Measurements

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
? In fact, we cannot
because
is, in general, a complex number, and complex
probabilities do not make any sense. Let us try again. Suppose that 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.

Because 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 Equation (20), as well as the fact that the eigenstates are mutually orthogonal. Note that all of the above results follow from the extremely useful (and easily proved) result

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) |