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| [[File:InfiniteSquareWellAnimation.gif|thumb|300px|right|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 eigenstate]]s, but (E,F) are not.]]
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| 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 physics|classical]] and quantum systems. In classical systems, for example a ball trapped inside a large box, the particle 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 [[nanometer]]s), 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.
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| The particle in a box model provides one of the very few problems in quantum mechanics which can be solved analytically, without approximations. This means that the observable properties of the particle (such as its energy and position) are related to the mass of the particle and the width of the well by simple mathematical expressions. Due to its simplicity, the model allows insight into quantum effects without the need for complicated mathematics. 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. See also: the [[history of quantum mechanics]].
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| == One-dimensional solution ==
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| [[File:Infinite potential well-en.svg|thumb|right|The barriers outside a one-dimensional box have infinitely large potential, while the interior of the box has a constant, zero potential.]]
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| 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.<ref name="Davies4">Davies, p.4</ref>
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| 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.<ref>Actually, any constant, finite potential <math>V_0</math> can be specified within the box. This merely shifts the energies of the states by <math>V_0</math>.</ref> This means that no forces act upon the particle inside the box and it can move freely in that region. However, infinitely large [[force]]s repel the particle if it touches the walls of the box, preventing it from escaping. The potential energy in this model is given as
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| :<math>V(x) =
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| \begin{cases}
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| 0, & 0 < x < L,\\
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| \infty, & \text{otherwise,}
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| \end{cases},
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| </math>
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| where <math>L</math> is the length of the box and <math>x</math> is the position of the particle within the box.
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| === Wavefunctions ===
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| 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.<ref name="Davies1">Davies, p. 1</ref>
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| The wavefunction <math>\psi(x,t)</math> can be found by solving the [[Schrödinger equation]] for the system
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| :<math>\mathrm{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),</math>
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| where <math>\hbar</math> is the [[reduced Planck constant]], <math>m</math> is the [[mass]] of the particle, <math>\mathrm{i}</math> is the [[imaginary unit]] and <math>t</math> is time.
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| 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]]:<ref name="Davies4" /><ref name = "Bransden157">Bransden and Joachain, p. 157</ref>
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| :<math>\psi(x,t) = [A \sin(kx) + B \cos(kx)]\mathrm{e}^{-\mathrm{i}\omega t},</math>
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| where <math>A</math> and <math>B</math> are arbitrary [[complex number]]s. The frequency of the oscillations through space and time are given by the [[wavenumber]] <math>k</math> and the [[angular frequency]] <math>\omega</math> respectively. These are both related to the total energy of the particle by the expression
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| :<math>E = \hbar\omega = \frac{\hbar^2 k^2}{2m},</math>
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| which is known as the [[dispersion relation]] for a free particle.<ref name = "Davies4" />
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| [[File:particle in a box wavefunctions 2.svg|thumb|right|upright|Initial wavefunctions for the first four states in a one-dimensional particle in a box]]
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| The size (or [[amplitude]]) of the wavefunction at a given position is related to the probability of finding a particle there by <math>P(x,t) = |\psi(x,t)|^2</math>. The wavefunction must therefore vanish everywhere beyond the edges of the box.<ref name="Davies4" /><ref name="Bransden157" /> Also, the amplitude of the wavefunction may not "jump" abruptly from one point to the next.<ref name="Davies4" /> These two conditions are only satisfied by wavefunctions with the form
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| :<math>\psi_n(x,t) =
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| \begin{cases}
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| A \sin(k_n x)\mathrm{e}^{-\mathrm{i}\omega_n t}, & 0 < x < L,\\
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| 0, & \text{otherwise,}
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| \end{cases}
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| </math>
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| where <math>n</math> is a positive integer. The wavenumber is restricted to certain, specific values given by<ref name = "Davies5">Davies p. 5</ref>
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| :<math>k_n = \frac{n \pi}{L}, \quad \mathrm{where} \quad n = \{1,2,3,4,\ldots\},</math>
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| where <math>L</math> is the size of the box.{{#tag:ref|The simplest solutions, <math>k_n=0</math> or <math>A=0</math> both yield the trivial wavefunction <math>\psi(x)=0</math>, which describes a particle that does not exist anywhere in the system.<ref name="Bransden158">Bransden and Joachain, p.158</ref>}} Negative values of <math>n</math> are neglected, since they give wavefunctions identical to the positive <math>n</math> solutions except for a physically unimportant sign change.<ref name="Bransden158" />
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| Finally, the unknown constant <math>A</math> may be found by [[Normalizable wavefunction|normalizing the wavefunction]] so that the total probability density of finding the particle in the system is 1. It follows that
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| :<math>\left| A \right| = \sqrt{\frac{2 }{L}}.</math>
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| 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.
