851HW1_09 - HOMEWORK ASSIGNMENT 1 Due Monday PHYS851...

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HOMEWORK ASSIGNMENT 1: Due Monday, 9/14/09 PHYS851 Quantum Mechanics I, Fall 2009 1. What is the relationship between ( ψ | φ ) and ( φ | ψ ) ? What is the relationship between the matrix elements of ˆ M and the matrix elements of ˆ M ? Assuming that H = H , what is ( n | H | m ) in terms of ( m | H | n ) ? 2. Use the matrix representation and summation notation to prove that ( AB ) = B A , where A and B are both operators. Use summation notation to expand ( φ | AB | ψ ) in terms of the constituent matrix elements and vector components? 3. Consider the discrete orthonormal basis {| m )} , m = 1 , 2 , 3 ,...,M that spans an M -dimensional Hilbert space, H M . (a) Show that the identity operator, ˆ I = m | m )( m | , satisfies ˆ I 2 = ˆ I . (b) Form a new projector, ˆ P , by removing the state | 3 ) , i.e. ˆ P = m negationslash =3 | m )( m | . Does ˆ P 2 = ˆ P ? Is P also the identity operator? (c) Compute the trace of ˆ P and compare it to the trace of ˆ I . (d) Based on the previous results, formulate the two necessary and sufficient conditions for an
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