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 n | φ n ) but that ( φ n | φ n ) negationslash = 1. Prove that | a n ) = c | φ n ) is also an eigenstate of A . What is its eigenvalue? What should c be so that ( a n | a n ) =1? 2. [10 pts] Assume that | φ n ) and | a n ) are degenerate eigenstates of A with eigenvalue a n . They sat- isfy ( φ n | φ n ) = ( a n | a n ) = 1 and ( φ n | a n ) negationslash = 1. Show that the states | a n , 1 ) = | a n ) and | a n , 2 ) = | φ n )−| a n )( a n | φ n ) ||| φ n )−| a n )( a n | φ n )|| are both eigenstates of A with eigenvalue a n , and are mutually orthogonal. Now assume that the system is in an arbitrary state | ψ ) 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. 3. [20 pts] Consider the wavefunction φ ( x ) ≡ ( x | φ ) = N e ( x - x 0 ) 2 2 σ 2 . What should the value of N be so that ( φ | φ ) = 1?
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