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 (normalized) state which is not an eigenstate of . Can we identify the transition probability to a final eigenstate with the inner product ? Unfortunately, this is not possible 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 (1.21), as well as the fact that the eigenstates are mutually orthogonal. Note that all of the previous results follow from the extremely useful result

(See Exercise 9.) The relative probability of a transition to an eigenstate , which is equivalent to the relative probability of a measurement of yielding the result , is

(1.58) |

The absolute probability is clearly

(1.59) |

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

(1.60) |