Physics 6210/Spring 2007/Lecture 11
Lecture 11
Relevant sections in text:
§
1.7, 2.1
Product observables
We have seen how to build the Hilbert space for a composite system via the tensor
product construction. Let us now see how to build the observables. Let
A
1
be an observable
for system 1 and
B
2
an observable for system 2. We deﬁne the corresponding observables
for the composite system as follows. Consider the following linear operators on product
vectors:
A
1
⊗
I

α
i ⊗ 
β
i
= (
A
1

α
i
)
⊗ 
β
i
,
I
⊗
B
2

α
i ⊗ 
β
i
=

α
i ⊗
(
B
2

β
i
)
.
We extend the deﬁnitions of these operators to general states by expanding the state in a
product basis and demanding linearity of the operator. Thus deﬁned, these operators are
Hermitian. Moreover, they commute (good exercises!):
[
A
1
⊗
I, I
⊗
B
2
] = 0
,
which means that one can still determine with statistical certainty the subsystem observ
ables within the composite system. For example, if we are looking at 2 particles, it is
possible to ascertain both particle 1’s position and particle 2’s momentum with arbitrarily
good statistical accuracy. Likewise, when using this contruction to build a model for a
particle moving in 3d from our 1d model, we end up with all the position variables com
muting among themselves and all the momentum variables commuting among themselves.
Of course, the commutation relations obtained in the subsystems appear in the combined
system.
Exercise
Show that if [
A
1
, B
1
] =
C
1
then
[
A
1
⊗
I, B
1
⊗
I
] =
C
1
×
I,
thus explaining the generalization of the canonical commutation relations we obtained for
a particle in 3d.
Please take note that it is conventional in many references to simply drop the
⊗
I
or
I
⊗
factors when denoting the extension of the subsystem operators to the composite system.
Indeed, it is common to also drop the
⊗
symbol entirely!
Also note that not every observable of the system is of the above form. As usual, any
selfadjoint operator can represent an observable. Thus there will be observables that refer
1
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to the composite system as a whole (
e.g.,
the total energy) and are not observables of
either of the subsystems alone.
Returning to our example of a particle moving in 2d. We have position operators for
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 Spring '07
 M
 Physics, mechanics, Hilbert space, time evolution

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