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Conjugated Polyene Systems
Conjugated polyene systems can be modeled using particle in a box. The conjugated system of electrons can be modeled as a one dimensional box with length equal to the total bond distance from one terminus of the polyene to the other. In this case each pair of electrons in each π bond corresponds to one energy level. The energy difference between two energy levels, nf and ni is:

$$\Delta E = \frac{(n_f^2 - n_i^2) h^2}{8mL^2}$$

The difference between the ground state energy, n, and the first excited state, n+1, corresponds to the energy required to excite the system. This energy has a specific wavelength, and therefore color of light, related by:

$$\lambda = \frac{hc}{\Delta E}$$

A common example of this phenomenon is in β-carotene. β-carotene (C40H56) is a conjugated polyene with an orange color and a molecular length of approximately 3.8nm (though its chain length is only approximately 2.4nm). Due to β-carotene's high level of conjugation, electrons can travel freely along the length of the molecule, allowing one to model it as a one-dimensional particle in a box. β-carotene has 11 carbon-carbon double bonds in conjugation ; since each of those double bonds contains two π-electrons, β-carotene has 22 π-electrons. With two electrons per energy level, β-carotene can be treated as a particle in a box at energy level n=11. Therefore, we can calculate the minimum energy needed to excite an electron to the next energy level, n=12, as follows (recalling that the mass of an electron is 9.109 × 10-31 kg ):

$$\Delta E = \frac{(n_f^2 - n_i^2) h^2}{8 m L^2}$$

$$= \frac{(12^2 - 11^2) h^2}{8 m L^2}$$

$$= 2.3658\times10^{-19} J$$

Using our previous relation of wavelength to energy, recalling both Planck's constant h and the speed of light c:

$$\lambda = \tfrac{ hc }{ \Delta E }$$

$$= 8303 AU$$

This number describes the wavelength of light, in atomic units, required to induce a transition from beta-carotene's ground state to its first excited state. We can now determine the wavelength in nanometers, using that 1 atomic unit is 0.529177 × 10-10 m :

$$8303 AU \times \left ( \frac{0.529177 \times 10^{-10} m}{1AU} \right ) = 4.394 \times 10^{-7} m $$

$$(4.394 \times 10^{-7}m) \times \left ( \frac{10^9 nm}{1 m} \right ) = 439 nm $$

So the calculated wavelength using this method is 439 nm, while the observed wavelength is 450 nm. Therefore, the particle in a box model can be used to predict the coloration of conjugated molecules such as β-carotene.

Quantum Well Laser
The particle in a box model can be applied to quantum well lasers, 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 Å thick), quantum confinement effects can be observed. 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

Specifically, the quantum well’s 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$$(1/m) d\phi/dz$$, 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. 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. This effect is known as the quantum confinement and has led to numerous applications of quantum dots such as the quantum well laser.

Researchers at Princeton University have recently built a quantum well laser which is no bigger than a grain of rice. The laser is powered by a single electron which 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.

The quantum well laser is heavily based on the interaction between light and electrons. This relationship is a key component in quantum mechanical theories which 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.

Quantum Dots
Quantum dots are extremely small semiconductors (on the scale of nanometers). They display quantum confinement in that the electrons cannot escape the “dot”, thus allowing particle-in-a-box approximations to be applied. Their behavior can be described by three-dimensional particle-in-a-box energy quantization equations.

The energy gap of a quantum dot is the energy gap between its valence and conduction bands. This energy gap (E(r)) is equal to the band gap of the bulk material (Egap) plus the energy equation derived from particle-in-a-box, which gives the energy for electrons and holes. This can be seen in the following equation, where m*e and m*h are the effective masses of the electron and hole, r is radius of the dot, and h is Plank's constant:

$$\bigtriangleup E(r)=E_{gap}+\left ( \frac{h^2}{8r^2} \right )(\frac{1}{m^*_e}+\frac{1}{m^*_h})$$

Hence, the energy gap of the quantum dot is inversely proportional to the square of the “length of the box,” i.e. the the radius of the quantum dot.

Manipulation of the band gap allows for the absorption and emission of specific wavelengths of light, as energy is inversely proportional to wavelength. The smaller the quantum dot, the larger the band gap and thus the shorter the wavelength absorbed.

Different semiconducting materials are used to synthesize quantum dots of different sizes and therefore emit different wavelengths of light. Materials that normally emit light in the visible region are often used and their sizes are fine-tuned so that certain colors are emitted. Typical substances used to synthesize quantum dots are cadmium (Cd) and selenium (Se). For example, when the electrons of two nanometer CdSe quantum dots relax after excitation, blue light is emitted. Similarly, red light is emitted in four nanometer CdSe quantum dots.

Quantum dots have a variety of functions including but not limited to fluorescent dyes, transistors, LEDs, solar cells, and medical imaging via optical probes.

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.

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.