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Unformatted text preview: HOMEWORK ASSIGNMENT 1 PHYS851 Quantum Mechanics I, Fall 2009 1. [10 pts]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 . Assume that H † = H what is ( n  H †  m ) in terms of ( m  H  n ) ? ( ψ  φ ) = ( φ  ψ ) ∗ . ( m  ˆ M †  n ) = ( n  ˆ M  m ) ∗ . ( n  ˆ H †  m ) = ( n  ˆ H  m ) † . Also, because H † = H , we have ( n  H †  m ) = ( n  H  m ) . Putting these together gives ( n  H  m ) = ( m  H  n ) , which is an equivalent definition of Hermiticity This says firstly, the diagonal elements of a hermitian operator must be real in every possible basis, and secondly, the offdiagonal elements come in complexconjugate pairs, such that H nm = H ∗ mn . Thus we can tell just by looking at it whether a matrix is Hermitian or not. 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? step 1: by definition, we have ( ˆ A ˆ B ) † = ( A ∗ B ∗ ) T step 2: by the standard rules of matrix algebra, we have ( 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. Consider the discrete orthonormal basis { m )} , m = 1 , 2 , 3 ,... ,M that spans an Mdimensional Hilbert space, H M . (a) Show that the identity operator, I = ∑ m  m )( m  , satisfies I 2 = I . I 2 = ( ∑ m  m )( m  )( ∑ n  n )( n  ) = ∑ mn  m )( m  n )( n  = ∑ mn  m ) δ m,n ( n  = ∑ m  m )( m  = I (b) Form a new projector, I !3 , by removing the state  3 ) , i.e. I !3 := ∑ m negationslash =3  m )( m  . Does I !3 2 = I !3 ? Is I !3 also the identity operator? I 2 !3 = ∑ m,n negationslash =3  m )( m  n )( n  = ∑ m,n negationslash =3  m ) δ m,n ( n  = ∑ m negationslash =3  m )( m  = I !3 , so yes to first question....
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This note was uploaded on 12/02/2010 for the course PHYSICS 851 taught by Professor M.moore during the Fall '09 term at Michigan State University.
 Fall '09
 M.Moore
 mechanics, Work

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