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348-problem9 - Chem 540 Instructor Nancy Makri PROBLEM 9...

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Chem. 540 Instructor: Nancy Makri PROBLEM 9 Suppose that 1 φ and 2 φ are two degenerate eigenstates of an operator ˆ A , but they are not chosen to be orthogonal; however, they are normalized to unity. Let the overlap of these states be 1 2 φ φ λ = . a) Construct new mutually orthogonal states 1 Φ and 2 Φ that correspond to the same eigen- value of ˆ A . There are many ways to do this, but perhaps the simplest is to keep 1 Φ as is and replace 2 Φ by its component that is orthogonal to 1 Φ . To do this, subtract from 2 Φ its component along 1 Φ . (You may write the coefficient of this component as a variable and de- termine its value by requiring zero overlap with
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