Question

Make a rough sketches of the wave functions at the top of the $\mathbf{n}\mathbf{=}\mathbf{2}$ band and the bottom of the $\mathbf{n}\mathbf{=}\mathbf{3}$ band for a one dimensional “crystal” consisting of seven finite wells. Explain why these two states of roughly equal wave number have vastly different energies.

Short answer

The sketch of the wavefunction is obtained and the explanation for the energy differences are explained.

Step-by-step solution

Determine the concept:

Consider the top of $\mathit{n}\mathbf{}\mathbf{=}\mathbf{}\mathbf{2}$band consists of the 14 anitnodes distributed in a way that all the nods are at the separation of the wells and the potential energy is maximum with the potneital energy of the state being low and the total energy is different from the kinetic energy. Consider the bottom of the 3 band has in total 15 anitnodes. The distribution of the antinodes is located at the point where the potential energy of the walls is maximum and the state is high. Thus, the total energy is at higher value than the kinetic energy.

Determine the sketch

Consider from the discussion it is clear that the two states have the difference of one anti-node and the high difference of energy.

Consider the sketch for the wavefunction at the top of the $n=2$and bottom of $n=3$band is shown below.