Angular Momentum Operator Algebra
Michael Fowler 10/29/07
Preliminaries: Translation and Rotation Operators
As a warm up to analyzing how a wave function transforms under rotation, we review the effect
of
linear translation
on a single particle wave function
( )
x
ψ
.
We have already seen an example
of this: the coherent states of a simple harmonic oscillator discussed earlier were (at
t
= 0)
identical to the ground state except that they were centered at some point displaced from the
origin. In fact, the operator creating such a state from the ground state is a translation operator.
The
translation operator
T
(
a
) is
defined
at that operator which when it acts on a wave function
ket
()
x
gives the ket corresponding to that wave function moved over by
a
, that is,
( ) ( )
( )
,
Ta
x
x a
ψψ
=−
so, for example, if
x
is a wave function centered at the origin,
T
(
a
) moves it to be centered at
the point
a
.
We have written the wave function as a ket here to emphasize the parallels between this
operation and some later ones, but it is simpler at this point to just work with the wave function
as a function, so we will drop the ket bracket for now.
The form of
T
(
a
) as an operator on a
function is made evident by rewriting the Taylor series in operator form:
(
)
() ()
22
2
2!
.
d
a
dx
da
d
xa
x
x
dx
dx
ex
x
−
−=
−
+
−
=
=
…
Now for the quantum connection: the differential operator appearing in the exponential is in
quantum mechanics proportional to the momentum operator ( ˆ
/
pi
d
d
x
= −
=
) so the translation
operator
( )
ˆ /
.
iap
Ta e
−
=
=
An important special case is that of an infinitesimal translation,
( )
ˆ /
ˆ
1/
ip
Te
i
p
ε
εε
−
==
−
=
=
.
The momentum operator
is said to be the
generator
of the translation.
ˆ
p
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(
A note on possibly confusing notation
: Shankar writes (page 281)
( )
.
Tx
x
ε
=+
Here
x
denotes a deltafunction type wave function centered at
x
. It might be better if he had written
()
00
Txx
, then we would see right away that this translates into the wave function
transformation
() ( ) ( )
xx
x
δδ
−=
−
−
, the sign of
now obviously consistent with our
usage above.)
It is important to be clear about whether the
system
is being translated by
a
, as we have done
above or whether, alternately, the
coordinate axes
are being translated by
a,
that latter would
result in the
opposite
change in the wave function. Translating the coordinate axes, along with
the apparatus and any external fields by
−
a
relative to the wave function would of course give
the same physics as translating the wave function by +
a
.
In fact, these two equivalent operations
are analogous to the time development of a wave function being described either by a
Schrödinger picture, in which the bras and kets change in time, but not the operators, and the
Heisenberg picture in which the operators develop but the bras and kets do not change.
To
pursue this analogy a little further, in the “Heisenberg” case
() () [ ]
ˆˆ
1/
/
ˆ
ˆ
ˆ
ˆ
,/
ip
xT
x
T
e
x
e
x
i
p
x
x
εε
ˆ
−−
→
=
==
=
and
is unchanged since it commutes with the operator.
So there are two possible ways to deal
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 Fall '07
 MichaelFowler
 Angular Momentum, Momentum, wave function, Rotation Operator

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