Question

The graph in FIGURE EX40.16 shows the potential-energy function U(x of a particle. Solution of the Schrödinger equation finds that the n=3 level has $E_3=0.5eV$and that the n=6 level has $E_6=2.0eV$.

a. Redraw this figure and add to it the energy lines for the n=3 and n=6 states.

b. Sketch the n=3 and n=6 wave functions. Show them as oscillating about the appropriate energy line.

Short answer

(a) The 3 energy level of energy 0.5eV and 6 energy level of energy 2.30eV is shown in the figure.

(b)

Step-by-step solution

Step 1. Given information

(a) The third energy level of energy $0.5eV$and 6 energy level of energy $2.0eV$is shown in the figure.

Step 2. To sketch the n=3 and n=6 wave functions,

For n=3, the wave function has 3 antinodes and 2 nodes. As ${\mathrm{E}}_3<\mathrm{V}\text{after}x=1$, then the function decays exponentially inside the classical forbidden region.

For n=6, the wave function has 6 antinodes and 5 nodes. As the kinetic energy is larger in the region$0<x<1$. Therefore the wave function will have shorter wave length and shorted amplitude. And in the region $2<x<3$, the wave function will have larger wave length and large amplitude.

Step 3. The sketch