851HW1_09_Solutions

# 851HW1_09_Solutions - HOMEWORK ASSIGNMENT 1 PHYS851 Quantum...

• Notes
• 3

This preview shows pages 1–2. Sign up to view the full content.

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?

This preview has intentionally blurred sections. Sign up to view the full version.

This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern