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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 3D real space is given by  r  vector r vector r , also called the norm of the vector. For complexvalued vectors in Ndimensions 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 ddimensional space, but they are not orthonormal. Thedimensional 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.
 Fall '08
 MichaelMoore
 mechanics, Work

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