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.

••• The Heisenberg uncertainty principle applies to photons as well as to material particles. Thus, a photon confined to a very tiny box (size Δx) necessarily has a large uncertainty in momentum and energy (recall that for photons, E = pc) and hence a large average energy. Since energy is equivalent to mass, according to E = mc2, a confined photon can create a large gravitational field. If Δx is sufficiently small, the energy density will be sufficiently large to create a black hole (Section 2.10). This size Δx is called the Planck length and defines the scale at which gravity and quantum mechanics are inextricably mixed. (The branch of physics that tries to combine quantum mechanics with gravity is called string theory.) (a) Show that the escape speed for a star of mass M and radius R is . (b) By equating vesc for a black hole to the speed of light, derive a formula for the Planck length in terms of c, G, and ℏ, and (c) show that its value is fantastically small, of order 10−35 m. (d) What is the diameter of a proton (about 2 fm) in Planck lengths?