Problems for Edward F. Redish |
(a) If |f_{1}>, |f_{2}> is an orthonormal basis, what are the conditions on the coefficients {a_{ij} so that |e_{2}> = a_{21} |f_{1}> + a_{22} |f_{2}> is also an orthonormal bases?
(b) Construct the 2x2 matrices A_{ij} = <f_{i}|e_{j}> and B_{ij} = <e_{i}|f_{j}>.
(c) Prove that these are the matrices that transform the coordinates of an arbitrary vector in the f-basis to its representation in the e-basis and back.
(d) Calculate the matrix product (AB)_{ij} = Σ_{k} A_{ik} B_{kj} in two ways; first, by putting your matrices in explicitly from part (b) and multiplying them, second, by writing the A and B matrix elements as Dirac inner products and isolating the k summation.
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Last revision 28. October, 2004.