Problems for
Intermediate Methods in Theoretical Physics

Edward F. Redish

Changing Bases

Consider two bases for a 2-D complex linear inner product space: |f1>, |f2> and |e1>, |e2> where
|e1> = a11 |f1> + a12 |f2>
|e2> = a21 |f1> + a22 |f2>

(a) If |f1>, |f2> is an orthonormal basis, what are the conditions on the coefficients {aij so that |e2> = a21 |f1> + a22 |f2> is also an orthonormal bases?

(b) Construct the 2x2 matrices Aij = <fi|ej> and Bij = <ei|fj>.

(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 = Σk Aik Bkj 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.

This page prepared by

Edward F. Redish
Department of Physics
University of Maryland
College Park, MD 20742
Phone: (301) 405-6120

Last revision 28. October, 2004.