7—Operators and Matrices 2037.13 A cube of uniform volume mass density, massm, and sideahas one corner at the origin of the coordinate
system and the adjacent edges are placed along the coordinate axes. Compute the components of the tensor of
inertia. Do it directly and by using the parallel axis theorem to check your result.
Ans:ma^2
2 / 3 − 1 / 4 − 1 / 4
− 1 /4 2/ 3 − 1 / 4
− 1 / 4 − 1 /4 2/ 3
7.14 Compute the cube of Eq. ( 11 ) to find the trigonometric identities for the cosine and sine of triple angles in
terms of single angle sines and cosines. Compare the results of problem3.9.
7.15( On the vectors of column matrices, the operators are matrices. For the two dimensional case takeM =
a b
c d
)
and find its components in the basis(
1
1
)
and(
1
− 1
)
.
What is the determinant of the resulting matrix? Ans:M 11 = (a+b+c+d)/ 2.
7.16 Show that the tensor of inertia, Eq. ( 3 ), satisfies~ω 1 .I(~ω 2 ) =I(~ω 1 ).~ω 2. What does this identity tell you
about the components of the operator when you use the ordinary orthonormal basis? First determine in such a
basis what~e 1 .I(~e 2 )is. This identity is to linear operators (tensors) what Eq. (5.12) is to Fourier series.
7.17 Use the definition of the center of mass to show that the two cross terms in Eq. ( 17 ) are zero.
7.18 Prove the Perpendicular Axis Theorem. This says that for a mass that lies flat in a plane, the moment
of inertia about an axis perpendicular to the plane equals the sum of the two moments of inertia about the two
perpendicular axes that lie in the plane and that intersect the third axis.
7.19 Verify in the conventional, non-matrix way that Eq. ( 36 ) really does provide a solution to the original second
order differential equation ( 34 ).
7.20 The Pauli spin matrices are
σx=(
0 1
1 0
)
, σy=(
0 −i
i 0)
, σz=(
1 0
0 − 1
)
Show thatσxσy=iσzand the same for cyclic permutations of the indicesx,y,z. Compare the productsσxσy
andσyσxand the other pairings of these matrices.