HW1_Solutions - HOMEWORK ASSIGNMENT 1 PHYS851 Quantum...

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Unformatted text preview: HOMEWORK ASSIGNMENT 1 PHYS851 Quantum Mechanics I, Fall 2008 1. [10 pts]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 ) ? ( | ) = ( | ) . ( m | M | n ) = ( n | M | m ) . ( m | H | n ) = ( 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 | ) ? ( A B ) = ( A B ) T ( A B ) T mn = ( A B ) nm = k A nk B km = k ( B ) T mk ( A ) T kn = k B mk A kn = ( B A ) mn 3. [10 pts] 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 N-dimensions this generalizes to || || radicalbig ( | ) . Expand | ) onto the basis of states {| n )} , n = 1 , 2 ,... N , and express its magnitude in terms of the components n = ( n | ) . || || 2 = ( | ) = n ( | n )( n | ) = n n n = n | n | 2 || || = radicalbig n | n | 2 4. [20 pts] 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. The-dimensional space, but they are not orthonormal....
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This note was uploaded on 10/25/2010 for the course PHYSICS PHYS 851 taught by Professor Michaelmoore during the Fall '08 term at Michigan State University.

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HW1_Solutions - HOMEWORK ASSIGNMENT 1 PHYS851 Quantum...

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