Physics 263: Problem Set #16

1. (BTM 8.3.1.) Don't worry, the first one (eq. 8.3.1) is by far the worst and then the others are quick. The idea is to write M and N in terms of their elements M11, M12, and so on, and carry out the determinants. In the first one there are 8 terms, some of which cancel, and the rest factor into the product of |M||N|, as desired. You just have to slug it out. The others are done the same way, but only involve one matrix on each side of the equation. The matrix transpose and exchange are defined following equation (8.3.4).
2. (BTM 8.3.4.) This exercise is designed to convince you to use MATLAB. :) Actually, it's an application of Eq. (8.3.12) for the inverse of a matrix, applied to the 3x3 matrices formed by the coefficients on the left side. You can find the determinant by applying Eq. (8.3.9), which is illustrated in Eq. (8.3.11). Take a look here for an example of calculating matrix cofactors.
3. (BTM 8.4.2 and 8.4.3.) The first proof follows directly from Eq.(8.4.11), which defines the adjoint of a matrix (set i=j). The second proof is very similar to the proof of Eq.(8.4.4), which is given in (8.4.5) through (8.4.10).
4. (BTM 8.4.5.) To show a matrix is unitary, you can show that the matrix times its adjoint is the identity matrix. To find the adjoint, take the complex conjugate of each matrix element and then the transpose of the matrix. A unimodular complex number is one of the form e^{i theta}. If you write the determinant in the form r e^{i theta}, then you just need to show that r=1.
5. (BTM 8.4.9.) Working to second order means that you can throw out any term that has M and L to a total power of three. That is, M³ is gone but also L²M, and so on.
6. (BTM 8.4.10.) The game is to show that the adjoint of U is e^-iH by following the hint given in the problem.