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

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

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This Q in Eq. 82b) is the matrix representation of the operator Q\ that is, the ji element of Q is O I G I O of Eq. 82a). The density matrix provides a convenient description of a mixed state, since the results of any measurement can be predicted by the use of Eq. 82b). We must give the properties of the density matrix, and we do so with a view to its application to the polarization experiments. Under these circumstances the matrix refers to mixed spin states of a system of particles. The important properties of the density matrix are as follows.

1/2 2 The parity of these angular-momentum functions is important. 7). For our spherical functions this corresponds to replacing θ and φ by π — θ and φ — π. We have that cos(7R — Θ) = — C O S 0 , ( i WCOS(TT Ν d COS(K <-ir(-d—Y V = Θ)/ — Wcos0/ Θ) ' and sin(7R — Θ) = sin θ . In addition, we have that pirn (φ—π) __ ^ J^m ^ΐτηφ Then we find on making the replacement in Eq. 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).

The new state | } =D-»|0>, where D~ is the reciprocal of Z>, the rotation operator for the passive trans­ formation, and is obtained from D by changing a -> — α, β —• —β and γ —> — γ. 66) e x p ( _ ) sin £ . 10 Rotations (II) The common exp( — iy/2) has been dropped, as it will be an unobservable phase factor. However, we must remember that it represents the transformation property of the state for rotations around the direction in which the spin points.

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