Particle in a box

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Some trajectories of a particle in a box according to Newton's laws of classical mechanics (A), and according to the Schrödinger equation of quantum mechanics (B–F). In (B–F), the horizontal axis is position, and the vertical axis is the real part (blue) and imaginary part (red) of the wavefunction. The states (B,C,D) are energy eigenstates, but (E,F) are not.

In quantum mechanics, the particle in a box model (also known as the infinite potential well or the infinite square well) describes a particle free to move in a small space surrounded by impenetrable barriers. The model is mainly used as a hypothetical example to illustrate the differences between classical and quantum systems. In classical systems, for example, a particle trapped inside a large box can move at any speed within the box and it is no more likely to be found at one position than another. However, when the well becomes very narrow (on the scale of a few nanometers), quantum effects become important. The particle may only occupy certain positive energy levels. Likewise, it can never have zero energy, meaning that the particle can never "sit still". Additionally, it is more likely to be found at certain positions than at others, depending on its energy level. The particle may never be detected at certain positions, known as spatial nodes.

The particle in a box model is one of the very few problems in quantum mechanics which can be solved analytically, without approximations. Due to its simplicity, the model allows insight into quantum effects without the need for complicated mathematics. It serves as a simple illustration of how energy quantizations (energy levels), which are found in more complicated quantum systems such as atoms and molecules, come about. It is one of the first quantum mechanics problems taught in undergraduate physics courses, and it is commonly used as an approximation for more complicated quantum systems.

One-dimensional solution

The barriers outside a one-dimensional box have infinitely large potential, while the interior of the box has a constant, zero potential.

The simplest form of the particle in a box model considers a one-dimensional system. Here, the particle may only move backwards and forwards along a straight line with impenetrable barriers at either end. The walls of a one-dimensional box may be visualised as regions of space with an infinitely large potential energy. Conversely, the interior of the box has a constant, zero potential energy. This means that no forces act upon the particle inside the box and it can move freely in that region. However, infinitely large forces repel the particle if it touches the walls of the box, preventing it from escaping. The potential energy in this model is given as

${\displaystyle V(x)={\begin{cases}0,&x_{c}-{\tfrac {L}{2}}<x<x_{c}+{\tfrac {L}{2}},\\\infty ,&{\text{otherwise,}}\end{cases}},}$

where L is the length of the box, x_c is the location of the center of the box and x is the position of the particle within the box. Simple cases include the centered box (x_c = 0 ) and the shifted box (x_c = L/2 ).

Position wave function

In quantum mechanics, the wavefunction gives the most fundamental description of the behavior of a particle; the measurable properties of the particle (such as its position, momentum and energy) may all be derived from the wavefunction. The wavefunction ${\displaystyle \psi (x,t)}$ can be found by solving the Schrödinger equation for the system

${\displaystyle i\hbar {\frac {\partial }{\partial t}}\psi (x,t)=-{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}}{\partial x^{2}}}\psi (x,t)+V(x)\psi (x,t),}$

where ${\displaystyle \hbar }$ is the reduced Planck constant, ${\displaystyle m}$ is the mass of the particle, ${\displaystyle i}$ is the imaginary unit and ${\displaystyle t}$ is time.

Inside the box, no forces act upon the particle, which means that the part of the wavefunction inside the box oscillates through space and time with the same form as a free particle:

${\displaystyle \psi (x,t)=[A\sin(kx)+B\cos(kx)]\mathrm {e} ^{-i\omega t},}$

(1)

where ${\displaystyle A}$ and ${\displaystyle B}$ are arbitrary complex numbers. The frequency of the oscillations through space and time is given by the wavenumber ${\displaystyle k}$ and the angular frequency ${\displaystyle \omega }$ respectively. These are both related to the total energy of the particle by the expression

${\displaystyle E=\hbar \omega ={\frac {\hbar ^{2}k^{2}}{2m}},}$

which is known as the dispersion relation for a free particle. Here one must notice that now, since the particle is not entirely free but under the influence of a potential (the potential V described above), the energy of the particle given above is not the same thing as ${\displaystyle {\frac {p^{2}}{2m}}}$ where p is the momentum of the particle, and thus the wavenumber k above actually describes the energy states of the particle, not the momentum states (i.e. it turns out that the momentum of the particle is not given by ${\displaystyle p=\hbar k}$). In this sense, it is quite dangerous to call the number k a wavenumber, since it is not related to momentum like "wavenumber" usually is. The rationale for calling k the wavenumber is that it enumerates the number of crests that the wavefunction has inside the box, and in this sense it is a wavenumber. This discrepancy can be seen more clearly below, when we find out that the energy spectrum of the particle is discrete (only discrete values of energy are allowed) but the momentum spectrum is continuous (momentum can vary continuously) and in particular, the relation ${\displaystyle E={\frac {p^{2}}{2m}}}$ for the energy and momentum of the particle does not hold. As said above, the reason this relation between energy and momentum does not hold is that the particle is not free, but there is a potential V in the system, and the energy of the particle is ${\displaystyle E=T+V}$, where T is the kinetic and V the potential energy.

