The Quantum Mechanics Solver 14 uniquely illustrates the application of quantum mechanical concepts to various fields of modern physics. It aims at encouraging the reader to apply quantum mechanics to research problems in fields such as molecular physics, condensed matter physics or laser physics. Advanced undergraduates and graduate students will find a rich and challenging source of material for further exploration. This book consists of a series of problems concerning present-day experimental or theoretical questions on quantum mechanics | 128 13 Quantum Cryptography This reduces to a -i i a ii b -ii A b b i b . i d b b i az 1 az l - az 1 K 1 .P 1 B ffz 1 where the spin state of a is irrelevant. . For the first and second configurations we can summarize the results as follows 6a 6b Alice Bob Probability 0 0 0 0 0 n 2 0 n 2 h 2 h 2 p 1 -h 2 -h 2 p 1 h 2 h 2 p 1 2 -h 2 h 2 p 1 2 The results for 6a n 2 0b 0 are identical to those of 6a 0 0b n 2 similarly the case 6a n 2 6b n 2 is identical to 6a 0 6b 0 one actually recovers the same result for any 6a 0b . In the two cases a and d where 6a 0b . when they measure along the same axis Alice and Bob are sure to find the same result. . a Concerning the findings of Alice and of the spy we have Alice Spy Probability h 2 h 2 cos2 6s 2 h 2 -h 2 sin2 6s 2 -h 2 h 2 sin2 6s 2 -h 2 -h 2 cos2 6 s 2 b Concerning the findings of Bob and of the spy Spy Bob Probability h 2 h 2 cos2 6s 2 h 2 -h 2 sin2 6s 2 -h 2 h 2 sin2 6s 2 -h 2 -h 2 cos2 6s 2 c The probability that Alice and Bob find the same result has actually been calculated in questions b c we simply have p 6s 2 1 cos2 6S . d Amazingly enough the two expectation values are the same. On one hand one has f2n P 0s d0s 2n 3 4. On the other since P 0 1 and P n 2 1 2 on the average p 3 4 if the values 6s 0 and 6s n 2 are chosen with equal probabilities. Solutions 129 Section The Quantum Cryptography Procedure . Necessarily if 0a db the results of Alice and Bob must be the same. If a single measurement done along the same axis 0a gives different results for Alice and Bob a spy is certainly operating at least in an ideal experiment . If ha db on the average half of the results are the same half have opposite signs. . The only chance for the spy to remain invisible is that Alice and Bob always find the same results when they choose the same axis. For each pair of spins there is a probability 1 2 that they choose the same axis and there is in this case a probability 1 4 that they do
