Continuous spectra

Up to now, we have studiously avoided dealing with observables possessing eigenvalues which lie in a continuous range, rather than having discrete values. The reason for this is because continuous eigenvalues imply a ket space of nondenumerably infinite dimension. Unfortunately, continuous eigenvalues are unavoidable in quantum mechanics. In fact, the most important observables of all, namely position and momentum, generally have continuous eigenvalues. Fortunately, many of the results we obtained previously for a finite-dimensional ket space with discrete eigenvalues can be generalized to ket spaces of nondenumerably infinite dimensions.

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

$$\xi \vert\xi'\rangle =\xi' \vert\xi'\rangle.$$ (84)

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

$$\langle \xi'\vert\xi''\rangle = \delta(\xi'-\xi''),$$ (85)

where $\delta(x)$ denotes the famous Dirac delta-function. Note that there are clearly a nondenumerably infinite number of mutually orthogonal eigenstates of $\xi$. Hence, the dimensionality of ket space is nondenumerably infinite. Note, also, that eigenstates corresponding to a continuous range of eigenvalues cannot be normalized so that they have unit norms. In fact, these eigenstates have infinite norms: i.e., they are infinitely long. This is the major difference between eigenstates in a finite-dimensional and an infinite-dimensional ket space. The extremely useful relation (54) generalizes to

$$\int d\xi' \vert\xi'\rangle\langle \xi'\vert = 1.$$ (86)

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

$$\vert A\rangle = \int d\xi' \vert\xi'\rangle\langle \xi'\vert A\rangle,$$ (87)

$$\langle A\vert = \int d\xi' \langle A\vert\xi'\rangle \langle \xi'\vert,$$ (88)

$$\langle A\vert A\rangle = \int d\xi' \langle A\vert\xi'\rangle\langle \xi'\vert A\rangle = \int d\xi' \vert\langle A\vert\xi'\rangle\vert^2.$$ (89)

These results also follow simply from Eq. (86). We have seen that it is not possible to normalize the eigenstates $\vert\xi'\rangle$ such that they have unit norms. Fortunately, this convenient normalization is still possible for a general state vector. In fact, according to Eq. (89), the normalization condition can be written

$$\langle A\vert A\rangle =\int d\xi' \vert\langle A\vert\xi'\rangle\vert^2 = 1.$$ (90)

We have now studied observables whose eigenvalues can take a discrete number of values as well as those whose eigenvalues can take any value. There are number of other cases we could look at. For instance, observables whose eigenvalues can only take a finite range of values, or observables whose eigenvalues take on a finite range of values plus a set of discrete values. Both of these cases can be dealt with using a fairly straight-forward generalization of the previous analysis (see Dirac, Cha. II and III).