Initial wavefunctions for the first four states in a one-dimensional particle in a box

The size (or amplitude) of the wavefunction at a given position is related to the probability of finding a particle there by ${\displaystyle P(x,t)=|\psi (x,t)|^{2}}$. The wavefunction must therefore vanish everywhere beyond the edges of the box. Also, the amplitude of the wavefunction may not "jump" abruptly from one point to the next. These two conditions are only satisfied by wavefunctions with the form

${\displaystyle \psi _{n}(x,t)={\begin{cases}A\sin \left(k_{n}\left(x-x_{c}+{\tfrac {L}{2}}\right)\right)\mathrm {e} ^{-i\omega _{n}t}\quad &x_{c}-{\tfrac {L}{2}}<x<x_{c}+{\tfrac {L}{2}}\\0&{\text{otherwise}}\end{cases}},}$

where 

${\displaystyle k_{n}={\frac {n\pi }{L}}}$,

and

${\displaystyle E_{n}=\hbar \omega _{n}={\frac {n^{2}\pi ^{2}\hbar ^{2}}{2mL^{2}}}}$,

where n is a positive integer (1,2,3,4...). For a shifted box (x_c = L/2), the solution is particularly simple. The simplest solutions, ${\displaystyle k_{n}=0}$ or ${\displaystyle A=0}$ both yield the trivial wavefunction ${\displaystyle \psi (x)=0}$, which describes a particle that does not exist anywhere in the system. Negative values of ${\displaystyle n}$ are neglected, since they give wavefunctions identical to the positive ${\displaystyle n}$ solutions except for a physically unimportant sign change. Here one sees that only a discrete set of energy values and wavenumbers k are allowed for the particle. Usually in quantum mechanics it is also demanded that the derivative of the wavefunction in addition to the wavefunction itself be continuous; here this demand would lead to the only solution being the constant zero function, which is not what we desire, so we give up this demand (as this system with infinite potential can be regarded as a nonphysical abstract limiting case, we can treat it as such and "bend the rules"). Note that giving up this demand means that the wavefunction is not a differentiable function at the boundary of the box, and thus it can be said that the wavefunction does not solve the Schrödinger equation at the boundary points ${\displaystyle x=0}$ and ${\displaystyle x=L}$ (but does solve everywhere else).

Finally, the unknown constant ${\displaystyle A}$ may be found by normalizing the wavefunction so that the total probability density of finding the particle in the system is 1. It follows that

${\displaystyle \left|A\right|={\sqrt {\frac {2}{L}}}.}$

Thus, A may be any complex number with absolute value √2/L; these different values of A yield the same physical state, so A = √2/L can be selected to simplify.

It is expected that the eigenvalues, i.e., the energy ${\displaystyle E_{n}}$ of the box should be the same regardless of its position in space, but ${\displaystyle \psi _{n}(x,t)}$ changes. Notice that ${\displaystyle x_{c}-{\tfrac {L}{2}}}$ represents a phase shift in the wave function, This phase shift has no effect when solving the Schrödinger equation, and therefore does not affect the eigenvalue.

If we set the origin of coordinates to the left edge of the box, we can rewrite the spacial part of the wave function succinctly as:

${\displaystyle \psi _{n}(x)={\begin{cases}{\sqrt {\frac {2}{L}}}\sin(k_{n}x)\quad {}{\text{for }}n{\text{ even}}\\{\sqrt {\frac {2}{L}}}\cos(k_{n}x)\quad {}{\text{for }}n{\text{ odd}}\end{cases}}}$.

Momentum wave function

The momentum wavefunction is proportional to the Fourier transform of the position wavefunction. With ${\displaystyle k=p/\hbar }$ (note that the parameter k describing the momentum wavefunction below is not exactly the special k_n above, linked to the energy eigenvalues), the momentum wavefunction is given by

${\displaystyle \phi _{n}(p,t)={\frac {1}{\sqrt {2\pi \hbar }}}\int _{-\infty }^{\infty }\psi _{n}(x,t)e^{-ikx}\,dx={\sqrt {\frac {L}{\pi \hbar }}}\left({\frac {n\pi }{n\pi +kL}}\right)\,{\textrm {sinc}}\left({\tfrac {1}{2}}(n\pi -kL)\right)e^{-ikx_{c}}e^{i(n-1){\tfrac {\pi }{2}}}e^{-i\omega _{n}t},}$

where sinc is the cardinal sine sinc function, sinc(x)=sin(x)/x. For the centered box (x_c= 0), the solution is real and particularly simple, since the phase factor on the right reduces to unity. (With care, it can be written as an even function of p.)

