As we see in Figures 10.23, in a one dimensional crystal of finite wells, top of the band states closely resemble infinite well states. In fact, the famous particle in a box energy formula gives a fair value for the energies of the band to which they belong. (a) If for nin that formula you use the number of anitnodes in the whole function, what would you use for the box length L? (b) If, instead, the n in the formula were taken to refer to band n, could you still use the formula? If so, what would you use for L? (c) Explain why the energies in a band do or do not depend on the size of the crystal as a whole.

Consider the formula for the relation between the energy of the particle in the box as follows:

$\mathbf{E}\mathbf{=}\frac{{\mathbf{n}}^{\mathbf{2}}{\mathbf{h}}^{\mathbf{2}}}{\mathbf{8}{\mathbf{mL}}^{\mathbf{2}}}$

Here, E is the energy, n is the number of antinodes, h is the planck’s constant and m is the mass.

Consider the crystal is considered as the box and the length L is taken as the length of the whole crystal. As the length L covers the all the n antinodes.

Consider the equation that determines the energy as:

$E=\frac{{\left(nN\right)}^2{\mathrm{\pi h}}^2}{2m{d}^2}$

Resolve the equation as:

$$\begin{gathered} E=\frac{{\left(n\right)}^2{\mathrm{\pi h}}^2}{2m{\left({\displaystyle \frac{W}{N}}\right)}^2} \\ E=\frac{{\left(n\right)}^2{\mathrm{\pi h}}^2}{2m{a}^2} \end{gathered}$$

Hence, for n to be the band the use of L is the length of the single atom as the n corresponds to the number of antinodes in the single atom.

Consider that the band energy is clustered around the corresponding single atom energy and it do not depend on the number of atoms in the crystal this is why the band energy is independent of the size of the crystal.