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# ps3 - Physics 115A Third Problem Set Harry Nelson Office...

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Physics 115A Third Problem Set Harry Nelson Office Hours (This Week) Th 1:30-3:00pm TA: Antonio Boveia Office Hours M 9-10am, Fr 1-3pm PLC Grader: Victor Soto Office Hours Th 11:00-12:30pm PLC due Monday, January 27, 2003 1. The three kets | 1 ) , | 2 ) , and | 3 ) are represented, in a particular basis as: | 1 ) . = 1 / 2 3 / 2 2 3 / 2 2 , | 2 ) . = - 3 / 2 1 / 2 2 1 / 2 2 , | 3 ) . = 0 1 / 2 - 1 / 2 . (a) Find the matrices that represent | 1 )( 1 | , | 2 )( 2 | , | 1 )( 1 | + | 2 )( 2 | , and | 1 )( 1 | + | 2 )( 2 | + | 3 )( 3 | . (b) Use the matrix representation to evaluate the action of | 1 )( 1 | + | 2 )( 2 | on | 3 ) , and interpret the result. (c) Evaluate and interpret the trace of | 1 )( 1 | + | 2 )( 2 | . 2. Use an insertion of the ‘decomposition of unity,’ which is the relationship n k =1 | k )( k | = I , to derive the ‘matrix multiplication’ relationship between the matrix elements of a sequence of two linear operators, ΛΩ , with matrix elements in the basis of ( i | ΛΩ | j ) , and the matrix elements of the individual operators ( i | Λ | k ) and ( k | Ω | j ) . 3. In class, and on page 22 of your text, the representation of the rotation operator
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