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Unformatted text preview: Physics
601
Homework
3­­­Due
Friday
September
24
 
 
 Goldstein
problems

8.1,

8.6,
8.9,
8.26,
8.23
 
 In
addition:
 
 1. For
a
particle
in
3
dimensions
the
angular
momentum
operator
is
given
by
 L = x × p 
where
 x 
and
 p 
satisfy
are
canonical
(i.e.
they
satisfy
the
canonical
 Poisson
bracket
relations.
 a. Show
that
 [ Li , L j ]PB = εijk Lk 
where
i,j,k

take
on
the
values
x,y,z.

(Note
 that
this
is
isomorphic
to
the

commutators
for
the
angular
 € € € momentum
in
quantum
mechanics.

For
those
with
mathematically
 inclinations,
this
is
the
Lie
algebra
SO(3).)

 € b. Show
that
 [ L, L2 ]PB = 0 

(that
is
that [ Li , L2 ]PB = 0 
for
all
i)
where
 L2 = L ⋅ L 
.
 c. Show
that
 [ L, f ( r)]PB = 0 

where
f
is
an
arbitrary
function
and
 r€ x ⋅ x .
 = € Note
that
parts
b.
&c.
reflect
a
deeper
result
 [ L, s]PB = 0 
for
any
scalar
s.

This
 € reflects
the
fact
that
the
angular
momentum
is
the
generator
of
rotations.



 € 
€ 2. Label
our
canonical
variables
by
a
phase‐space
vector



 € 
 € q1 q2 ... qn η= 
which
satisfies
canonical
Poisson
brackets
 [ηi ,η j ]PB = J ij 
 p1 p2 ... € pn1 
 now
suppose
that

there
is
a
one
parameter
continuous
family
of

 canonical
transformations
depending
on
one
parameter
ε :
 ξ (η;ε) 

. a. 
Show
that
 ξ (η;ε) 

satisfies
 [ξ i ,ξ j ]PB = J ij 
if
it
satisfies
the
conditions

 dξ i = [ξ i , g]PB with ξ (η;0) = η 
for
some

g.

Note
that
this
is
the
same
 dε € form
as
usual
Hamiltonian
time

with
t
replaced
by
ε and
H
replaced
 € € by
g.

The
function
g
is
called
the
generator
of
the
transformation.



 
 € € b. The
time
evolution
of
the
phase‐space
position
under
Hamiltonian
 gives
a
transformation
from
an
initial
point
in
phase
space
to
a
 
 
 subsequent
one.

That
is
 η( t ) is
really
a
function
the
time
and
the
 initial
conditions:
 η(η0 , t ) .

Note
that
the
initial
conditions
are
a
set
of
 phase
space
variable
which
are
canonical.

Now
let
me
define
a
 canonical
transformation
 ξ (η, T ) 
which
has
the
same
functional
 € relation
as

 η(η0 , t ) .

That
is
 ξ (η, T ) corresponds
to
the
value
the
point
 € in
phase
space
that
any
point
in
phase
space
evolves
into
from
 η 
a
 time
T
later.

Using
part
a)
show
that
for
any
T
,
 η(ξ , T ) 

satisfies
the
 € canonical
Poission
bracket
relation
with
H
acting
as
the
generator
of
 € € time
translations.
 € € 
 
 
 
 ...
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This note was uploaded on 12/29/2011 for the course PHYSICS 601 taught by Professor Hassam during the Fall '11 term at Maryland.

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