Particle in a box
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 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 seen 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 formula_1 where "L" is the length of the box, "x" 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" = 0) and the shifted box ("x" = "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 formula_2 can be found by solving the Schrödinger equation for the system formula_3 where formula_4 is the reduced Planck constant, formula_5 is the mass of the particle, formula_6 is the imaginary unit and formula_7 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: formula_12 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 formula_13 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 formula_14). 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 formula_15 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 formula_16, where "T" is the kinetic and "V" the potential energy. The size (or amplitude) of the wavefunction at a given position is related to the probability of finding  a particle there by formula_17. 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 formula_18 where formula_19 and formula_20 where "n" is a positive integer (1, 2, 3, 4, ...). For a shifted box ("x" = "L"/2), the solution is particularly simple. The simplest solutions, formula_21 or formula_22 both yield the trivial wavefunction formula_23, which describes a particle that does not exist anywhere in the system. Negative values of formula_24 are  neglected, since they give wavefunctions identical to the positive formula_24 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 formula_26 and formula_27 (but does solve it everywhere else). Finally, the unknown constant formula_8 may be found by normalizing the wavefunction so that the total probability density of finding the particle in the system is 1. Mathematically, formula_29 (The particle must be "somewhere") It follows that formula_30 Thus, "A" may  be any complex number with absolute value ; these different values of "A" yield the same physical state, so "A" = can be selected to simplify. It is expected that the "eigenvalues", i.e., the energy formula_31 of the box should be the same regardless of its position in space, but formula_32 changes. Notice that formula_33 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 center of the box, we can rewrite the spacial part of  the wave function succinctly as: formula_34 Momentum wave function. The momentum wavefunction is proportional to the Fourier transform of the position wavefunction. With formula_35 (note that the parameter describing the momentum wavefunction below is not exactly the special above, linked to the energy eigenvalues), the momentum wavefunction is given by formula_36 where sinc is the cardinal sine sinc function, . For the centered box (), 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 .) 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 , the momentum can, when measured, also attain "other values" beyond formula_37. Hence, it also appears that, since the energy is formula_38 for the "n"th eigenstate, the relation formula_39 does not strictly hold for the measured momentum ; the energy eigenstate formula_40 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 () above: peculiarly, it has no well-defined momentum before measurement! Position and momentum probability distributions. In classic  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 formula_41 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 formula_42 Thus, for any value of "n" greater than one, there are regions within the box for which formula_43, 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 formula_44 For the steady state particle in a box, it can be shown that the average position is always formula_45, 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 formula_46. The variance in the position is a measure of the uncertainty in position of the particle: formula_47 The probability density for finding a particle with a given momentum is derived from the wavefunction as formula_48. As with position, the probability density  for finding the particle at a given momentum depends upon its state, and is given by formula_49 where, again, formula_35. The expectation value for the momentum is then calculated to be zero, and the variance in the momentum is calculated to be: formula_51 The uncertainties in position and momentum (formula_52 and formula_53) are defined as being equal to the square root of their respective variances, so that: formula_54 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 formula_4 which obeys the Heisenberg uncertainty principle, which  states that the product will be greater than or equal to formula_56 Another measure of uncertainty in position is the information entropy of the probability distribution "H": formula_57 where "x" is an arbitrary reference length. Another measure of uncertainty in momentum is the information entropy of the probability distribution "H": formula_58 formula_59 where "γ" is Euler's constant. The quantum mechanical entropic uncertainty principle states that for formula_60 formula_61 (nats) For formula_62, the sum of the position and momentum entropies yields: formula_63 (nats) which satisfies the quantum entropic uncertainty principle. Energy levels. The energies which correspond with each of the  permitted wavenumbers may be written as formula_64 The energy levels increase with formula_65, 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 formula_66 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 formula_67 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 formula_68, 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. Section::::(Hyper)rectangular walls. If a particle is trapped in a two-dimensional box, it  may freely move in the formula_69 and formula_70-directions, between barriers separated by lengths formula_71 and formula_72 respectively. For a centered box, the position wave function may be written including the length of the box as formula_73. 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 formula_74 formula_75 where the two-dimensional wavevector is given by formula_76 For a three dimensional box, the solutions are formula_77 formula_78 where the three-dimensional wavevector is given by: formula_79 In general for an "n"-dimensional  box, the solutions are formula_80 The n-dimensional momentum wave functions may likewise be represented by formula_81 and the momentum wave function for an n-dimensional centered box is then: formula_82 An interesting feature of the above solutions is that when two or more of the lengths are the same (e.g. formula_83), there are multiple wavefunctions corresponding to the same total energy. For example, the wavefunction with formula_84 has the same energy as the wavefunction with formula_85. 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. 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 their energy level. The energy difference between two energy levels, "n" and "n" is: formula_86 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: formula_87 A common example of  this phenomenon is in β-carotene. β-carotene (CH) 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 kg): formula_88 Using the previous relation of wavelength to energy, recalling both Planck's constant "h" and the speed of light "c": formula_89 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 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 formula_90, 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 used. 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 formula_91 is equal  to the gap of the bulk material formula_92 plus the energy equation derived particle-in-a-box, which gives the energy for electrons and holes. This can be seen in the following equation, where formula_93 and formula_94 are the effective masses of the electron and hole, formula_95 is radius of the dot, and formula_96 is Planck's constant: formula_97 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. Relativistic effects. The probability density does not go to zero at the nodes if relativistic effects are taken to account via Dirac equation.