Physics 6210/Spring 2007/Lecture 22
Lecture 22
Relevant sections in text:
§
3.1, 3.2
Rotations in quantum mechanics
Now we will discuss what the preceding considerations have to do with quantum me
chanics. In quantum mechanics transformations in space and time are “implemented” or
“represented” by unitary transformations on the Hilbert space for the system. The idea
is that if you apply some transformation to a physical system in 3d, the state of the
system is changed and this should be mathematically represented as a transformation of
the state vector for the system. We have already seen how time translations and spatial
translations are described in this fashion. Following this same pattern, to each rotation
R
we want to define a unitary transformation,
D
(
R
), such that if

ψ
is the state vector
for the system, then
D
(
R
)

ψ
represents the state vector after the system has undergone a
rotation characterized by
R
. The key requirement here is that the pattern for combining
two rotations to make a third rotation is “mimicked” by the unitary operators. For this
we require that the unitary operators
D
(
R
) depend continuously upon the rotation axis
and angle and satisfy
D
(
R
1
)
D
(
R
2
) =
e
iω
12
D
(
R
1
R
2
)
,
where
ω
12
is a real number, which may depend upon the choice of rotations
R
1
and
R
2
,
as its notation suggests. This phase freedom is allowed since the state vector
D
(
R
1
R
2
)

ψ
cannot be
physically
distinguished from
e
iω
12
D
(
R
1
R
2
)

ψ
.
If we succeed in constructing this family of unitary operators
D
(
R
), we say we have
constructed a “unitary representation of the rotation group up to a phase”, or a “projective
unitary representation of the rotation group”. You can think of all the
ω
parameters as
simply specifying, in part, some of the freedom one has in building the unitary represen
tatives of rotations.
(If the representation has all the
ω
parameters vanishing we speak
simply of a “unitary representation of the rotation group”.)
This possible phase freedom in the combination rule for representatives of rotations is
a purely quantum mechanical possibility and has important physical consequences. Inci
dentally, your text book fails to allow for this phase freedom in the general definition of
representation of rotations.
This is a pedagogical error, and an important one at that.
This error is quite ironic: the first example the text gives of the
D
(
R
) operators is for a
spin 1/2 system where the phase factors are definitely nontrivial, as we shall see.
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 Spring '07
 M
 mechanics, Angular Momentum, Hilbert space, Ji, rotations, state vector

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