An Introduction to Elementary Particles by W.S.C. WILLIAMS (Eds.)

By W.S.C. WILLIAMS (Eds.)

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19) this has eigen­ values {7(7 + 1) — j (j + \)}h . Λ ><ΛΛ μ ΐΛΛ>. / / / + This sum must normally be taken over all possible intermediate states; however, since Η and J commute with / and with / _ , all matrix ele­ ments are zero, except those connected to intermediate states having the same energy and total angular momentum. 6 The Matrix Elements of Angular Momentum eigenvalues of J differing by one unit. Therefore, from the orthonormality of states, we see that /+ \j,j } l y be matched with the eigenfunction 0>Λ + 1 |> and so on.

56) that the parity of the differential operator and of the exponential cancel and we are left with the parity the Legendre polynomial, P/(cos 0). This polynomial contains the following powers of cos#; /, / — 2, / — 4, . . , and therefore has parity (— 1)'. Thus the parity of the eigenfunction 7(/, l ) is (— 1)'. In quantum-mechnical language the spherical harmonic 7(/, l ) is the amplitude for finding a particle at a particular angle if it is in the state that is the eigenstate of L and L with eigenvalues /, / .

In strict analogy with the orbital angular momentum, we can construct a space with state vectors and operators. Such operators are S, corresponding to L (that is, S ,S , S corresponding to L , L L ). Then if |χ> is the vector describing the state of a particle in a pure spin state, it is an eigenvector of the operators S and S with eigenvalues s(s + \)H and s fi, respectively, where s is the spin of the parti­ cle and s its ζ component. 4 Spin and Total Angular Momentum 29 The operators S (S , S , S ) and S are assumed to commute among themselves in a completely analogous way to the orbital angular-momentum operators.

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