3502_HWkey2

3502_HWkey2 - Chem 3502/5502 Physical Chemistry II(Quantum...

This preview shows pages 1–3. Sign up to view the full content.

Chem 3502/5502 Physical Chemistry II (Quantum Mechanics) 3 Credits Fall Semester 2009 Laura Gagliardi Answers to Homework Set 2 From lecture 5: Consider a pair of degenerate, normalized eigenfunctions, φ 1 and φ 2 , of a Hermitian operator A with common eigenvalue a . Show that two new functions defined as u 1 = φ 1 and u 2 = φ 2 + S φ 1 are orthogonal, provided that S is properly chosen (i.e., determine what value of S is required to enforce orthogonality). Show that u 1 and u 2 remain degenerate with common eigenvalue a . We are told that φ 1 and φ 2 are degenerate eigenfunctions of A with eigenvalue a . The question is how to pick a value S such that the two functions u 1 = " 1 and u 2 = " 2 # S " 1 will be orthogonal to one another and still be eigenfunctions of A with eigenvalue a . To begin, we simply proceed from the definition of orthogonality (and practice our newfound mastery of Dirac notation 0 = u 1 u 2 = " 1 " 2 # S " 1 = " 1 " 2 # " 1 S " 1 = " 1 " 2 # S " 1 " 1 = " 1 " 2 # S " 1 2 = " 1 " 2 # S Where the last line follows from the normalization of φ 1 . From rearranging, we have S = " 1 " 2 This integral ( S ) is called the “overlap integral” between functions φ 1 and φ 2 . Notice that it is zero if the functions are already orthogonal, and it is the square modulus of φ 1 if the functions are identical (equal to one when the functions are normalized). Thus, S , ranges from 0 to 1 depending on how much the two functions “overlap” and hence its name.

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

View Full Document
HW2-2 This choice of S guarantees orthogonality, but we need to verify that u 1 and u 2 are eigenfunctions of A with eigenvalues a . Of course, by definition of u 1 this is a given, since it is unchanged from φ 1 , but what about u
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 12/18/2010 for the course CHEM 3502 taught by Professor Staff during the Fall '08 term at Minnesota.

Page1 / 4

3502_HWkey2 - Chem 3502/5502 Physical Chemistry II(Quantum...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online