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Unformatted text preview: PHYS 455 Answers to Homework II 1. Consider a system with a 3 dimensional state space which is described as 3 × 1 column vectors. Let the state of the particle be ψ = N 1 1 + i 2 i , and the following observable is measured on the system, A = 0 0 1 0 1 0 1 0 0 (a) Find a value for N such that ψ is normalized. k ψ k 2 = h ψ  ψ i =  N  2 7 = 1 ⇒  N  = 1 / √ 7 We can choose N = 1 / √ 7 . (b) (i) What are the possible outcomes of the measurement, (ii) what are the probabilities of each outcome and (iii) what is the final (collapsed) state in each case? Remember the projection operators to the eigensubspaces of A from the last homework. A has two eigenvalues, +1 and 1 . The corresponding projection operators are P + = 1 2 1 0 1 0 2 0 1 0 1 P = 1 2 1 1 1 0 1 . So, (i) the possible outcomes are +1 and 1 . (ii) Their corresponding probabilities are p + = k P + ψ k 2 = 9 14 , p = k P + ψ k 2 = 5 14 . (iii) The final states are ψ + = P + ψ = 1 2 √ 7 1 + 2 i 2 + 2 i 1 + 2 i , ψ = P ψ = 1 2 √ 7 1 2 i 1 + 2 i . Note that ψ ± are not normalized. If you want to use normalized states, then the final states are ψ nrmd + = 1 √ p + ψ + = 1 √ 18 1 + 2 i 2 + 2 i 1 + 2 i , ψ nrmd = 1 √ p ψ = e iφ √ 2 1 1 where e iφ = (1 2 i ) / √ 5 is an overall phase factor....
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This note was uploaded on 02/11/2012 for the course MATH 435 taught by Professor Starg during the Spring '11 term at Al Ahliyya Amman University.
 Spring '11
 starg
 Vectors

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