 B, C that satisfies ABC = I and correspond to the unique decomposition of U in Eq. (N22). Assume heuristically, for instance, A = R2(21) RR(-2x) B = Ry(2x)R2(2w) (N23) C=R2(20), where , X, y are rotation angles to be determined (the factor of two will lighten the calculations). It is clear that Eq. (N23) satisfies the condition ABC = I, or ABC = R2(2u)Ry(-2x)Ry(2x)R2(2w)Rz(20) (N24) if we impose the angle condition (N25) Pauli matrices, rotations, and unitary operators We shall now develop the product AXBXC to obtain AXBXC = =AXBXR2(20) = AXB = AXB (N26) = R2(21)/ cos 2 sin sin 2xe-i(w-4tv) cos 2xe-i(w-y-v), 2xe2ip 2xe2i(p+y) Appendix N Identifying this result with the definition in Eq. (N20), we obtain (N27) and, thus, (N28) The three unitary operators are then fully defined according to (N29) Commutation properties of rotation operators Two rotation operators R(), Rj() do not commute except when i = j. It is left as an easy exercise to verify that [Ri(20),Rj(24)]=-2i8ijkksin0sin = Q. (N30) Consider, for instance, the triple rotation A = with i # j. We would intuitively think that the last rotation R (-0) cancels the effect of the first rotation R (A) and, therefore, that A = R (Q) but, as we shall see, this is not the case. Indeed: = i(0)]} (N31) = R() which proves that, except for i = j(Eijk = 0), in the general case A # R(). It is also straightforward to show that the commutation of the rotation operators with the Pauli matrices satisfy [Ri(20),oj]2i8ijksinok = (N32) This result expresses the fact that the two operators Ri(20), oj do not commute, except in the specific cases i = j (Eiik = 0) and 20 = 0, 2 (Ri(0) = Ri(2)==) Appendix 0 (Chapter 17) Heisenberg uncertainty principle In this appendix, I provide a demonstration of the Heisenberg uncertainty principle.¹ According to this principle, the uncertainties AA, AB associated with two observables A, B, measured in the state 1/s), must satisfy the inequality: To prove Eq. (01), define where 2 is a real parameter. We then obtain = ((A+-iB+)(A+iB)y where we us