7—Operators and Matrices 174Show thatσxσy=iσz and the same for cyclic permutations of the indicesx,y,z. Compare the
productsσxσyandσyσxand the other pairings of these matrices.
7.21 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~
=
(
I−~σ.A~
)− 1
Evaluate this by two methods: (a) 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.
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. Ans:Icosθ−iσysinθ
7.24 For the moment of inertia about an axis, the integral is
∫
r^2 ⊥dm. State precisely what thism
function must be for this to make sense as a Riemann-Stieltjes integral, Eq. (1.28). For the case that
you have eight masses, allm 0 at the 8 corners of a cube of sidea, write explicitly what this function
is and evaluate the moment of inertia about an axis along one edge of the cube.
7.25The summation convention allows you to write some compact formulas. Evaluate these, assuming
that you’re dealing with three dimensions. Note Eq. (7.30). Define the alternating symbolijkto be
1: It 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.
7.26 The set of Hermite polynomials starts out as
H 0 = 1, H 1 = 2x, H 2 = 4x^2 − 2 , H 3 = 8x^3 − 12 x, H 4 = 16x^4 − 48 x^2 + 12,
(a) For the vector space of cubic polynomials inx, choose a basis of Hermite polynomials and compute
the matrix of components of the differentiation operator,d/dx.
(b) Compute the components of the operatord^2 /dx^2 and show the relation between this matrix and
the preceding one.