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.

•• In a region where the potential energy U(x) varies, the Schrödinger equation must usually be solved by numerical methods — generally with the help of a computer. The simplest such method, though not the most efficient, is called Euler’s method and can be described as follows: In solving the Schrödinger equation numerically, one needs to know the Values of ψ(x) and ψ′(x) at one point x0. For example, for the finite wells of Fig. 7.8 we know these values at x0 = 0 since we know that ψ(x) has the form ψ(x) = eαx for x ≤ 0. Suppose now we want to find ψ(x) at some point x > 0. We first divide the interval from x0 to x into n equal intervals each of width Δx:

Knowing ψ(x) and ψ′(x) at x = x0, we can use the Schrödinger equation to find approximate values of these two functions at x1, and from these, we can find values at x2 and so on until we know both functions at xn = x. To do this, we use the approximations

 ψ(xi + 1) ≈ ψ(xi) + ψ′(xi) Δx

and

 ψ′(xi + 1) ≈ ψ′(xi) + ψ″(xi) Δx

[The first approximation is just the definition of the derivative, and the second is the same approximation applied to ψ′(x).] Consider the equation ψ″(x) = −ψ(x), with the starting values ψ(0) = 0 and ψ′(0) = 1, and do the following (for which you don’t need a computer): (a) What is the exact solution of this equation with these initial conditions? What is the exact value of ψ(x) at x = 1? (b) Divide the interval from x = 0 to 1 into two equal intervals (n = 2), and use Euler’s method to find an approximate value for ψ(1). (c) Repeat with n = 3 and n = 4. Compare your results with the exact answer, and make a plot of the exact and approximate values for the case that n = 4. Note well how the approximate solution improves as you increase n.