851HW1_09_Solutions - HOMEWORK ASSIGNMENT 1 PHYS851 Quantum...

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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 off-diagonal elements come in complex-conjugate 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 M -dimensional 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. Based on their definitions, I !3 is clearly not equivalent to I , so no on second question?
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