7—Operators and Matrices 2077.39 The commutator of two matrices is defined to be[A,B] =AB−BA. Show that this commutator satisfies
the Jacobi identity.
[A,[B,C]] + [B,[C,A]] + [C,[A,B]] = 0
The anti-commutator of two matrices is{A,B}=AB+BA. Show that there is an identity like the Jacobi
identity, but with one of the two commutators (the inner one or the outer one) replaced by an anti-commutator.
I’ll leave it to you to figure out which.
7.40 Diagonalize each of the Pauli spin matrices of problem 20. That is, find their eigenvalues and specify the
respective eigenvectors as the basis in which they are diagonal.
7.41 What are the eigenvalues and eigenvectors of the rotation matrix Eq. ( 11 )? Translate the answer back into
a statement about rotating vectors, not just their components.
7.42 Same as the preceding problem, but replace the circular trigonometric functions with hyperbolic ones. Also
change the sole minus sign in the matrix to a plus sign. Draw pictures of what this matrix does to the basis
vectors. What is its determinant?
7.43 Compute the eigenvalues and eigenvectors of the matrix Eq. ( 15 ). Interpret each.
7.44 Look again at the vector space of problem6.36and use the basisf 1 ,f 2 ,f 3 that you constructed there. In
this basis, what are the components of the two operators described in that problem?
(b) What is the product of these two matrices? Do it in the order so that it represents the composition of the
first rotation followed by the second rotation.
(c) Find the eigenvectors of this product and from the result show that the combination of the two rotations is
a third rotation about an axis that you can now specify. Can you anticipate before solving it, what one of the
eigenvalues will be?
(d) Does a sketch of this rotation axis agree with what you should get by doing the two original rotations in
order?