Editing Potential well (section)

===Quantum mechanics view===
{{see also|Particle in a box}}

The electronic and optical properties of materials are affected by size and shape. Well-established technical achievements including quantum dots were derived from size manipulation and investigation for their theoretical corroboration on quantum confinement effect.<ref>{{cite journal|pmid=9983472|year=1996|last1=Norris|first1=DJ|last2=Bawendi|first2=MG|title=Measurement and assignment of the size-dependent optical spectrum in CdSe quantum dots|volume=53|issue=24|pages=16338–16346|journal=Physical Review B|bibcode = 1996PhRvB..5316338N |doi = 10.1103/PhysRevB.53.16338 }}</ref> The major part of the theory is the behaviour of the [[exciton]] resembles that of an atom as its surrounding space shortens. A rather good approximation of an exciton's behaviour is the 3-D model of a [[particle in a box]].<ref>{{cite journal|doi=10.1063/1.445676|title=A simple model for the ionization potential, electron affinity, and aqueous redox potentials of small semiconductor crystallites|year=1983|last1=Brus|first1=L. E.|journal=The Journal of Chemical Physics|volume=79|issue=11|pages=5566–5571|bibcode = 1983JChPh..79.5566B }}</ref> The solution of this problem provides a sole{{clarify|date=January 2016}} mathematical connection between energy states and the dimension of space. Decreasing the volume or the dimensions of the available space, increases the energy of the states. Shown in the diagram is the change in electron energy level and [[bandgap]] between nanomaterial and its bulk state.

The following equation shows the relationship between energy level and dimension spacing:

:<math>\psi_{n_x,n_y,n_z} = \sqrt{\frac{8}{L_x L_y L_z}} \sin \left( \frac{n_x \pi x}{L_x} \right) \sin \left( \frac{n_y \pi y}{L_y} \right) \sin \left( \frac{n_z \pi z}{L_z} \right)</math>

:<math>E_{n_x,n_y,n_z} = \frac{\hbar^2\pi^2}{2m} \left[ \left( \frac{n_x}{L_x} \right)^2 + \left( \frac{n_y}{L_y} \right)^2 + \left( \frac{n_z}{L_z} \right)^2 \right]</math>

Research results<ref>{{cite journal|doi=10.1088/0022-3719/14/20/004|title=Pressure-induced modifications of the energy band structure of crystalline CdS|year=1981|last1=Kunz|first1=A B|last2=Weidman|first2=R S|last3=Collins|first3=T C|journal=Journal of Physics C: Solid State Physics|volume=14|issue=20|pages=L581|bibcode = 1981JPhC...14L.581K }}</ref> provide an alternative explanation of the shift of properties at nanoscale. In the bulk phase, the surfaces appear to control some of the macroscopically observed properties. However, in [[nanoparticles]], surface molecules do not obey the expected configuration{{which|date=January 2016}} in space. As a result, surface tension changes tremendously.