PHZ 3113 Fall 2010
Homework #7, Due Friday, November 12
1. Compute
Z ∞
dx
−∞
[1 + (x − a)2 ][1 + (x + a)2 ]
2. Let
1
Q= √
2
0
1
0
1
0
1
0
1
0
.
!
.
(a) Compute Q2 . Compute Q3 . What is Q to any odd power, Q2k+1 .
(b) What is sinh(ζQ)?
3. Show that two successive rotations about a given axis is equivalent to a single rotation
about the same axis. Find the net rotation angle.
4. Let the three matrices
J1 =
0
0
0
0
0
0 −1
1
0
!
J2 =
0
0
−1
0
0
0
1
0
0
!
J3 =
0 −1
1
0
0
0
0
0
0
!
form a vector J = (J1 , J2 , J3 ).
(a) The commutator of two matrices is defined to be [A, B] = AB − BA. Compute the
commutators [Ji , Jj ]. There are nine ij combinations, but only three need to be multiplied
out in detail. See if you can encapsulate your results using ijk .
(b) The action of a rotation by θ on a vector v is v 0 = exp(θ · J ) v. Show that for small ∆θ,
the change in v is ∆v = ∆θ × v.
(c) The action of a rotation can also be considered as acting on J by conjugation, such that
J 0 = exp(θ · J ) J exp(−θ · J ). Show that for small ∆θ, the change in J is ∆J = −∆θ × J .
(A change in J means a change in axis directions; confuse yourself by reading about “active”
and “passive” rotations.)