It can be seen that the momentum spectrum in this wave packet is continuous, and one may conclude that for the energy state described by the wavenumber k_n, the momentum can, when measured, also attain other values beyond ${\displaystyle p=\pm \hbar k_{n}}$.

Hence, it also appears that, since the energy is ${\displaystyle E_{n}={\frac {\hbar ^{2}k_{n}^{2}}{2m}}}$ for the nth eigenstate, the relation ${\displaystyle E={\frac {p^{2}}{2m}}}$ does not strictly hold for the measured momentum p; the energy eigenstate ${\displaystyle \psi _{n}}$ is not a momentum eigenstate, and, in fact, not even a superposition of two momentum eigenstates, as one might be tempted to imagine from equation (1) above: peculiarly, it has no well-defined momentum before measurement!

Position and momentum probability distributions

In classical physics, the particle can be detected anywhere in the box with equal probability. In quantum mechanics, however, the probability density for finding a particle at a given position is derived from the wavefunction as ${\displaystyle P(x)=|\psi (x)|^{2}.}$ For the particle in a box, the probability density for finding the particle at a given position depends upon its state, and is given by

${\displaystyle P_{n}(x,t)={\begin{cases}{\frac {2}{L}}\sin ^{2}(k_{n}(x-x_{c}+{\tfrac {L}{2}})),&x_{c}-L/2<x<x_{c}+L/2,\\0,&{\text{otherwise,}}\end{cases}}}$

Thus, for any value of n greater than one, there are regions within the box for which ${\displaystyle P(x)=0}$, indicating that spatial nodes exist at which the particle cannot be found.

In quantum mechanics, the average, or expectation value of the position of a particle is given by

${\displaystyle \langle x\rangle =\int _{-\infty }^{\infty }xP_{n}(x)\,\mathrm {d} x.}$

For the steady state particle in a box, it can be shown that the average position is always ${\displaystyle \langle x\rangle =x_{c}}$, regardless of the state of the particle. For a superposition of states, the expectation value of the position will change based on the cross term which is proportional to ${\displaystyle \cos(\omega t)}$.

The variance in the position is a measure of the uncertainty in position of the particle:

${\displaystyle \mathrm {Var} (x)=\int _{-\infty }^{\infty }(x-\langle x\rangle )^{2}P_{n}(x)\,dx={\frac {L^{2}}{12}}\left(1-{\frac {6}{n^{2}\pi ^{2}}}\right)}$

The probability density for finding a particle with a given momentum is derived from the wavefunction as ${\displaystyle P(x)=|\phi (x)|^{2}}$. As with position, the probability density for finding the particle at a given momentum depends upon its state, and is given by

${\displaystyle P_{n}(p)={\frac {L}{\pi \hbar }}\left({\frac {n\pi }{n\pi +kL}}\right)^{2}\,{\textrm {sinc}}^{2}\left({\tfrac {1}{2}}(n\pi -kL)\right)}$

where, again, ${\displaystyle k=p/\hbar }$. The expectation value for the momentum is then calculated to be zero, and the variance in the momentum is calculated to be:

${\displaystyle \mathrm {Var} (p)=\left({\frac {\hbar n\pi }{L}}\right)^{2}}$

The uncertainties in position and momentum (${\displaystyle \Delta x}$ and ${\displaystyle \Delta p}$) are defined as being equal to the square root of their respective variances, so that:

${\displaystyle \Delta x\Delta p={\frac {\hbar }{2}}{\sqrt {{\frac {n^{2}\pi ^{2}}{3}}-2}}}$

This product increases with increasing n, having a minimum value for n=1. The value of this product for n=1 is about equal to 0.568 ${\displaystyle \hbar }$ which obeys the Heisenberg uncertainty principle, which states that the product will be greater than or equal to ${\displaystyle \hbar /2}$

Another measure of uncertainty in position is the information entropy of the probability distribution H_x:

${\displaystyle H_{x}=\int _{-\infty }^{\infty }P_{n}(x)\log(P_{n}(x)x_{0})\,dx=\log \left({\frac {2L}{e\,x_{0}}}\right)}$

where x₀ is an arbitrary reference length.

