Simple Nature - Light and Matter

conservation of energy and momentum, the difference between thephoton’s initial and final wavelengths must be 2h/mc, where m isthe mass of the electron. The experimental verification of this typeof “pool-ball” behavior by Arthur Compton in 1923 was taken asdefinitive proof of the particle nature of light. Note that we’re notmaking any nonrelativistic approximations. To keep the algebrasimple, you should use natural units — in fact, it’s a good idea touse even-more-natural-than-natural units, in which we have not justc = 1 but also h = 1, and m = 1 for the mass of the electron. You’llalso probably want to use the relativistic relationship E 2 − p 2 = m 2 ,which becomes E 2 − p 2 = 1 for the energy and momentum of theelectron in these units.34 Generalize the result of problem 33 to the case where thephoton bounces off at an angle other than 180 ◦ with respect to itsinitial direction of motion.35 On page 869 we derived an expression for the probabilitythat a particle would tunnel through a rectangular barrier, i.e., aregion in which the interaction energy U(x) has a graph that lookslike a rectangle. Generalize this to a barrier of any shape. [Hints:First try generalizing to two rectangular barriers in a row, and thenuse a series of rectangular barriers to approximate the actual curveof an arbitrary function U(x). Note that the width and height ofthe barrier in the original equation occur in such a way that all thatmatters is the area under the U-versus-x curve. Show that this isstill true for a series of rectangular barriers, and generalize using anintegral.] If you had done this calculation in the 1930’s you couldhave become a famous physicist.36 Show that the wavefunction given in problem 30 is properlynormalized.37 Show that a wavefunction of the form Ψ = e by sin ax isa possible solution of the Schrödinger equation in two dimensions,with a constant potential. Can we tell whether it would apply to aclassically allowed region, or a classically forbidden one?38 Find the energy levels of a particle in a three-dimensional√rectangular box with sides of length a, b, and c.900 Chapter 13 Quantum Physics