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


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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} .

14). We shall deal later with techniques for describing the polarization, but at this point we wish to draw the reader's attention to the possibility of " alignment" that exists for particles of spin ft and greater (except photons). Consider an assembly of particles, each having spin ft, that has no net angular momentum. This can happen in several ways; for example: (1) All spin states are equally populated so that a measurement of spin component in a fixed direction yields the values ft, 0, and —ft with equal probability; this is an entirely unpolarized assembly.

The existence of f'(x') defines a function f'(x) that has the same value at the point P' (coordinate χ referred to F') as / ( χ + Δχ) has at this point (χ + Δχ referred to F). We can make a Taylor expansion of / ( x ) around χ to give f'(x) = f(x + Δχ) = f(x) - (y± - *|;)/(x) # + ··· For infinitesimal rotations, this can be written / ' ( x ) = (l +±L di)f(x). z For finite rotations, άφ -> φ and we make a summation of the infinitesimal rotations to give / (x) = , exp(+i0£ /*)/(x). n)/*]/(x). So far in this section, we have developed the rotation operator as an alge­ braic operator that is operating on real functions.

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