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.
•• A second-order differential equation like the Schrödinger equation has two independent solutions ψ1(x) and ψ2(x). These two solutions can be chosen in many ways, but once they are chosen, any solution can be expressed as a linear combination Aψ1(x) + Bψ2(x) (where A and B are constants, real or complex), (a) To illustrate this property, consider the differential equation ψ″ = −k2ψ where k is a constant. Prove that each of the three functions sin kx, cos kx, and eikx is a solution. (b) Show that each can be expressed as a combination of the other two.
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