.g. Hadronic states differing only in their u,d quark content) K K + sd su π π η π + ud uu, dd ud K K su sd
22 Isospin at Quark Level u = d = s = Other quark flavours (c,b,t) are also isospin singlets, like the strange quark Mesons containing u,d quarks can therefore have isospin or π π π + = ud = uu, dd = ud η = uu, dd isospin triplet I = isospin singlet I = Mesons containing only one u,d quark can only have isospin = K = us + K = ds K = us K = ds form an isospin doublet form a separate isospin doublet
23 The Eightfold Way Meson Octet (u,d,s) S = K K + isospin doublet π S = π η π + isospin triplet S = - Isospin singlet K K isospin doublet Q = - Q = Q = +
24 The Eightfold Way Baryon Octet (u,d,s) S = n p isospin doublet Σ S = - S = -2 Σ Ξ Λ Isospin singlet Ξ + Σ isospin triplet isospin doublet Q = - Q = Q = +
25 Application example: pion-nucleon scattering Six elastic πn πn processes (e.g. same particles in and out of scattering) (a) π + + p π + + p (b) π + p π + p (c) π + p π + p (d) π + + n π + + n (e) π + n π + n ( f ) π + n π + n and 4 charge-exchange processes ( ) ( ) + + g π + n π + p h π + p π + n ( ) ( ) i π + n π + p j π + p π + n Pions carry I = and nucleons, I = /2 so the total isospin can be either /2 or 3/2 Since isospin is conserved (in these strong interactions) there are two distinct amplitudes: M for I = /2 and M 3 for I = 3/2!
26 Pion-Nuclon Scattering Amplitudes π + + p : 3 3 = π π π p : = p : = n : = See Perkins Introduction to High Energy Physics section on Isospin in the pion nucleon system for more detail. (Section number varies from edition to edition; it is 3.3 in the 4th edition) π π n : = n : 3 3 = ( a) π + p π + p ( f ) π + n π + n Both are pure I = 3/2 on either side of the reaction. Both proceed purely }via the I = 3/2 amplitude M 3
27 Pion-Nucleon Scattering Amplitudes 3 2 π + p : = π + n : = ( c) π + p π + p M c = 3 M M ( ) j π + p π + n M j = M M Cross-section σ for some process is proportional to M 2
28 Relative Cross-sections for πn Scattering σ a :σ c :σ j = 9 M 3 2 : M 3 + 2M 2 : 2 M 3 M 2 At E CM = 232 MeV there is a dramatic bump (increase) in the pion-nucleon scattering cross-section, do to a πn resonance ( the Δ ). Δ carries isospin 3/2, therefore: Expect the scattering at this energy to be dominated by the isospin 3/2 channel σ a :σ c :σ j = 9 :: 2 This is listed amongst the particles on the inside cover of your text Generally, experimentally, it is easiest to measure the total cross-section, so processes (c) and (j) are combined to yield: ( ) ( π + p) = σ a σ tot π + + p σ tot σ c + σ j = 3
29 Pion-proton Scattering Cross-sections π + p π p
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