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Continuous Spectra

Up to now, we have studiously avoided dealing with observables possessing eigenvalues that lie in a continuous range, rather than having discrete values. The reason for this is that continuous eigenvalues imply a ket space of non-denumerably 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. (See the following chapter.) Fortunately, many of the results that we obtained previously for a finite-dimensional ket space with discrete eigenvalues can be generalized to ket spaces of non-denumerably infinite dimensions.

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

$\displaystyle \xi \,\vert\xi'\rangle =\xi'\, \vert\xi'\rangle.$ (1.87)

But, $ \xi'$ now takes a continuous range of values. Let us assume, for the sake of simplicity, that $ \xi'$ can take any value. The orthonormality condition (1.53) generalizes to

$\displaystyle \langle \xi'\vert\xi''\rangle = \delta(\xi'-\xi''),$ (1.88)

where $ \delta(x)$ denotes the famous Dirac delta function [32,92], and satisfies

$\displaystyle \delta(x\neq 0)$ $\displaystyle =0,$ (1.89)
$\displaystyle \int_{-\infty}^\infty dx\,\delta(x)$ $\displaystyle =1,$ (1.90)
$\displaystyle \int_{-\infty}^\infty dx\,f(x)\,\delta(x-x')$ $\displaystyle =f(x')$ (1.91)

for any function, $ f(x)$ , that is well behaved at $ x=x'$ . The Dirac delta function is a generalized function [104] that can be realized in many equivalent limiting forms. For instance,

$\displaystyle \delta(x)$ $\displaystyle = \lim_{\eta\rightarrow 0} \frac{1}{\pi}\,\frac{\eta}{x^{\,2}+\eta^{\,2}},$ (1.92)
$\displaystyle \delta(x)$ $\displaystyle =\lim_{\eta\rightarrow 0} \frac{1}{\pi}\,\frac{\sin(x/\eta)}{x}.$ (1.93)

Note from Equations (1.87) and (1.88) that there are a non-denumerably infinite number of mutually orthogonal eigenstates of $ \xi$ . Hence, the dimensionality of ket space is non-denumerably infinite. Furthermore, eigenstates corresponding to a continuous range of eigenvalues cannot be normalized such that they have unit norms. In fact, it is clear from Equation (1.88), together with the well-known fact that $ \delta(0)\rightarrow \infty$ , that these eigenstates have infinite norms. In other words, 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 (1.57) generalizes to

$\displaystyle \int d\xi' \,\vert\xi'\rangle\langle \xi'\vert = 1.$ (1.94)

(See Exercise 18.) 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 (1.54)-(1.56) generalize to give

$\displaystyle \vert A\rangle$ $\displaystyle = \int d\xi'\,\vert\xi'\rangle\langle \xi'\vert A\rangle,$ (1.95)
$\displaystyle \langle A\vert$ $\displaystyle = \int d\xi'\,\langle A\vert\xi'\rangle \langle \xi'\vert,$ (1.96)
$\displaystyle \langle A\vert A\rangle$ $\displaystyle = \int d\xi'\,\langle A\vert\xi'\rangle\langle \xi'\vert A\rangle = \int d\xi' \,\vert\langle A\vert\xi'\rangle\vert^{\,2},$ (1.97)

respectively. These results also follow simply from Equation (1.94). 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 ket. In fact, according to Equation (1.97), the normalization condition can be written

$\displaystyle \langle A\vert A\rangle =\int d\xi' \,\vert\langle A\vert\xi'\rangle\vert^{\,2} = 1.$ (1.98)

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 that 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 [32].

next up previous
Next: Exercises Up: Fundamental Concepts Previous: Uncertainty Relation
Richard Fitzpatrick 2016-01-22