In Section 5.3, we learned that to be normalizable, a wave function (l) must not itself di...

In Section 5.3, we learned that to be normalizable, a wave function (l) must not itself diverge and (2) must fall to 0 faster than |x| -1/2 as x gets large. Nevertheless, we find two functions that slightly violate these requirements very useful. Consider the quantum mechanical plane wave Aei(kx-ωt) ancj me wejrti function ψx0 (x) pictured in Figure 5.19, which we here call by its proper name, the Dirac delta function, (a) Which of the two normalizability requirements is violated by the plane wave, and winch by the Dirac delta function? (b) Normalization of the plane wave could be accomplished if it were simply truncated, restricted to the region —b<x<+b, being identically 0 outside. What would then be the relationship between b and A, and what would happen to A as b approaches infinity? (c) Rather than an infinitely tall and narrow spike like the Dirac delta function, consider a function that is 0 everywhere except the narrow region -ε<x< +ε, where its value is a constant B. This too could be normalized. What would be the relationship between ε and B, and what would happen to B as ε approaches 0? (What we get is not exactly the Dirac delta function, but the distinction involves comparing infinities, a dangerous business that we will avoid.) (d) As we see, the two "exceptional" functions may be viewed as limits of normalizable ones. In those limits, they are also complementary to each other in terms of their position and momentum uncertainties. Without getting into calculations, describe how they are complementary.