A problem listed for a given section requires an understanding of that section and earlier...

A problem listed for a given section requires an understanding of that section and earlier sections, but not of later sections. Within each section, problems are listed in approximate order of difficulty. A single dot (•) indicates straightforward problems involving just one main concept and sometimes requiring no more than substitution of numbers in the appropriate formula. Two dots (••) identify problems that are slightly more challenging and usually involve more than one concept. Three dots (•••) indicate problems that are distinctly more challenging, either because they are intrinsically difficult or involve lengthy calculations. Needless to say, these distinctions are hard to draw and are only approximate.

•• For theoretical purposes the best measure of the spread Δx of a pulse is the rms spread defined in (6.30), although this rms spread is sometimes rather less than one might guess. To illustrate this, think about the following: Consider a pulse whose probability density P(x) = |Ψ(x, t)|2 (at one fixed time t) is as shown in Fig. 1(a). (The particle represented by this rectangular pulse is equally likely to be found anywhere between x = −a and x = a. Since the total probability of finding the particle anywhere must be 1, the height of the pulse shown must be 1/2a.) Since this pulse is centered at x = 0, the rms spread (6.30) is

Find Δx for the rectangular pulse.

FIGURE 1