HW1 - HOMEWORK ASSIGNMENT 1 PHYS851 Quantum Mechanics I,...

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Unformatted text preview: HOMEWORK ASSIGNMENT 1 PHYS851 Quantum Mechanics I, Fall 2008 1. What is the relationship between ( | ) and ( | ) ? What is the relationship between the matrix elements of M and the matrix elements of M ? Assume that H = H what is ( m | H | n ) in terms of ( m | H | n ) ? 2. Prove that ( AB ) = B A , where A and B are both operators (Hint: Switch to a matrix formalism and use summation notation). What is ( | AB | ) ? 3. The magnitude of a vector in 3-D real space is given by || r || vector r vector r , also called the norm of the vector. For complex-valued vectors in M-dimensions this generalizes to || || radicalbig ( | ) . Expand | ) onto the basis of states {| n )} , n = 1 , 2 ,... M , and express its magnitude in terms of the components c n = ( n | ) . 4. Suppose you start out with a set of d linearly independent vectors; | e 1 ) , | e 2 ) ,... , | e d ) ; in a d- dimensional space, but they are not orthonormal. Thedimensional space, but they are not orthonormal....
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