(A) This is the ground state wave function for a particle in the infinite well. It is clearly wrong here, because it is $0$ for $x < a$ and $x > b$ .

(B) There cannot be sinusiodal oscillations in the region $a < x < b$ because the particle's energy is less than the potential energy in that region.

(C) This is the correct answer.

(D) The particle's energy is greater than the potential energy in the region's $x < a$ and $x > b$ . Therefore, the wave function must be a sinusoidal function in those regions, not a decaying exponential.

(E)

Solution to 1996 Problem 18