Quantum Mechanics for Mathematicians

B.9 Chapters 18 and 19

Problem 1:
Starting with the Lie algebraso(3), with basisl 1 ,l 2 ,l 3 , consider new basis
elements given by leavingl 3 alone and rescaling

l 1 →l 1 /R, l 2 →−l 2 /R

whereRis a real parameter. Show that in the limitR→ ∞these new basis
elements satisfy the Lie bracket relations for the Lie algebra ofE(2). Because
of this, the groupE(2) is sometimes said to be a “contraction” ofSO(3).

Problem 2:
For the case of the groupE(2), show that in any representationπ′of its Lie
algebra, there is a Casimir operator

|P|^2 =π′(p 1 )π′(p 1 ) +π′(p 2 )π′(p 2 )

that commutes with all the Lie algebra representation operators (i.e., with
π′(p 1 ), π′(p 2 ), π′(l)).
For the case of the groupE(3), similarly show that there are two Casimir
operators.
|P|^2 =π′(p 1 )π′(p 1 ) +π′(p 2 )π′(p 2 ) +π′(p 3 )π′(p 3 )

and
L·P=π′(l 1 )π′(p 1 ) +π′(l 2 )π′(p 2 ) +π′(l 3 )π′(p 3 )

that commute with all the Lie algebra representation operators.

Problem 3:
Show that theE(3) Casimir operatorL·Pacts trivially on theE(3) repre-
sentation on free-particle wavefunctions of energyE >0.

B.10 Chapters 21 and 22

Problem 1:
Consider the classical Hamiltonian function for a particle moving in a central
potential

h=

1

2 m

(p^21 +p^22 +p^23 ) +V(r)

where
r^2 =q 12 +q^22 +q 32

• Show that the angular momentum functionsljsatisfy

{lj,h}= 0

and note that this implies that theljare conserved functions along clas-

sical trajectories.