851HW2_09 - PHYS851 Quantum Mechanics I Fall 2007 HOMEWORK...

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PHYS851 Quantum Mechanics I, Fall 2007 HOMEWORK ASSIGNMENT 2: Postulates of Quantum Mechanics 1. [10 pts] Assume that A | φ n a = a n | φ n a but that A φ n | φ n a n = 1. Prove that | a n a = c | φ n a is also an eigenstate of A . What is its eigenvalue? What should c be so that A a n | a n a =1? 2. [10 pts] Assume that | φ n a and | a n a are degenerate eigenstates of A with eigenvalue a n . They sat- isfy A φ n | φ n a = A a n | a n a = 1 and A φ n | a n a n = 1. Show that the states | a n , 1 a = | a n a and | a n , 2 a = | φ n a−| a n aA a n | φ n a ||| φ n a−| a n aA a n | φ n a|| are both eigenstates of A with eigenvalue a n , and are mutually orthogonal. Now assume that the system is in an arbitrary state | ψ a when A is measured. Write an expression for the probability to obtain the result a n . Write down the state of the system immediately after the measurement, assuming that a n was obtained by random chance.
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851HW2_09 - PHYS851 Quantum Mechanics I Fall 2007 HOMEWORK...

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