Problem Set 1
Phy 315 - Fall 2006
Assigned: Thursday, September 7 Due: Thursday, September 14
Problem 1: Noether’s Theorem and Galilean Boosts
Consider the following Lagrangian for N particles moving in three-dimensions:
L=
N
X
i=1
N
X
1
mi ṙ2i −
V (ri − rj ) .
2
i6=j
A Galilean boost is defined by r0i = ri − vt. Consider an infinitesmal boost and,
a) work out the change in the Lagrangian,
δL =
d
Ω.
dt
What is Ω?
b) What is the conserved quantity, G, associated with Galilean invariance?
c) Verify that the G you found in part b) is the generator of Galilean boosts, i.e.
{ri , G}P B = δri
{pi , G}P B = δpi .
where {, }P B denotes the Poisson bracket and δri and δpi are the variations of the
coordinates and momenta, respectively, under infinitesmal Galilean boosts.
Problem 2: Angular Momentum and Generators of Rotations
a) Using the commutations for angular momentum, L, and position, r, show that
eiL·Θ re−iL·Θ = r + Θ × r .
to lowest order in Θ.
b) Show that







x

 x cos θ − y sin θ 

.
eiLz θ y e−iLz θ = x sin θ + y cos θ



z



z
Problem 3: Quantum Runge-Lenz Vector
For the central force problem in three-dimensions,
L=
e2
p2
−
2m
r
(1)
the quantum mechanical operator corresponding to the Runge-Lenz vector is
A=
L×p−p×L
r
+ m e2 .
2
r
(2)
Note that in order to obtain a Hermitian operator from the classical expression for A we
must make the substitution:
L×p→
L×p−p×L
2
(3)
a) Compute the following commutation relations:
[Li , Aj ] = ???
[Ai , Aj ] = ??? .
b) Show that
A · A = 2mE (L2 + h̄2 ) + m2 e4
(4)
This agrees with the classical result in the limit h̄ → 0, the term proportional to h̄2 is a
quantum correction.
Problem 4: Rotations in Four Dimensions
Consider a 4-dimensional space with coordinates (x, y, z, w).
a) Show that the operators
Li = ijk xj pk
Ki = w p i − x i p w ,
generate rotations in this space by showing that the transformations generated by these
operators leave the four dimensional radius, defined by R2 = x2 + y 2 + z 2 + w 2 , invariant.
b) Compute the commutators
[Li , Kj ] = ???
[Ki , Kj ] = ??? .
and show they same as those found in part b) of Problem 3, up to an overall rescaling of Ai .