Commutation
The operators
A
and
B
commute:
AB
=
BA
[
A, B
] = 0.

Ψ
a
±
is an eigenstate of
A
with eigenvalue
a
.
A

Ψ
a
±
=
a

Ψ
a
±
Then
AB

Ψ
a
±
=
BA

Ψ
a
±
=
aB

Ψ
a
±
A
(
B

Ψ
a
±
) =
a
(
B

Ψ
a
±
)
So
B

Ψ
a
±
is an eigenstate of
A
with eigenvalue
a
.
If the state is nondegenerate (unique eigenstate for a particular eigenvalue) then
B

Ψ
a
±
must be proportional to

Ψ
a
±
so that
B

Ψ
a
±
=
b

Ψ
a
±
.
Thus

Ψ
a
±
is a
simultaneous eigentstate
of
A
and
B
with eigenvalues
a
and
b
.
We can call that state

Ψ
a,b
±
.
If a system is in this state then it has deﬁnite values for the observables
A
and
B
.
The deﬁnite values are
a
and
b
.
The operators
S
x
,
S
y
, and
S
z
do not commute; there is no state which is simultaneously an
eigenstate of any two of these.
The commutation realtions among the
S
operators is of the form
[
S
x
, S
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 Spring '10
 ROTHBERG
 Linear Algebra, Matrices, Angular Momentum, Fundamental physics concepts, QR algorithm

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