How is the color difference of CdS nano-particles related to the "particle in a box" problem in quantum mechanics? In my lecture on quantum mechanics, I was told that nano-particles of CdS with different sizes of have different colors and that this fact is a direct consequence of the "particle in a box" problem. Can someone clarify (and possibly justify) this claim? 
 A: The short answer is:
Nano-particle could only absorb a certain spectrum of light. The spectrum is the spectrum of particle in a box(treating the absorbed photon as the particle in the box.) with a certain frequency broadening mechanism.
After absorbing it for a while, the nano-particle could possibly emit the photon causing it to have some color.
A: We assume that the reader knows, from elementary quantum mechanics, that the allowed energy levels of a particle of mass $m$ in an infinite potential well (a.k.a. particle in a box) of width $a$:
$$
V(x)=\begin{cases}0\hspace{1cm} &0\leq x\leq a\\
\infty &\mathrm{otherwise}
\end{cases}
$$
are given by 
$$
E_n=\frac{n^2\pi^2\hbar^2}{2ma^2}\hspace{1cm}n=1,2,3,\dots
$$
In particular, the energy difference between the two lowest energy levels is
$$ \Delta E_{2-1}=\frac{3\pi^2\hbar^2}{2ma^2}$$
Now, let us assume for simplicity that the color of these nanoparticles is due to the transition from the first excited state to the ground state. This is probably not true, but it's obvious that for different transitions similar expressions for the transition energies hold and we're just trying to give a sketch of how the phenomenon arises, rather than a detailed, quantitative result.
It makes sense to assume that a nano-particle can be (roughly) modeled by a "particle in a box" potential, with the (effective) width being a monotonously increasing function $f(a)$ of the particle size $a$. The simplest possible choice is of course $f(a)=a$, but something different would work too. Under these simple assumptions, it's clear that the transition energies, and hence the colors of the nanoparticles themselves, depend on the size of the particle under consideration. The difference is something like
$$\Delta E^{\mathrm{size 1}}_{2-1}-\Delta E^{\mathrm{size 2}}_{2-1}=
\frac{3\pi^2\hbar^2}{2}\bigg(\frac{1}{m_1f(a_1)^2}-\frac{1}{m_2f(a_2)^2}\bigg) $$
From here, it is not hard (once all parameters are fixed) to determine the difference in wavelengths emitted by particles of different sizes (and masses). It is in this sense that one can say that the different colors of nanoparticles of various sizes can be attributed to the "particle in a box" problem in quantum mechanics.
