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HWkey2 - Chem 3502/5502 Physical Chemistry II(Quantum...

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Chem 3502/5502 Physical Chemistry II (Quantum Mechanics) 3 Credits Fall Semester 2010 Laura Gagliardi Answers to Homework Set 2 To be collected on September 24 2010 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 = φ 1 and u 2 = φ 2 S φ 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 = φ 1 φ 2 S φ 1 = φ 1 φ 2 − φ 1 S φ 1 = φ 1 φ 2 S φ 1 φ 1 = φ 1 φ 2 S φ 1 2 = φ 1 φ 2 S Where the last line follows from the normalization of φ 1 . From rearranging, we have S = φ 1 φ 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.
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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 2 ?
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