Another measure of uncertainty in momentum is the information entropy of the probability distribution H_p:

${\displaystyle H_{p}(n)=\int _{-\infty }^{\infty }P_{n}(p)\log(P_{n}(p)p_{0})\,dp}$

${\displaystyle \lim _{n\to \infty }H_{p}(n)=\log \left({\frac {4\pi \hbar \,e^{2(1-\gamma )}}{L\,p_{0}}}\right)}$

where γ is Euler's constant. The quantum mechanical entropic uncertainty principle states that for ${\displaystyle x_{0}\,p_{0}=\hbar }$

${\displaystyle H_{x}+H_{p}(n)\geq \log(e\,\pi )\approx 2.14473...}$ (nats)

For ${\displaystyle x_{0}\,p_{0}=\hbar }$, the sum of the position and momentum entropies yields:

${\displaystyle H_{x}+H_{p}(\infty )=\log(8\pi \,e^{1-2\gamma })\approx 3.06974...}$ (nats)

which satisfies the quantum entropic uncertainty principle.

Energy levels

The energy of a particle in a box (black circles) and a free particle (grey line) both depend upon wavenumber in the same way. However, the particle in a box may only have certain, discrete energy levels.

The energies which correspond with each of the permitted wavenumbers may be written as

${\displaystyle E_{n}={\frac {n^{2}\hbar ^{2}\pi ^{2}}{2mL^{2}}}={\frac {n^{2}h^{2}}{8mL^{2}}}}$.

The energy levels increase with ${\displaystyle n^{2}}$, meaning that high energy levels are separated from each other by a greater amount than low energy levels are. The lowest possible energy for the particle (its zero-point energy) is found in state 1, which is given by

${\displaystyle E_{1}={\frac {\hbar ^{2}\pi ^{2}}{2mL^{2}}}.}$

The particle, therefore, always has a positive energy. This contrasts with classical systems, where the particle can have zero energy by resting motionlessly. This can be explained in terms of the uncertainty principle, which states that the product of the uncertainties in the position and momentum of a particle is limited by

${\displaystyle \Delta x\Delta p\geq {\frac {\hbar }{2}}}$

It can be shown that the uncertainty in the position of the particle is proportional to the width of the box. Thus, the uncertainty in momentum is roughly inversely proportional to the width of the box. The kinetic energy of a particle is given by ${\displaystyle E=p^{2}/(2m)}$, and hence the minimum kinetic energy of the particle in a box is inversely proportional to the mass and the square of the well width, in qualitative agreement with the calculation above.

Higher-dimensional boxes

(Hyper)rectangular walls

The wavefunction of a 2D well with n_x=4 and n_y=4

If a particle is trapped in a two-dimensional box, it may freely move in the ${\displaystyle x}$ and ${\displaystyle y}$-directions, between barriers separated by lengths ${\displaystyle L_{x}}$ and ${\displaystyle L_{y}}$ respectively. For a centered box, the position wave function may be written including the length of the box as ${\displaystyle \psi _{n}(x,t,L)}$. Using a similar approach to that of the one-dimensional box, it can be shown that the wavefunctions and energies for a centered box are given respectively by

${\displaystyle \psi _{n_{x},n_{y}}=\psi _{n_{x}}(x,t,L_{x})\psi _{n_{y}}(y,t,L_{y})}$,

${\displaystyle E_{n_{x},n_{y}}={\frac {\hbar ^{2}k_{n_{x},n_{y}}^{2}}{2m}}}$,

where the two-dimensional wavevector is given by

${\displaystyle \mathbf {k_{n_{x},n_{y}}} =k_{n_{x}}\mathbf {\hat {x}} +k_{n_{y}}\mathbf {\hat {y}} ={\frac {n_{x}\pi }{L_{x}}}\mathbf {\hat {x}} +{\frac {n_{y}\pi }{L_{y}}}\mathbf {\hat {y}} }$.

For a three dimensional box, the solutions are

${\displaystyle \psi _{n_{x},n_{y},n_{z}}=\psi _{n_{x}}(x,t,L_{x})\psi _{n_{y}}(y,t,L_{y})\psi _{n_{z}}(z,t,L_{z})}$,

${\displaystyle E_{n_{x},n_{y},n_{z}}={\frac {\hbar ^{2}k_{n_{x},n_{y},n_{z}}^{2}}{2m}}}$,

where the three-dimensional wavevector is given by:

${\displaystyle \mathbf {k_{n_{x},n_{y},n_{z}}} =k_{n_{x}}\mathbf {\hat {x}} +k_{n_{y}}\mathbf {\hat {y}} +k_{n_{z}}\mathbf {\hat {z}} ={\frac {n_{x}\pi }{L_{x}}}\mathbf {\hat {x}} +{\frac {n_{y}\pi }{L_{y}}}\mathbf {\hat {y}} +{\frac {n_{z}\pi }{L_{z}}}\mathbf {\hat {z}} }$.

