Solved: Protons, neutrons, and many other particles are

Chapter 40, Problem 40.66

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Protons, neutrons, and many other particles are made of more fundamental particles called quarks and antiquarks (the antimatter equivalent of quarks). A quark and an antiquark can form a bound state with a variety of different energy levels, each of which corresponds to a different particle observed in the laboratory. As an example, the c particle is a low-energy bound state of a so-called charm quark and its antiquark, with a rest energy of 3097 MeV; the c12S2 particle is an excited state of this same quarkantiquark combination, with a rest energy of 3686 MeV. A simplified representation of the potential energy of interaction between a quark and an antiquark is U1x2 = A0 x 0 , where A is a positive constant and x represents the distance between the quark and the antiquark. You can use the WKB approximation (see Challenge 40.64) to determine the bound-state energy levels for this potential-energy function. In the WKB approximation, the energy levels are the solutions to the equation L b a 22m3E - U1x24 dx = nh 2 1n = 1, 2, 3, c2 Here E is the energy, U1x2 is the potential-energy function, and x = a and x = b are the classical turning points (the points at which E is equal to the potential energy, so the Newtonian kinetic energy would be zero). (a) Determine the classical turning points for the potential U1x2 = A0 x 0 and for an energy E. (b) Carry out the above integral and show that the allowed energy levels in the WKB approximation are given by En = 1 2m a 3mAh 4 b 2>3 n2>3 1n = 1, 2, 3, c2 (Hint: The integrand is even, so the integral from -x to x is equal to twice the integral from 0 to x.) (c) Does the difference in energy between successive levels increase, decrease, or remain the same as n increases? How does this compare to the behavior of the energy levels for the harmonic oscillator? For the particle in a box? Can you suggest a simple rule that relates the difference in energy between successive levels to the shape of the potential-energy function?