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Additional info for New perspectives in quantum systems in chemistry and physics, part 2
Using the results in (21) gives e m si = MT(1 + L a ) ' n = MTILmbi, si b - (V~)-av (1 + La) (42) W i t h these choices the explicit forms for the translationally invariant coordinates are L H t~ = x~ + a ~] x~ + ~ v~xy j=l (43) j=l H t~ = ~--~x y V ~ , j=l i---- 1 , 2 , . . , H - 1 (44) Some Mathematical Problems in the Description of Dissociating Molecules 29 and for translational invariance of ti, from (43), H (1+La) =-Evj j=1 Their inverses are xi = X + tz - L H-1 a E te + ~ bj tn (1 + La) j=1 j=1 (45) -1 46 xt - X + E tj (V")_ jz j=1 If the relations (36) to (39) are satisfied then it is easy to show that (35) becomes e n P(tetnXT) = (t t XT) where P V )zj (H)tij = ((V) Pe 0 ( 0 0 0 H 0 0 1 ) (47) 2j = 1, 2, .
In the case considered above, where upspin and down-spin states are found with equal probability pt = p2 = ], it is evident that at = a~. = ~/~. But, however general the context, a probability is always the square of an amplitude. One of the triumphs of quantum mechanics is that it allows us to deal with states of a system in which the value of some quantity is not known, in terms of states in which its value is perfectly definite. What conclusion can be drawn from all this? e. in the analyser) a state vector is rotated until it points along the axis corresponding to one of the observable values (sl,s2) - which are called 'eigenvalues'.
Our future work on charge transfer collisions will concentrate on fine-structure effects and on polyatomic targets, which present a whole new range of challenges for the type of methodology presented here. ; Ralmondi, M. Phys. Rev. A 1992, 46, 3846. ; Zygelman, B. J. Phys. B. 1994, 27, L73. ; Cooper, D . L . J . Phys. B.