In general for an n-dimensional box, the solutions are

${\displaystyle \psi =\prod _{i}\psi _{n_{i}}(x_{i},t,L_{i})}$

The n-dimensional momentum wave functions may likewise be represented by ${\displaystyle \phi _{n}(x,t,L_{x})}$ and the momentum wave function for an n-dimensional centered box is then:

${\displaystyle \phi =\prod _{i}\phi _{n_{i}}(k_{i},t,L_{i})}$

An interesting feature of the above solutions is that when two or more of the lengths are the same (e.g. ${\displaystyle L_{x}=L_{y}}$), there are multiple wavefunctions corresponding to the same total energy. For example, the wavefunction with ${\displaystyle n_{x}=2,n_{y}=1}$ has the same energy as the wavefunction with ${\displaystyle n_{x}=1,n_{y}=2}$. This situation is called degeneracy and for the case where exactly two degenerate wavefunctions have the same energy that energy level is said to be doubly degenerate. Degeneracy results from symmetry in the system. For the above case two of the lengths are equal so the system is symmetric with respect to a 90° rotation.

More complicated wall shapes

The wavefunction for a quantum-mechanical particle in a box whose walls have arbitrary shape is given by the Helmholtz equation subject to the boundary condition that the wavefunction vanishes at the walls. These systems are studied in the field of quantum chaos for wall shapes whose corresponding dynamical billiard tables are non-integrable.

Applications

Because of its mathematical simplicity, the particle in a box model is used to find approximate solutions for more complex physical systems in which a particle is trapped in a narrow region of low electric potential between two high potential barriers. These quantum well systems are particularly important in optoelectronics, and are used in devices such as the quantum well laser, the quantum well infrared photodetector and the quantum-confined Stark effect modulator. It is also used to model a lattice in the Kronig-Penney model and for a finite metal with the free electron approximation.

Conjugated polyenes

β-carotene is a conjugated polyene

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, n_f and n_i is:

${\displaystyle \Delta E={\frac {(n_{f}^{2}-n_{i}^{2})h^{2}}{8mL^{2}}}}$

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:

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

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 3.8 nm (though its chain length is only approximately 2.4 nm). Due to β-carotene's high level of conjugation, electrons are dispersed throughout 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; each of those double bonds contains two π-electrons, therefore 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, the minimum energy needed to excite an electron to the next energy level can be calculated, n=12, as follows (recalling that the mass of an electron is 9.109 × 10⁻³¹ kg):

${\displaystyle \Delta E={\frac {(n_{f}^{2}-n_{i}^{2})h^{2}}{8mL^{2}}}}$

${\displaystyle ={\frac {(12^{2}-11^{2})h^{2}}{8mL^{2}}}}$

${\displaystyle =2.3658\times 10^{-19}{\text{ J}}}$

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

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

${\displaystyle =0.00000084{\text{ m}}=840{\text{ nm}}}$

This indicates that β-carotene primarily absorbs light in the infrared spectrum, therefore it would appear white to a human eye. However the observed wavelength is 450 nm, indicating that the particle in a box is not a perfect model for this system.

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${\displaystyle (1/m)d\phi /dz}$, 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 ${\displaystyle \Delta E(r)}$ is equal to the band gap of the bulk material ${\displaystyle E_{\text{gap}}}$ 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 ${\displaystyle m_{e}^{*}}$ and ${\displaystyle m_{h}^{*}}$ are the effective masses of the electron and hole, ${\displaystyle r}$ is radius of the dot, and ${\displaystyle h}$ is Planck's constant:

${\displaystyle \Delta E(r)=E_{\text{gap}}+\left({\frac {h^{2}}{8r^{2}}}\right)\left({\frac {1}{m_{e}^{*}}}+{\frac {1}{m_{h}^{*}}}\right)}$

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.

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.

Teleology

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Plato and Aristotle, depicted here in The School of Athens, both developed philosophical arguments addressing the universe's apparent order (logos)

Teleology (from τέλος, telos, 'end', 'aim', or 'goal,' and λόγος, logos, 'explanation' or 'reason') or finality is a reason or explanation for something as a function of its end, purpose, or goal. A purpose that is imposed by a human use, such as that of a fork, is called extrinsic.

Natural teleology, common in classical philosophy, though controversial today, contends that natural entities also have intrinsic purposes, irrespective of human use or opinion. For instance, Aristotle claimed that an acorn's intrinsic telos is to become a fully grown oak tree. Though ancient atomists rejected the notion of natural teleology, teleological accounts of non-personal or non-human nature were explored and often endorsed in ancient and medieval philosophies, but fell into disfavor during the modern era (1600–1900).

In the late 18th century, Immanuel Kant used the concept of telos as a regulative principle in his Critique of Judgment (1790). Teleology was also fundamental to the philosophy of Karl Marx and G. W. F. Hegel.

