Editing Particle in a box (section)

=== Quantum well laser ===
The particle in a box model can be applied to [[quantum well laser]]s, which are laser diodes consisting of one semiconductor “well” material sandwiched between two other semiconductor layers of different material . Because the layers of this sandwich are very thin (the middle layer is typically about 100&nbsp;Å thick), [[quantum confinement]] effects can be observed.<ref>{{Cite book|title=Quantum Well Lasers|last=Zory|first=Peter|publisher=Academic Press Unlimited|year=1993|location=San Diego}}</ref> The idea that quantum effects could be harnessed to create better laser diodes originated in the 1970s. The quantum well laser was patented in 1976 by R. Dingle and C. H. Henry.<ref>U.S. Patent #3,982,207, issued September 21, 1976, Inventors R. Dingle and C. H. Henry, "Quantum Effects in Heterostructure Lasers", filed March 7, 1975.</ref>

Specifically, the quantum wells behavior can be represented by the particle in a finite well model. Two boundary conditions must be selected. The first is that the wave function must be continuous. Often, the second boundary condition is chosen to be the derivative of the wave function must be continuous across the boundary, but in the case of the quantum well the masses are different on either side of the boundary. Instead, the second boundary condition is chosen to conserve particle flux as <math>(1/m) d\phi/dz</math>, which is consistent with experiment. The solution to the finite well particle in a box must be solved numerically, resulting in wave functions that are sine functions inside the quantum well and exponentially decaying functions in the barriers.<ref>{{Cite book|title=Confined Electrons and Photons: New Physics and Applications|last=Miller|first=David|publisher=Plenum Press|year=1995|editor-last=Burstein|editor-first=Elias|location=New York|pages=675–702|editor-last2=Weisbuch|editor-first2=Claude}}</ref> This quantization of the energy levels of the electrons allows a quantum well laser to emit light more efficiently than conventional semiconductor lasers.

Due to their small size, quantum dots do not showcase the bulk properties of the specified semi-conductor but rather show quantised energy states.<ref name="Inorganic chemistry">{{cite book|last1=Miessler|first1=G. L.|title=Inorganic chemistry|date=2013|publisher=Pearson|location=Boston|isbn=978-0321811059|pages=235–236|edition=5}}</ref> This effect is known as the quantum confinement and has led to numerous applications of quantum dots such as the quantum well laser.<ref name="Inorganic chemistry"/>

Researchers at Princeton University have recently built a quantum well laser that is no bigger than a grain of rice.<ref name="Princeton University">{{cite web|last1=Zandonella|first1=Catherine|title=Rice-sized laser, powered one electron at a time, bodes well for quantum computing|url=https://www.princeton.edu/main/news/archive/S42/13/37M75/index.xml?section=topstories|website=Princeton University|access-date=8 November 2016}}</ref> The laser is powered by a single electron that passes through two quantum dots; a double quantum dot. The electron moves from a state of higher energy, to a state of lower energy whilst emitting photons in the microwave region. These photons bounce off mirrors to create a beam of light; the laser.<ref name="Princeton University"/>

The quantum well laser is heavily based on the interaction between light and electrons. This relationship is a key component in quantum mechanical theories that include the De Broglie Wavelength and Particle in a box. The double quantum dot allows scientists to gain full control over the movement of an electron, which consequently results in the production of a laser beam.<ref name="Princeton University"/>

==== 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" />