7—Operators and Matrices 2047.21 Interpret~σ.A~asσxAx+σyAy+σzAzand prove that
~σ.A~σ~ .B~=A~.B~+i~σ.A~×B~where the first term on the right has to include the identity matrix for this to make sense.
7.22 Evaluate the matrix
I
I−~σ.A~
(What dimensions or units mustA~have for this to make sense?) You can evaluate this a couple of ways: You
may assume thatA~is in some sense small enough for you to manipulate by infinite series methods. This then
becomes a geometric series that you can sum. Use the results of the preceding problem.
(b) You can manipulate the algebra directly without series. I suggest that you recall the sort of manipulation that
allows you to write the complex number 1 /(1−i)without anyi’s in the denominator.
Do it both ways, perhaps using one to guide the other. I suppose you could do it a third way, writing out the
2 × 2 matrix and explicitly inverting it, but I definitely don’t recommend this.
7.23 Evaluate the sum of the infinite series defined bye−iσyθ. Where have you seen this result before? The first
term in the series must be interpreted as the identity matrix.
7.24 For the moment of inertia about an axis, the integral is
∫
r^2 ⊥dm. State precisely what thismfunction must
be for this to make sense as a Riemann-Stieljes integral, Eq. (1.21). For the case that you have eight masses, all
m 0 at the 8 corners of a cube, write explicitly what this function is and evaluate the moment of inertia about an
axis along one edge of the cube.
7.25 The summation convention allows you to write some compact formulas. Evaluate these, assuming that
you’re dealing with three dimensions. Note Eq. ( 24 ). Define the alternating symbolijkto be (1) This is totally
anti-symmetric. That is, interchange any two indices and you change the sign of the value. (2) 123 = 1.
[E.g. 132 =− 1 , 312 = +1]
δii, ijkAjBk, δijijk, δmnAmBn, Smnumvn, unvn,
ijkmnk=δimδjn−δinδjm
Multiply the last identity byAjBmCnand interpret.