851HW2_09_Solutions

851HW2_09_Solutions - PHYS851 Quantum Mechanics I, Fall...

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Unformatted text preview: PHYS851 Quantum Mechanics I, Fall 2009 HOMEWORK ASSIGNMENT 2: Postulates of Quantum Mechanics 1. [10 pts] Assume that A | n ) = a n | n ) but that ( n | n ) negationslash = 1. Prove that | a n ) = c | n ) is also an eigenstate of A . What is its eigenvalue? What should c be so that ( a n | a n ) =1? Solution: A | a n ) = Ac | n ) = cA | n ) = ca n | n ) = a n ( c | n ) ) = a n | a n ) . ( a n | a n ) = | c | 2 ( n | n ) . Setting equal to unity requires c = 1 radicalbig ( n | n ) . 2. [10 pts] Assume that | n ) and | a n ) are degenerate eigenstates of A with eigenvalue a n . They sat- isfy ( n | n ) = ( a n | a n ) = 1 and ( n | a n ) negationslash = 1. Show that the states | a n , 1 ) = | a n ) and | a n , 2 ) = | n )| a n )( a n | n ) ||| n )| a n )( a n | n )|| are both eigenstates of A with eigenvalue a n , and are mutually orthogonal. Now assume that the system is in an arbitrary state | ) 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. Solution: A | a n , 1 ) = A | a n ) = a n | a n ) Let N = ||| n ) | a n )( a n | n )|| , A | a n , 2 ) = A | n ) | a n )( a n | n ) N = A | n ) A | a n )( a n | n ) N = a n | n ) a n | a n )( a n | n ) N = a n | n ) | a n )( a n | n ) N ( a n , 1 | a n , 2 ) = ( a n | | n ) | a n )( a n | n ) N = ( a n | n ) ( a n | a n )( a n | n ) N = ( a n | n ) ( a n | n ) N = 0 . P ( a n ) = ( | parenleftBig | a n , 1 )( a n , 1 | + | a n , 2 )( a n , 2 | parenrightBig | ) = |( a n , 1 | )| 2 + |( a n , 2 | )| 2 | ) = 1 radicalbig P ( a n ) parenleftBig | a n , 1 )( a n , 1 | + | a n , 2 )( a n , 2 | parenrightBig | )...
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851HW2_09_Solutions - PHYS851 Quantum Mechanics I, Fall...

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