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Measurements
We have seen that a measurement of some observable of a microscopic
system causes the system to jump into one of the eigenstates of . The
result of the measurement is the associated eigenvalue (or some function of
this quantity). It is impossible to determine into which eigenstate a
given system will jump, but it is possible to predict the probability of
such a transition. So, what is the probability that a system in some
initial state makes a transition to an eigenstate
of an observable , as a result of a measurement made on the system?
Let us start with the simplest case. If the system is initially in an eigenstate
then the transition probability to a eigenstate
corresponding to a different eigenvalue is zero, and the transition probability
to the same eigenstate is unity. It is convenient to normalize
our eigenkets such that they all have unit norms. It follows from the
orthogonality property of the eigenkets that

(50) 
where
is unity if , and zero otherwise.
For the moment, we are assuming that the eigenvalues of
are all different.
Note that the probability of a transition from an initial eigenstate
to a final eigenstate is the same as
the value of the inner product
. Can we use this correspondence
to obtain a general rule for
calculating transition probabilities? Well, suppose that the system is initially
in a state which is not an eigenstate of . Can we
identify the transition probability to a final eigenstate with
the inner product
? The straight answer is ``no'',
since
is, in general, a complex number, and complex
probabilities do not make much sense. Let us try again. How about if we
identify the transition probability with the modulus squared of the
inner product,
? This
quantity is definitely a positive
number (so it could be a probability). This guess also gives the right answer for
the transition probabilities between eigenstates. In fact, it is the
correct guess.
Since the eigenstates of an observable form a complete
set, we can express any given state as a linear combination of them.
It is easily demonstrated that
where the summation is over all the different eigenvalues of
, and use has been
made of Eq. (20), and
the fact that the eigenstates are mutually orthogonal. Note that all of the
above results follow from the extremely useful (and easily proved) result

(54) 
where 1 denotes the identity operator.
The relative
probability of a transition to an eigenstate , which is
equivalent to
the relative probability of a measurement of yielding the result , is

(55) 
The absolute probability
is clearly

(56) 
If the ket is normalized such that its norm is unity, then this
probability simply reduces to

(57) 
Next: Expectation values
Up: Fundamental concepts
Previous: Observables
Richard Fitzpatrick
20060216