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Continuous Eigenvalues

In the previous two sections, it was tacitly assumed that we were dealing with operators possessing discrete eigenvalues and square-integrable eigenstates. Unfortunately, some operators--most notably, and --possess eigenvalues which lie in a continuous range and non-square-integrable eigenstates (in fact, these two properties go hand in hand). Let us, therefore, investigate the eigenstates and eigenvalues of the displacement and momentum operators.

Let be the eigenstate of corresponding to the eigenvalue . It follows that (277)

for all . Consider the Dirac delta-function . We can write (278)

since is only non-zero infinitesimally close to . Evidently, is proportional to . Let us make the constant of proportionality unity, so that (279)

Now, it is easily demonstrated that (280)

Hence, satisfies the orthonormality condition (281)

This condition is analogous to the orthonormality condition (263) satisfied by square-integrable eigenstates. Now, by definition, satisfies (282)

where is a general function. We can thus write (283)

where , or (284)

In other words, we can expand a general wavefunction as a linear combination of the eigenstates, , of the displacement operator. Equations (283) and (284) are analogous to Eqs. (261) and (264), respectively, for square-integrable eigenstates. Finally, by analogy with the results in Sect. 4.9, the probability density of a measurement of yielding the value is , which is equivalent to the standard result . Moreover, these probabilities are properly normalized provided is properly normalized [cf., Eq. (265)]: i.e., (285)

Finally, if a measurement of yields the value then the system is left in the corresponding displacement eigenstate, , immediately after the measurement: i.e., the wavefunction collapses to a spike-function'', , as discussed in Sect. 3.16.

Now, an eigenstate of the momentum operator corresponding to the eigenvalue satisfies (286)

It is evident that (287)

Now, we require to satisfy an analogous orthonormality condition to Eq. (281): i.e., (288)

Thus, it follows from Eq. (210) that the constant of proportionality in Eq. (287) should be : i.e., (289)

Furthermore, according to Eqs. (202) and (203), (290)

where [see Eq. (203)], or (291)

In other words, we can expand a general wavefunction as a linear combination of the eigenstates, , of the momentum operator. Equations (290) and (291) are again analogous to Eqs. (261) and (264), respectively, for square-integrable eigenstates. Likewise, the probability density of a measurement of yielding the result is , which is equivalent to the standard result . The probabilities are also properly normalized provided is properly normalized [cf., Eq. (221)]: i.e., (292)

Finally, if a mesurement of yields the value then the system is left in the corresponding momentum eigenstate, , immediately after the measurement.   Next: Stationary States Up: Fundamentals of Quantum Mechanics Previous: Measurement
Richard Fitzpatrick 2010-07-20