Physics 580
Handout 6
7 September 2010
Quantum Mechanics I
webusers.physics.illinois.edu/
∼
goldbart/
Prof. P. M. Goldbart
3135 (& 2115) ESB
University of Illinois
1) Conservation of angular momentum in Hamiltonian mechanics
: A particle moves
in three dimensions and its state is specified by the position
q
and the momentum
p
. Under
an infinitesimal rotation of the state,
q
and
p
transform according to:
q
→
q
+
δ
q
=
q
+
a
×
q
p
→
p
+
δ
p
=
p
+
a
×
p
where
a
is a vector of infinitesimal magnitude describing a rotation through angle

a

about
the axis specified by
a
/

a

.
a) Show that
G
≡
a
·
(
q
×
p
) is the generator of this infinitesimal transformation, and that
the transformation is canonical.
b) Consider the hamiltonian
H
=
1
2
m

p

2
+
U
(

q

)
.
Show that the Poisson bracket
{H
, G
}
vanishes for arbitrary infinitesimal
a
, and hence
that angular momentum is conserved.
c) The angular momentum vector
L
has cartesian components
L
μ
(with
µ
= 1
,
2
,
3).
Evaluate the Poisson brackets
{
L
μ
, L
ν
}
between them. Explain why the transformation
to a pair of components of angular momentum cannot be a canonical transformation.
2) Hamiltonian mechanics and Poisson brackets – only parts (g) to (l) are manda
tory
: As Hamilton discovered, there are significant virtues associated with making yet an
other reformulation of mechanics, beyond the
Newton
⇒
Lagrange
⇒
Stationary Action
reformulations that we have already encountered. Amongst these virtues are: a set of dy
namical equations – the socalled canonical equations – that is even more simple and sym
metrical in its structure than Lagrange’s equations; the freedom to make transformations not
only between convenient sets of generalised coordinates, but also to make socalled canonical
transformations,
i.e.
, transformations that mix the generalised momenta and coordinates
under which the canonical equations retain their form (thus breaking the link between coor
dinates and velocities that is inherent in the Lagrangian framework); and a formulation for
classical mechanics that provides the most straightforward platform from which to embark
upon quantum mechanics. In Hamilton’s formulation the coordinates and momenta appear
on what is an essentially equal footing.
Recall that in the Lagrangian formulation of mechanics we focus on the generalised
coordinates
{
q
j
}
n
j
=1
and the associated generalised velocities
{
˙
q
j
}
n
j
=1
.
In particular, the
1
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dynamics is governed by the Lagrangian function
L
, which is a function of the generalised
coordinates, the generalised velocities and, possibly, the time.
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 Fall '10
 Staff
 mechanics, Angular Momentum, Momentum, Work, Hamiltonian mechanics, Poisson bracket, Poisson brackets

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