Contemporary philosophers and scientists are still in debate as to whether teleological axioms are useful or accurate in proposing modern philosophies and scientific theories. An example of the reintroduction of teleology into modern language is the notion of an attractor. Another instance is when Thomas Nagel (2012), though not a biologist, proposed a non-Darwinian account of evolution that incorporates impersonal and natural teleological laws to explain the existence of life, consciousness, rationality, and objective value. Regardless, the accuracy can also be considered independently from the usefulness: it is a common experience in pedagogy that a minimum of apparent teleology can be useful in thinking about and explaining Darwinian evolution even if there is no true teleology driving evolution. Thus it is easier to say that evolution "gave" wolves sharp canine teeth because those teeth "serve the purpose of" predation regardless of whether there is an underlying non-teleologic reality in which evolution is not an actor with intentions. In other words, because human cognition and learning often rely on the narrative structure of stories (with actors, goals, and immediate (proximal) rather than ultimate (distal) causation (see also proximate and ultimate causation), some minimal level of teleology might be recognized as useful or at least tolerable for practical purposes even by people who reject its cosmologic accuracy. Its accuracy is upheld by Barrow and Tippler (1986), whose citings of such teleologists as Max Planck and Norbert Wiener are significant for scientific endeavor.

History

In western philosophy, the term and concept of teleology originated in the writings of Plato and Aristotle. Aristotle's 'four causes' give special place to the telos or "final cause" of each thing. In this, he followed Plato in seeing purpose in both human and subhuman nature.

Etymology

The word teleology combines Greek telos (τέλος, from τελε-, 'end' or 'purpose') and logia (-λογία, 'speak of', 'study of', or 'a branch of learning"'). German philosopher Christian Wolff would coin the term, as teleologia (Latin), in his work Philosophia rationalis, sive logica (1728).

Platonic

In the Phaedo, Plato through Socrates argues that true explanations for any given physical phenomenon must be teleological. He bemoans those who fail to distinguish between a thing's necessary and sufficient causes, which he identifies respectively as material and final causes:

Imagine not being able to distinguish the real cause, from that without which the cause would not be able to act, as a cause. It is what the majority appear to do, like people groping in the dark; they call it a cause, thus giving it a name that does not belong to it. That is why one man surrounds the earth with a vortex to make the heavens keep it in place, another makes the air support it like a wide lid. As for their capacity of being in the best place they could be at this very time, this they do not look for, nor do they believe it to have any divine force, but they believe that they will some time discover a stronger and more immortal Atlas to hold everything together more, and they do not believe that the truly good and 'binding' binds and holds them together.

— Plato, Phaedo, 99

Plato here argues that while the materials that compose a body are necessary conditions for its moving or acting in a certain way, they nevertheless cannot be the sufficient condition for its moving or acting as it does. For example, if Socrates is sitting in an Athenian prison, the elasticity of his tendons is what allows him to be sitting, and so a physical description of his tendons can be listed as necessary conditions or auxiliary causes of his act of sitting. However, these are only necessary conditions of Socrates' sitting. To give a physical description of Socrates' body is to say that Socrates is sitting, but it does not give us any idea why it came to be that he was sitting in the first place. To say why he was sitting and not not sitting, we have to explain what it is about his sitting that is good, for all things brought about (i.e., all products of actions) are brought about because the actor saw some good in them. Thus, to give an explanation of something is to determine what about it is good. Its goodness is its actual cause—its purpose, telos or "reason for which."

Aristotelian

Aristotle argued that Democritus was wrong to attempt to reduce all things to mere necessity, because doing so neglects the aim, order, and "final cause", which brings about these necessary conditions:

Democritus, however, neglecting the final cause, reduces to necessity all the operations of nature. Now, they are necessary, it is true, but yet they are for a final cause and for the sake of what is best in each case. Thus nothing prevents the teeth from being formed and being shed in this way; but it is not on account of these causes but on account of the end.…

— Aristotle, Generation of Animals 5.8, 789a8–b15

In Physics, using eternal forms as his model, Aristotle rejects Plato's assumption that the universe was created by an intelligent designer. For Aristotle, natural ends are produced by "natures" (principles of change internal to living things), and natures, Aristotle argued, do not deliberate:

It is absurd to suppose that ends are not present [in nature] because we do not see an agent deliberating.

— Aristotle, Physics, 2.8, 199b27-9

These Platonic and Aristotelian arguments ran counter to those presented earlier by Democritus and later by Lucretius, both of whom were supporters of what is now often called accidentalism:

Nothing in the body is made in order that we may use it. What happens to exist is the cause of its use.

— Lucretius, De rerum natura [On the Nature of Things] 4, 833

Economics

A teleology of human aims played a crucial role in the work of economist Ludwig von Mises, especially in the development of his science of praxeology. More specifically, Mises believed that human action (i.e. purposeful behavior) is teleological, based on the presupposition that an individual's action is governed or caused by the existence of their chosen ends. In other words, individuals select what they believe to be the most appropriate means to achieve a sought after goal or end. Mises also stressed that, with respect to human action, teleology is not independent of causality: "No action can be devised and ventured upon without definite ideas about the relation of cause and effect, teleology presupposes causality."

