Continuous Spectra

Suppose that is an observable with continuous eigenvalues. We can still write the eigenvalue equation as

(84) |

But, can now take a continuous range of values. Let us assume, for the sake of simplicity, that can take any value. The orthogonality condition (50) generalizes to

where denotes the famous

(86) |

for any function, , that is well-behaved at . Note that there are clearly a nondenumerably infinite number of mutually orthogonal eigenstates of . Hence, the dimensionality of ket space is nondenumerably infinite. Furthermore, eigenstates corresponding to a continuous range of eigenvalues

Note that a summation over discrete eigenvalues goes over into an integral over a continuous range of eigenvalues. The eigenstates must form a complete set if is to be an observable. It follows that any general ket can be expanded in terms of the . In fact, the expansions (51)-(53) generalize to

(88) | ||

(89) | ||

(90) |

These results also follow simply from Equation (87). We have seen that it is not possible to normalize the eigenstates such that they have unit norms. Fortunately, this convenient normalization is still possible for a general state vector. In fact, according to Equation (90), the normalization condition can be written

(91) |

We have now studied observables whose eigenvalues take a discrete number of values, as well as those whose eigenvalues take a continuous range of values. There are a number of other cases we could look at. For instance, observables whose eigenvalues can take a (finite) continuous range of values, plus a set of discrete values. Such cases can be dealt with using a fairly straightforward generalization of the previous analysis (see Dirac, Chapters II and III).