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152 chapter 7algorithms in which decisions on what steps to follow are made based on thecurrent values of variables, and the program quits only when it thinks it has solvedthe problem. (We already did some of this when we summed a power series untilthe terms became small.) Writing this type of program is usually interesting becausewe must foresee how to have the computer act intelligently in all possible situations,and running them is very much like an experiment in which it is hard topredict what the computer will come up with.7.8 Quantum States in a Square Well (Problem)Probably the most standard problem in quantum mechanics 1 is to solve for theenergies of a particle of mass m bound within a 1-D square well of radius a:{−V0 , for |x|≤a,V (x)=0, for |x|≥a.(7.20)As shown in quantum mechanics texts [Gott 66], the energies of the bound statesE = −E B < 0 within this well are solutions of the transcendental equations√ (√ )10 − EB tan 10 − EB = √ E B (even), (7.21)√ (√ )10 − EB cotan 10 − EB = √ E B (odd), (7.22)where even and odd refer to the symmetry of the wave function. Here we havechosen units such that ¯h =1, 2m =1, a =1, and V 0 =10. Your problem is to1. Find several bound-state energies E B for even wave functions, that is, thesolution of (7.21).2. See if making the potential deeper, say, by changing the 10 to a 20 or a 30,produces a larger number of, or deeper bound states.7.9 Trial-and-Error Roots via the Bisection AlgorithmTrial-and-error root finding looks for a value of x at whichf(x)=0,where the 0 on the right-hand side (RHS) is conventional (an equation such as10 sin x =3x 3 can easily be written as 10 sin x − 3x 3 =0). The search procedure starts1 We solve this same problem in §9.9 using an approach that is applicable to almost anypotential and which also provides the wave functions. The approach of this section worksonly for the eigenenergies of a square well.−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 152