7—Operators and Matrices 173and repeat the problem, finding components of the first and second derivative operators. Verify an
example explicitly to check that your matrix reproduces the effect of differentiation on a polynomial of
your choice. Pick one that will let you test your results.
7.10 What is the determinant of the inverse of an operator, explaining why?
Ans: 1 /det(original operator)
7.11 Eight identical point massesmare placed at the corners of a cube that has one corner at the origin
of the coordinates and has its sides along the axes. The side of the cube is length=a. In the basis that
is placed along the axes as usual, compute the components of the inertia tensor. Ans:I 11 = 8ma^2
7.12 For the dumbbell rotating about the off-axis axis in Eq. (7.19), what is the time-derivative ofL~?
In very short timedt, what new direction does~Ltake and what then isd~L? That will tell youdL/dt~.
Prove that this is~ω×~L.
7.13 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 (a) directly and (b) 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. (7.13) 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 take
M=
(
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 , and the determinant
isstillad−bc.
7.16 Show that the tensor of inertia, Eq. (7.3), 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.
7.17 Use the definition of the center of mass to show that the two cross terms in Eq. (7.21) 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. (7.61) really does provide a solution to the
original second order differential equation (7.59).
7.20 The Pauli spin matrices are