Physics 263: Problem Set #17

1. (BTM 8.4.16.) It's actually kind of hard to "show" that the Pauli matrices are Hermitian, since it is apparent by inspection! Just describe what taking the adjoint involves and then state the result. In any case, the main point here is to play with the matrices to get familiar with them. If you can show that a matrix is Hermitian AND equal to its inverse, then it much be unitary! (Check the definition.)
2. (BTM 8.4.17 and 8.4.18.) For the first one, write out the dot product in terms of the Pauli matrices and then multiply the whole thing out. Apply the results of the last problem to simplify to the desired form (e.g., products of Pauli matrices either give the identity or another Pauli matrix). Follow the instructions for the second part and remember what the cross product of a vector with itself yields. The exponential of a matrix is defined by the Taylor expansion; group the even and odd powers together separately and use your result from 8.4.17.
3. (BTM 8.4.19.) These proofs are most easily carried out by writing the matrix multiplications and the trace in terms of components. Thus, MN = M_ij N_jk, using the Einstein summation convention (repeated indices are summed over) to reduce clutter. Then Tr MN = M_ij N_ji. Do the same with the other side of the equation and you'll see that they are the same. For the third one, apply the results of the second one!
4. (BTM 9.1.1.) The answers are given in the back of the book, and they are a strong guide. To test whether they are linear vector spaces, step through the list of axioms until you agree with them all or find one that doesn't work (one is enough!).
5. (BTM 9.1.2.) Once again, the answers are in the back. In this case, Shankar wants you to list all of the axioms that go wrong (if any do).
6. (BONUS: BTM 9.1.3.) Follow Shankar's hints and you should find your way through each of the proofs. These properties seem obvious when you consider any of the specific examples!