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212 chapter 99.9 Unit II. Binding A Quantum ParticleProblem: In this unit is we want to determine whether the rules of quantummechanics are applicable inside a nucleus. More specifically, you are told thatnuclei contain neutrons and protons (nucleons) with mass mc 2 ≃ 940 MeV and thata nucleus has a size of about 2 fm. 5 Your explicit problem is to see if these experimentalfacts are compatible, first, with quantum mechanics and, second, with theobservation that there is a typical spacing of several million electron volts (MeV)between the ground and excited states in nuclei.This problem requires us to solve the bound-state eigenvalue problem for the 1-D, time-dependent Schrödinger equation. Even though this equation is an ODE,which we know how to solve quite well by now, the extra requirement that we needto solve for bound states makes this an eigenvalue problem. Specifically, the boundstaterequirement imposes boundary conditions on the form of the solution, whichin turn means that a solution exists only for certain energies, the eigenenergies oreigenvalues.If this all sounds a bit much for you now, rest assured that you do not need tounderstand all the physics behind these statements. What we want is for you to gainexperience with the technique of conducting a numerical search for the eigenvalue inconjunction with solving an ODE numerically. This is how one solves the numericalODE eigenvalue problem. In §20.2.1, we discuss how to solve the equivalent, but moreadvanced, momentum-space eigenvalue problem as a matrix problem. In Chapter 18,PDE Waves: String, Quantum Packet, and we study the related problem of themotion of a quantum wave packet confined to a potential well. Further discussions ofthe numerical bound-state problem are found in [Schd 00, Koon 86].9.10 Theory: The Quantum Eigenvalue ProblemQuantum mechanics describes phenomena that occur on atomic or subatomicscales (an elementary particle is subatomic). It is a statistical theory in whichthe probability that a particle is located in a region dx around the point x isP = |ψ(x)| 2 dx, where ψ(x) is called the wave function. If a particle of energy Emoving in one dimension experiences a potential V (x), its wave function is determinedby an ODE (a PDE if greater than 1-D) known as the time-independentSchrödinger equation 6 :−¯h 22md 2 ψ(x)dx 2 + V (x)ψ(x)=Eψ(x). (9.46)5 A fermi (fm) equals 10 −13 cm =10 −15 m, and ¯hc ≃ 197.32 MeV fm.6 The time-dependent equation requires the solution of a partial differential equation, asdiscussed in Chapter 18, “PDE Waves: String, Quantum Packet, and E&M.”−101COPYRIGHT 2008, PRINCET O N UNIVE R S I T Y P R E S SEVALUATION COPY ONLY. NOT FOR USE IN COURSES.ALLpup_06.04 — 2008/2/15 — Page 212