W.S.C. WILLIAMS (Eds.)'s An Introduction to Elementary Particles PDF


ISBN-10: 0127567569

ISBN-13: 9780127567563

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Thus the direction of the polarization in this case is not the direction in which all the spins point but the direction in which every measurement of spin component yields the value ft/2. 14 The Density Matrix It is evident that we require a technique capable of describing fully the state of polarization or alignment of a system of particles. The tech­ nique is that of the density matrix (Fano, 1957). We examine this technique by returning to some of the basic concepts of quantum mechanics, in particular to the concept of a state of a system.

3. (c) If j = j + j — 2; then there are three ways of composing a total state vector with this J eigenvalue; we do not give them, as they are analogous to Eqs. 34). 36). z a b z 2 z This process continues; any value of j selected can have its state vector expressed as a linear sum of eigenstates of J . As we decrease the value of j , more J eigenstates are required until j = \ j — j |, after which no more are needed. When j = — \ j — Λ Ι > the number of eigenstates re­ quired begins to decrease until only one is required when j — — \ j + Λ I· What we have said up to now can be reduced to the simple statement that \jaJaz Jb,jbz} = Σ Cj \ j j = j + j} .

3) In view of the normalization undertaken in (2) we have to modify the expectation value of Eq. 83) Tvp (4) By a comparison of Eq. 3, we see that the diagonal elements of the density matrix are the prob­ ability of finding any member of the system in a pure state described by one of the state vectors such as | / ) . Thus the diagonal elements of the matrix are positive definite or zero: Pnn > 0. (5) A matrix with Ν rows and Ν columns requires 2N real quanti­ ties; the Hermitian density matrix therefore requires N , and if Tr ρ = 1, Ν — \ parameters suffice.

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

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