Assuming reason and action to be predominantly influenced by ideological credence, Mises derived his portrayal of human motivation from Epicurean teachings, insofar as he assumes "atomistic individualism, teleology, and libertarianism, and defines man as an egoist who seeks a maximum of happiness" (i.e. the ultimate pursuit of pleasure over pain). "Man strives for," Mises remarks, "but never attains the perfect state of happiness described by Epicurus." Moreover, expanding upon the Epicurean groundwork, Mises formalized his conception of pleasure and pain by assigning each specific meaning, allowing him to extrapolate his conception of attainable happiness to a critique of liberal versus socialist ideological societies. It is there, in his application of Epicurean belief to political theory, that Mises flouts Marxist theory, considering labor to be one of many of man's 'pains', a consideration which positioned labor as a violation of his original Epicurean assumption of man's manifest hedonistic pursuit. From here he further postulates a critical distinction between introversive labor and extroversive labor, further divaricating from basic Marxist theory, in which Marx hails labor as man's "species-essense", or his "species-activity".

Postmodern philosophy

Teleological-based "grand narratives" are renounced by the postmodern tradition, where teleology may be viewed as reductive, exclusionary, and harmful to those whose stories are diminished or overlooked.

Against this postmodern position, Alasdair MacIntyre has argued that a narrative understanding of oneself, of one's capacity as an independent reasoner, one's dependence on others and on the social practices and traditions in which one participates, all tend towards an ultimate good of liberation. Social practices may themselves be understood as teleologically oriented to internal goods, for example practices of philosophical and scientific inquiry are teleologically ordered to the elaboration of a true understanding of their objects. MacIntyre's After Virtue (1981) famously dismissed the naturalistic teleology of Aristotle's 'metaphysical biology', but he has cautiously moved from that book's account of a sociological teleology toward an exploration of what remains valid in a more traditional teleological naturalism.

Hegel

Historically, teleology may be identified with the philosophical tradition of Aristotelianism. The rationale of teleology was explored by Immanuel Kant (1790) in his Critique of Judgement and made central to speculative philosophy by G. W. F. Hegel (as well as various neo-Hegelian schools). Hegel proposed a history of our species which some consider to be at variance with Darwin, as well as with the dialectical materialism of Karl Marx and Friedrich Engels, employing what is now called analytic philosophy—the point of departure being not formal logic and scientific fact but 'identity', or "objective spirit" in Hegel's terminology.

Individual human consciousness, in the process of reaching for autonomy and freedom, has no choice but to deal with an obvious reality: the collective identities (e.g. the multiplicity of world views, ethnic, cultural, and national identities) that divide the human race and set different groups in violent conflict with each other. Hegel conceived of the 'totality' of mutually antagonistic world-views and life-forms in history as being 'goal-driven', i.e. oriented towards an end-point in history. The 'objective contradiction' of 'subject' and 'object' would eventually 'sublate' into a form of life that leaves violent conflict behind. This goal-oriented, teleological notion of the "historical process as a whole" is present in a variety of 20th-century authors, although its prominence declined drastically after the Second World War.

Ethics

Teleology significantly informs the study of ethics, such as in:

• Business ethics: People in business commonly think in terms of purposeful action, as in, for example, management by objectives. Teleological analysis of business ethics leads to consideration of the full range of stakeholders in any business decision, including the management, the staff, the customers, the shareholders, the country, humanity and the environment.
• Medical ethics: Teleology provides a moral basis for the professional ethics of medicine, as physicians are generally concerned with outcomes and must therefore know the telos of a given treatment paradigm.

Consequentialism

The broad spectrum of consequentialist ethics—of which utilitarianism is a well-known example—focuses on the end result or consequences, with such principles as John Stuart Mill's 'principle of utility': "the greatest good for the greatest number." This principle is thus teleological, though in a broader sense than is elsewhere understood in philosophy.

In the classical notion, teleology is grounded in the inherent nature of things themselves, whereas in consequentialism, teleology is imposed on nature from outside by the human will. Consequentialist theories justify inherently what most people would call evil acts by their desirable outcomes, if the good of the outcome outweighs the bad of the act. So, for example, a consequentialist theory would say it was acceptable to kill one person in order to save two or more other people. These theories may be summarized by the maxim "the end justifies the means."

Deontologicalism

Consequentialism stands in contrast to the more classical notions of deontological ethics, of which examples include Immanuel Kant's categorical imperative, and Aristotle's virtue ethics—although formulations of virtue ethics are also often consequentialist in derivation.

