Compatible Observables

Suppose, however, that the eigenstates of
are not eigenstates of
.
Is it still possible to measure both observables simultaneously? Let us again
make an observation of
that throws the system into an eigenstate
, with eigenvalue
.
We can now make a second observation to determine
.
This will throw the system into one of the (many) eigenstates of
that
depend on
. In principle, each of these eigenstates is
associated with a different result of the measurement. Suppose that the
system is thrown into an eigenstate
, with the eigenvalue
.
Another measurement of
will throw the system into one of the (many)
eigenstates of
that depend on
.
Each eigenstate is again associated with a different possible
result of the measurement. It is clear that if the observables
and
do not possess simultaneous eigenstates then if the value
of
is known (i.e., the system is in an eigenstate of
) then the
value of
is uncertain (i.e., the system is not in an eigenstate
of
), and vice versa. We say that the two observables are
*incompatible*.

We have seen that the condition for two observables and to be simultaneously measurable is that they should possess simultaneous eigenstates (i.e., every eigenstate of should also be an eigenstate of ). Suppose that this is the case. Let a general eigenstate of , with eigenvalue , also be an eigenstate of , with eigenvalue . It is convenient to denote this simultaneous eigenstate . We have

(1.66) | ||

(1.67) |

We can left-multiply the first equation by , and the second equation by , and then take the difference. The result is

(1.68) |

for each simultaneous eigenstate. Recall that the eigenstates of an observable must form a complete set. It follows that the simultaneous eigenstates of two observables must also form a complete set. Thus, the previous equation implies that

(1.69) |

where is a general ket. The only way that this can be true is if

(1.70) |

We conclude that the condition for two observables and to be simultaneously measurable is that they should commute.