Sketch the n=4 wave function for the potential energy shown in FIGURE EX40.13.

Considering the three factors to sketch the graph for n=4 wave function.

(1) If the speed and kinetic energy of the particle decreases, the de Broglie wavelength increases because de Broglie wavelength is inversely dependent on speed of the particle. Hence, the spacing between the nodes of the wave function $\psi \left(x\right)$will increases in regions where the potential energy $U\left(x\right)$is larger (or where the kinetic energy is smaller).

(2) The classical particle is more likely to be found where it moves more slowly. In quantum mechanics, the probability of finding the particle increases when the amplitude of the wave function increases. Consequently, the amplitude of the wave function is larger in regions where the potential energy $U\left(x\right)$is larger.

(3) The wave function for quantum state n has (n-1) nodes and n, antinodes. Therefore, the wave function has four antinodes for n=4 quantum state.