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| The above solution is for the specific case of a box situated between <math>0</math> and <math>L</math>. It's expected that the ''eigenvalues'', i.e., the energy <math>E_n</math> of the box should be the same regardless of its position in space, but <math>\psi_n(x,t)</math> changes. This is represented by a more general case of <math>\psi_n(x,t)</math>:
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| :<math>\psi_n(x,t) =
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| \begin{cases}
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| \sqrt{\frac{2 }{L}} \sin(k_n x - \frac{n \pi x_0}{L})\mathrm{e}^{-\mathrm{i}\omega_n t}, & x_0 < x < x_0 + L,\\
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| 0, & \text{otherwise,}
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| \end{cases}
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| </math>
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| Where <math>x_0</math> is the initial position. Notice that <math>\frac{n \pi x_0}{L}</math> represents a phase shift in the wave function, and <math>\psi_n(x,t)</math> simplifies to the above case when <math>x_0 = 0</math>. Also, the phase shift has no effect when solving the Schrödinger equation, thus not affecting the ''eigenvalue''.
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| The momentum wavefunction is proportional to the [[Fourier transform]] of the position wavefunction. With <math>k=p/\hbar</math> and <math>\omega_n=\frac{\pi h n^2}{4 L^2 m}</math>,
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| :<math>\phi_n(p,t)=\frac{1}{\sqrt{2\pi\hbar}}\int_{-\infty}^\infty \psi_n(x,t)e^{-ikx}\,dx = | |
| \sqrt{\frac{\pi L}{\hbar}}\,\,\frac{n\left(1-(-1)^ne^{-ikL}\right) e^{-i \omega_n t}}{\pi ^2 n^2-k^2 L^2}</math>
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| === Position and momentum ===
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| 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 <math>P(x) = |\psi(x)|^2.</math> 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
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| :<math>P_n(x) =
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| \begin{cases}
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| \frac{2 }{L}\sin^2\left(\frac{n\pi x}{L}\right); & 0 < x < L \\
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| 0; & \text{otherwise}.
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| \end{cases}
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| </math>
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| Thus, for any value of ''n'' greater than one, there are regions within the box for which <math>P(x)=0</math>, indicating that ''spatial nodes'' exist at which the particle cannot be found.
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| In quantum mechanics, the average, or [[expectation value]] of the position of a particle is given by
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| :<math>\langle x \rangle = \int_{-\infty}^{\infty} x P_n(x)\,\mathrm{d}x.</math>
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| For the steady state particle in a box, it can be shown that the average position is always <math>\langle x \rangle = L/2</math>, 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 <math>cos(\omega t)</math>.
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| The variance in the position is a measure of the uncertainty in position of the particle: | |
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| :<math>\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)</math>
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| The probability density for finding a particle with a given momentum is derived from the wavefunction as <math>P(x) = |\phi(x)|^2.</math>. As with position, the probability density for finding the particle at a given momentum depends upon its state, and is given by
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| :<math>P_n(p)=\frac{2 \pi L}{\hbar}\, \frac{n^2 \left(1-(-1)^n \cos (k L)\right)}{\left(k^2 L^2-\pi ^2 n^2\right)^2}</math>
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| where, again, <math>k=p/\hbar</math>. The expectation value for the momentum is then calculated to be zero, and the variance in the momentum is calculated to be:
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| :<math>\mathrm{Var}(p)=\left(\frac{\hbar n\pi}{L}\right)^2</math>
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| The uncertainties in position and momentum (<math>\Delta x</math> and <math>\Delta p</math>) are defined as being equal to the square root of their respective variances, so that:
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| :<math>\Delta x \Delta p = \frac{\hbar}{2} \sqrt{\frac{n^2\pi^2}{3}-2}</math>
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| 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 <math>\hbar</math> which obeys the [[Heisenberg uncertainty principle]], which states that the product will be greater than or equal to <math>\hbar/2</math>
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| === Energy levels ===
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| [[File:Confined particle dispersion - positive.svg|thumb|upright|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.]]
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| The energies which correspond with each of the permitted wavenumbers may be written as<ref name="Davies5" />
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| :<math>E_n = \frac{n^2\hbar^2 \pi ^2}{2mL^2} = \frac{n^2 h^2}{8mL^2}</math>.