In deontological ethics, the goodness or badness of individual acts is primary and a larger, more desirable goal is insufficient to justify bad acts committed on the way to that goal, even if the bad acts are relatively minor and the goal is major (like telling a small lie to prevent a war and save millions of lives). In requiring all constituent acts to be good, deontological ethics is much more rigid than consequentialism, which varies by circumstances.

Practical ethics are usually a mix of the two. For example, Mill also relies on deontic maxims to guide practical behavior, but they must be justifiable by the principle of utility.

Science

In modern science, explanations that rely on teleology are often, but not always, avoided, either because they are unnecessary or because whether they are true or false is thought to be beyond the ability of human perception and understanding to judge. But using teleology as an explanatory style, in particular within evolutionary biology, is still controversial.

Since the Novum Organum of Francis Bacon, teleological explanations in physical science tend to be deliberately avoided in favor of focus on material and efficient explanations. Final and formal causation came to be viewed as false or too subjective. Nonetheless, some disciplines, in particular within evolutionary biology, continue to use language that appears teleological in describing natural tendencies towards certain end conditions. Some suggest, however, that these arguments ought to be, and practicably can be, rephrased in non-teleological forms, others hold that teleological language cannot always be easily expunged from descriptions in the life sciences, at least within the bounds of practical pedagogy.

Biology

Apparent teleology is a recurring issue in evolutionary biology, much to the consternation of some writers.

Statements implying that nature has goals, for example where a species is said to do something "in order to" achieve survival, appear teleological, and therefore invalid. Usually, it is possible to rewrite such sentences to avoid the apparent teleology. Some biology courses have incorporated exercises requiring students to rephrase such sentences so that they do not read teleologically. Nevertheless, biologists still frequently write in a way which can be read as implying teleology even if that is not the intention. John Reiss (2009) argues that evolutionary biology can be purged of such teleology by rejecting the analogy of natural selection as a watchmaker. Other arguments against this analogy have also been promoted by writers such as Richard Dawkins (1987).

Some authors, like James Lennox (1993), have argued that Darwin was a teleologist, while others, such as Michael Ghiselin (1994), describe this claim as a myth promoted by misinterpretations of his discussions and emphasized the distinction between using teleological metaphors and being teleological.

Biologist philosopher Francisco Ayala (1998) has argued that all statements about processes can be trivially translated into teleological statements, and vice versa, but that teleological statements are more explanatory and cannot be disposed of. Karen Neander (1998) has argued that the modern concept of biological 'function' is dependent upon selection. So, for example, it is not possible to say that anything that simply winks into existence without going through a process of selection has functions. We decide whether an appendage has a function by analysing the process of selection that led to it. Therefore, any talk of functions must be posterior to natural selection and function cannot be defined in the manner advocated by Reiss and Dawkins.

Ernst Mayr (1992) states that "adaptedness…is an a posteriori result rather than an a priori goal-seeking." Various commentators view the teleological phrases used in modern evolutionary biology as a type of shorthand. For example, S. H. P. Madrell (1998) writes that "the proper but cumbersome way of describing change by evolutionary adaptation [may be] substituted by shorter overtly teleological statements" for the sake of saving space, but that this "should not be taken to imply that evolution proceeds by anything other than from mutations arising by chance, with those that impart an advantage being retained by natural selection." Likewise, J. B. S. Haldane says, "Teleology is like a mistress to a biologist: he cannot live without her but he's unwilling to be seen with her in public."

Selected-effects accounts, such as the one suggested by Neander (1998), face objections due to their reliance on etiological accounts, which some fields lack the resources to accommodate. Many such sciences, which study the same traits and behaviors regarded by evolutionary biology, still correctly attribute teleological functions without appeal to selection history. Corey J. Maley and Gualtiero Piccinini (2018/2017) are proponents of one such account, which focuses instead on goal-contribution. With the objective goals of organisms being survival and inclusive fitness, Piccinini and Maley define teleological functions to be “a stable contribution by a trait (or component, activity, property) of organisms belonging to a biological population to an objective goal of those organisms.”

Cybernetics

Cybernetics is the study of the communication and control of regulatory feedback both in living beings and machines, and in combinations of the two.

Arturo Rosenblueth, Norbert Wiener, and Julian Bigelow (1943) had conceived of feedback mechanisms as lending a teleology to machinery. Wiener (1948) coined the term cybernetics to denote the study of "teleological mechanisms." In the cybernetic classification presented by Rosenblueth, Wiener, and Bigelow (1943), teleology is feedback controlled purpose.

The classification system underlying cybernetics has been criticized by Frank Honywill George and Les Johnson (1985), who cite the need for an external observability to the purposeful behavior in order to establish and validate the goal-seeking behavior. In this view, the purpose of observing and observed systems is respectively distinguished by the system's subjective autonomy and objective control.