Problems are listed in approximate order of difficulty. A single dot (•) indicates straigh...

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 charged pion π− usually decays into a muon and a neutrino,

but occasionally into an electron and a neutrino,

The relative frequency of the electron decay (compared to the muon decay) is on the order of 1 in 104 The large difference between the probabilities for these two decays can be explained in terms of the μ−e mass difference as follows: The theory of weak interactions predicts that the probability for either decay is proportional to (1 − v/c), where v is the speed of the outgoing μ− or e−. Calculate the quantity (1 − v/c) for each decay, and compute their ratio. Note that this ratio has the same order of magnitude as the observed relative frequency. (Use the result of Problem 1; mπ ≈ 140, mμ ≈ 106, and me ≈ 0.51 MeV/c2.)

Problem 1

• At the Stanford Linear Accelerator, electrons are accelerated to energies of 50 GeV (1 GeV = 109 eV). (a) If this energy were classical kinetic energy, what would be the electrons’ speed? (Take the electrons’ mass to be 0.5 MeV/c2.) (b) Calculate γ and hence find the electrons’ actual speed.