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| The energy levels increase with <math>n^2</math>, 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<ref name="Bransden159">Bransden and Joachain, p. 159</ref>
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| :<math>E_1 = \frac{\hbar^2\pi^2}{2mL^2}.</math>
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| The particle, therefore, always has a positive energy. This contrasts with classical systems, where the particle can have zero energy by resting motionless at the bottom of the box. 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
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| :<math>\Delta x\Delta p \geq \frac{\hbar}{2}</math>
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| It can be shown that the uncertainty in the position of the particle is proportional to the width of the box.<ref name="Davies15">Davies, p. 15</ref> Thus, the uncertainty in momentum is roughly inversely proportional to the width of the box.<ref name="Bransden159" /> The kinetic energy of a particle is given by <math>E=p^2/(2m)</math>, 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.<ref name="Bransden159" />
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| ==Higher-dimensional boxes==
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| [[Image:Particle2D.svg|thumb|320px|right|The wavefunction of a 2D well with n<sub>x</sub>=4 and n<sub>y</sub>=4]]
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| If a particle is trapped in a two-dimensional box, it may freely move in the <math>x</math> and <math>y</math>-directions, between barriers separated by lengths <math>L_x</math> and <math>L_y</math> respectively. Using a similar approach to that of the one-dimensional box, it can be shown that the wavefunctions and energies are given respectively by
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| :<math>\psi_{n_x,n_y} = \sqrt{\frac{4}{L_x L_y}} \sin \left( k_{n_x} x \right) \sin \left( k_{n_y} y\right)</math>,
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| :<math>E_{n_x,n_y} = \frac{\hbar^2 k_{n_x,n_y}^2}{2m}</math>,
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| where the two-dimensional [[wavevector]] is given by
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| :<math>\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}}</math>.
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| For a three dimensional box, the solutions are
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| :<math>\psi_{n_x,n_y,n_z} = \sqrt{\frac{8}{L_x L_y L_z}} \sin \left( k_{n_x} x \right) \sin \left( k_{n_y} y \right) \sin \left( k_{n_z} z \right)</math>,
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| :<math>E_{n_x,n_y,n_z} = \frac{\hbar^2 k_{n_x,n_y,n_z}^2}{2m}</math>,
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| where the three-dimensional wavevector is given by
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| :<math>\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}}</math>.
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| In general for an n-dimensional box, the solutions are
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| :<math> \psi = \sqrt{\frac{2^n}{\prod_{i} L_i}} \prod_{i}\sin(k_i i)</math>
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| An interesting feature of the above solutions is that when two or more of the lengths are the same (e.g. <math>L_x = L_y</math>), there are multiple wavefunctions corresponding to the same total energy. For example the wavefunction with <math>n_x = 2, n_y = 1</math> has the same energy as the wavefunction with <math>n_x = 1, n_y = 2</math>. This situation is called ''[[Degenerate energy level|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.
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| == Applications ==
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| {{Expand section|date=January 2010}}
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| 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.
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| == Relativistic Effects ==
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| {{Expand section|date=August 2013}}
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| The probability density does not go to zero at the nodes if relativistic effects are taken into account.<ref>{{cite journal|doi=10.1088/0143-0807/17/1/004 |title=Relativistic particle in a box |year=1996 |last1=Alberto |first1=P |last2=Fiolhais |first2=C |last3=Gil |first3=V M S |url=https://estudogeral.sib.uc.pt/bitstream/10316/12349/1/Relativistic%20particle%20in%20a%20box.pdf |format=PDF |journal=European Journal of Physics |volume=17 |pages=19–24 |bibcode=1996EJPh...17...19A}}</ref>
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| ==See also==
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| * [[Finite potential well]]
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| * [[Delta function potential]]
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| * [[Gas in a box]]
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| * [[Particle in a ring]]
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| * [[Particle in a spherically symmetric potential]]
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| * [[Quantum harmonic oscillator]]
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| * [[Delta potential well (QM)]]
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| * [[Semicircle potential well]]
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| * [[Configuration integral]] (statistical mechanics)
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| == References ==
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| {{reflist|2}}
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| == Bibliography ==
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| *{{cite book | last1=Bransden | first1=B. H. | last2 = Joachain | first2 = C. J. | title = Quantum mechanics | edition = 2nd | publisher = Pearson Education|location=Essex| year = 2000|isbn=0-582-35691-1}}
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| *{{cite book | last=Davies|first= John H.|title=The Physics of Low-Dimensional Semiconductors: An Introduction | publisher=Cambridge University Press|year=2006|isbn=0-521-48491-X|edition=6th reprint}}
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| *{{cite book | last=Griffiths|first= David J.|title=Introduction to Quantum Mechanics |edition=2nd | publisher=Prentice Hall |year=2004 |isbn=0-13-111892-7}}
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| ==External links==
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| * [http://scienceworld.wolfram.com/physics/InfiniteSquarePotentialWell.html Scienceworld] (Infinite Potential Well)
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| * [http://www.falstad.com/qm1d/ 1-D quantum mechanics java applet] simulates particle in a box, as well as other 1-dimensional cases.
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| * [http://www.falstad.com/qm2dbox/ 2-D particle in a box applet]
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| {{DEFAULTSORT:Particle In A Box}}
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| [[Category:Quantum models]]
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