By Stewart, McMinnis, Melesco.
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Extra info for 101 Database Exercises Text-Workbook
If it is possible for C to convert a pure state to a mixture, then we would have the following: Cei = ck ek . 86) k Since C is reversible, C above, which yields −1 exists, and we can apply it to both sides of the equation C −1 Cei = ei = ck C −1 ek . 87) k But this says that ei is a mixture,13 which it is not. Hence we must conclude that C cannot convert ek to a mixture. Beyond linearity We can always think of a quite general transformation on a mixture deﬁned by the following formula: q= q i p0 , p1 , .
They imagined that atoms (and they did not suspect at the time that atoms could be made of even smaller constituents) would be subject to the same the laws of Newtonian dynamics that worked so well on cannon balls and anvils sliding oﬀ rotating wedges with rough surfaces. It is just as well that they were wrong because, otherwise, nothing would work and we could not and would not be here to discuss these issues. Matter based on the principles of classical physics would cease to exist almost instantaneously.
The arithmetic mean of fi over the ensemble is f¯ = = 1 np0 f0 + np1 f1 + · · · + np7 f7 n n n 7 pi fi = i=0 pi fi = pi f (ei ) . 29) i . By deﬁning that f (p) = f¯, we obtain the following formula: p i ei f (p) = f = i pi f (ei ) . 31) i Now let us take two arbitrary states10 p1 = pi 1 e i and p2 = i i and evaluate f on their mixture: f (P1 p1 + P2 p2 ) = f pi 1 ei + P2 P1 i = pi 2 e i i P1 pi 1 + P2 pi 2 ei f i i i i p 1 f (ei ) + P2 = P1 P1 pi 1 + P2 pi 2 f (ei ) = i p 2 f (ei ) i pi 1 ei = P1 f = P1 f (p1 ) + P2 f (p2 ) .
101 Database Exercises Text-Workbook by Stewart, McMinnis, Melesco.