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.

••• (Section 7.8) If you have access to computer software with preprogrammed numerical solution of differential equations (for example, the function NDSolve in Mathematica), do the following: (a) Plot the Gaussian well of Problem 1, using units such that U0 = a = ℏ = 1 and taking m = 36. (The first three choices simply define a convenient system of units — for instance, a is the unit of length — and the last is chosen so that the well supports several bound states.) (b) We know that to be acceptable, the wave function must be proportional to eαx far to the left of the well. To ensure this, use the boundary conditions ψ(−3) = 1 and ψ′(−3) = α, and solve the Schrödinger differential equation for energy E = 0.1. Plot your solution, and confirm that it does not behave acceptably as x→ ∞. (c) Repeat for E = 0.2. What can you conclude about the energy of the ground state from the plots of parts (b) and (c)? (d) Repeat for two or three intermediate energies until you know the ground-state energy to two significant figures.

Problem 1

•• Consider the potential-energy function (the Gaussian well)

where U0 and a are positive constants. (a) Sketch U(x). (b) For 0 < E < U0, find the classical turning points (in terms of U0, E, and a).