A fellow student proposes that a possible wave function

Chapter 40, Problem 40.53

(choose chapter or problem)

A fellow student proposes that a possible wave function for a free particle with mass m (one for which the potentialenergy function U1x2 is zero) is c1x2 = e e+kx , x 6 0 e-kx , x 0 where k is a positive constant. (a) Graph this proposed wave function. (b) Show that the proposed wave function satisfies the Schrdinger equation for x 6 0 if the energy is E = -U2 k2>2m that is, if the energy of the particle is negative. (c) Show that the proposed wave function also satisfies the Schrdinger equation for x 0 with the same energy as in part (b). (d) Explain why the proposed wave function is nonetheless not an acceptable solution of the Schrdinger equation for a free particle. (Hint: What is the behavior of the function at x = 0?) It is in fact impossible for a free particle (one for which U1x2 = 0) to have an energy less than zero.