Lecture 31:
Symmetry II
PHY851 Quantum Mechanics I
Fall, 2008
M.G. Moore
Review
•
A symmetry is a transformation which leaves the
Hamiltonian unchanged
•
Transformations are described by Unitary
operators:
–
States can transform as:
–
Operators can transform as:
•
We can generate continuoussymmetry operators
from observables via:
–
Here ‘
G
’ is the observable and
!
is the degree of
transformation
–
Example: position shift operator
•
A system is invariant under the transformation
when:
•
In which case the
generator
,
G
is a
constant of
motion
U
=
"
†
UOU
O
=
!
U
"
( )
=
e
#
iG
T d
( )
=
e
"
i
h
Pd
H
,
G
[ ]
=
0
T d
( )
x
=
x
+
d
p
(
g
,
t
)
=
p
(
g
,0)
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View Full DocumentActive Transformations
•
Question:
when we want to make a symmetry
transformation, should we transform both the
state and all observables?
•
A symmetry transformation is an active
transformation, in that the physical properties
of the transformed system are intended to be
changed by the transformation
•
Example:
–
suppose we have a wavepacket that is centered at
.
x
=
x
0
.
–
Suppose we want to move the position of our origin
to
x
=
d
.
–
Let
x
_
be the position measured in the new
coordinate system. It is related to
x
via:
–
With respect to our new coordinate system, the
wavepacket of our state should now be centered at
x
_ =
x
0

d
d
x
x
!
=
"
Classical versus Quantum symmetry
transformations
•
Notice that in classical mechanics, the state and
of the observables are the same thing. So there
is no question as to which should be transformed.
•
In QM state and observables are separate
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 Fall '08
 MichaelMoore
 mechanics, Hilbert space, Observables, symmetry transformation

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