HW13 (2) - University of Colorado Department of Physics...

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University of Colorado, Department of Physics PHYS3220, Fall 09, HW#13 due Wed, Dec 2, 2PM at start of class 1. Matrix representation of the eigenvalue problem (Total: 20 pts) Suppose there are two observables A and B with corresponding Hermitian operators ˆ A and ˆ B . The eigenfunctions to ˆ A as well as to ˆ B form a complete set of basis functions, and satisfy the eigenvalue equations: ˆ A | a n > = a n | a n > and ˆ B | b n > = b n | b n > . (Here we are adopting a common practice of labeling an eigenstate by its eigenvalues, so | a n > is the eigenvector of ˆ A with eigenvalue a n . Do not confuse the eigenvalues, which are numbers, with their eigenvectors, which are vectors in Hilbert space). a) Since the eigenfunctions form a complete set, any wavefunction | Ψ( t ) > can be written as an expansion in either basis (we assume for simplicity a discrete basis): | Ψ( t ) > = X n c n ( t ) | a n > = X n d n ( t ) | b n > Derive a formula that expresses any particular d n in terms of the set of
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HW13 (2) - University of Colorado Department of Physics...

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