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Unformatted text preview: Chem. 540
Instructor: Nancy Makri PROBLEM 9
Suppose that φ1 and φ2 are two degenerate eigenstates of an operator A , but they are not
chosen to be orthogonal; however, they are normalized to unity. Let the overlap of these states be φ1 φ2 = λ .
a) Construct new mutually orthogonal states Φ1 and Φ 2 that correspond to the same eigen - ˆ
value of A . There are many ways to do this, but perhaps the simplest is to keep Φ1 as is and
replace Φ 2 by its component that is orthogonal to Φ1 . To do this, subtract from Φ 2 its
component along Φ1 . (You may write the coefficient of this component as a variable and de termine its value by requiring zero overlap with Φ1 .)
b) Are φ1 and φ2 ˆ
orthogonal to other eigenstates Φ n , n > 2 of A that have eigenvalues different from the eigenvalues of the two degenerate states? Are the new states Φ1 and Φ 2
orthogonal to these other states? c) Are the states Φ1 and Φ 2 uniquely defined? Hint: Take x approximately equidistant from the two holes, so |<x|1>|=|<x|2>|.
Problem 9: Subtract from one of the states its component along the other state (possibly us ing an unknown multiplicative factor to be determine from the orthogonality condition) ...
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This note was uploaded on 02/08/2012 for the course CHEM 540 taught by Professor Mccall during the Fall '08 term at University of Illinois, Urbana Champaign.
- Fall '08