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

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

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We wish to examine the effect upon / ( x ) of a rotation of this coordinate system. We make an infinitesimal rotation of the coordinates through an angle άφ around the ζ axis. The new coordinates of the same space point Ρ in the new coordinate system F' are given by x' = χ + y άφ , y' = y — χάφ, ζ' = ζ. There must exist a function f'(x') that has the same value as has f(x) when χ and x' describe the same space point from different coordinates. 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).

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. 58 II. Angular Momentum (1) The density matrix is Hermitian. This follows from the fact that (Q} is a real quantity and Q is a Hermitian operator; therefore Pirn — 9ml · (2) Since any particle of a system must be in one of the available states, we require Tr ρ = 1 .

Angular Momentum Suppose we have a system containing both spin and orbital angular momentum /: this is represented by the vector | /, l , s, s }. Then the ampli­ tude for finding the system with a particle at the angle θ, φ with spin state s', s/ is <0, φ, s', s/ I /, l , s, s y. Since the state vectors for the two parts can be considered separately, this amplitude becomes z z <0, φ I /, / Ζ z z ><5', s/ 1s s } = Υ/φ, 9 z φ)δ ,δ , , 55 . 9 The Pauli Spin Matrices The matrix elements of the components of the angular-momentum vector for the case of angular momentum \ lead to the Pauli spin matrices.

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