Editing Particle in a box (section)

==== Quantum dots ====
[[Quantum dot]]s are extremely small [[semiconductors]] (on the scale of nanometers).<ref name="chemed">{{cite journal|last2=Griffin|first2=G.A.|date=2008|title=Simple Syntheses of CdSe Quantum Dots|url=http://pubs.acs.org/doi/abs/10.1021/ed300568e|journal=Journal of Chemical Education|volume=85|issue=6|page=842|last1=Rice|first1=C.V.|access-date=5 November 2016|bibcode=2008JChEd..85..842R|doi=10.1021/ed085p842}}</ref> They display [[quantum confinement]] in that the electrons cannot escape the “dot”, thus allowing particle-in-a-box approximations to be used.<ref name="openlab">{{cite web|url=http://physicsopenlab.org/2015/11/20/quantum-dots-a-true-particle-in-a-box-system/|title=Quantum Dots : a True "Particle in a Box" System|date=20 November 2015|website=PhysicsOpenLab|access-date=5 November 2016}}</ref> Their behavior can be described by three-dimensional particle-in-a-box energy quantization equations.<ref name="openlab" />

The [[Band gap|energy gap]] of a quantum dot is the energy gap between its [[valence and conduction bands]]. This energy gap <math>\Delta E(r)</math> is equal to the  gap of the bulk material <math>E_{\text{gap}}</math> plus the energy equation derived particle-in-a-box, which gives the energy for electrons and [[Electron hole|holes]].<ref name="openlab" /> This can be seen in the following equation, where <math>m^*_e</math> and <math>m^*_h</math> are the effective masses of the electron and hole, <math>r</math> is radius of the dot, and <math>h</math> is the Planck constant:<ref name="openlab" />
<math display="block">\Delta E(r) = E_{\text{gap}}+\left ( \frac{h^2}{8r^2} \right ) \left( \frac{1}{m^*_e}+\frac{1}{m^*_h} \right)</math>

Hence, the energy gap of the quantum dot is inversely proportional to the square of the "length of the box", i.e. the radius of the quantum dot.<ref name="openlab" />

Manipulation of the band gap allows for the absorption and emission of specific wavelengths of light, as energy is inversely proportional to wavelength.<ref name="chemed" /> The smaller the quantum dot, the larger the band gap and thus the shorter the wavelength absorbed.<ref name="chemed" /><ref name="washington">{{cite web|url=http://courses.washington.edu/overney/NME498_Material/NME498_Lectures/Lecture12_Reid_Quantum_Confinement.pdf|title=Quantum Confinement|publisher=University of Washington|last1=Overney|first1=René M.|access-date=5 November 2016|archive-date=2 December 2016|archive-url=https://web.archive.org/web/20161202034143/http://courses.washington.edu/overney/NME498_Material/NME498_Lectures/Lecture12_Reid_Quantum_Confinement.pdf|url-status=dead}}</ref>

Different semiconducting materials are used to synthesize quantum dots of different sizes and therefore emit different wavelengths of light.<ref name="washington" /> Materials that normally emit light in the visible region are often used and their sizes are fine-tuned so that certain colors are emitted.<ref name="chemed" /> Typical substances used to synthesize quantum dots are cadmium (Cd) and selenium (Se).<ref name="chemed" /><ref name="washington" /> For example, when the electrons of two nanometer CdSe quantum dots [[Emission spectrum|relax after excitation]], blue light is emitted. Similarly, red light is emitted in four nanometer CdSe quantum dots.<ref name="Zahn">{{cite web|url=http://www.osiconference.org/osi2015/presentations/Tu2.3%20Zahn.pdf|title=Surface and Interface Properties of Semiconductor Quantum Dots by Raman Spectroscopy|publisher=Technische Universität Chemnitz|last1=Zahn|first1=Dietrich R.T.|access-date=5 November 2016|archive-date=1 December 2016|archive-url=https://web.archive.org/web/20161201211210/http://www.osiconference.org/osi2015/presentations/Tu2.3%20Zahn.pdf|url-status=dead}}</ref><ref name="chemed" />

Quantum dots have a variety of functions including but not limited to fluorescent dyes, [[transistor]]s, [[LED]]s, [[solar cells]], and medical imaging via optical probes.<ref name="chemed" /><ref name="openlab" />

One function of quantum dots is their use in lymph node mapping, which is feasible due to their unique ability to emit light in the near infrared (NIR) region. Lymph node mapping allows surgeons to track if and where cancerous cells exist.<ref name="Medicine">{{cite journal|last2=Ebenstein|first2=Yuval|date=2009|title=Quantum Dots for In Vivo Small-Animal Imaging|journal=Journal of Nuclear Medicine|volume=50|issue=4|pages=493–496|last1=Bentolila|first1=Laurent A.|doi=10.2967/jnumed.108.053561|pmid=19289434|pmc=3081879}}</ref>

Quantum dots are useful for these functions due to their emission of brighter light, excitation by a wide variety of wavelengths, and higher resistance to light than other substances.<ref name="Medicine" /><ref name="chemed" />