thumb|200px|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 wave function. 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 the movement of a free particle 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 that 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

thumb|right|The barriers outside a one-dimensional box have infinitely large potential, while the interior of the box has a constant, zero potential. Shown is the shifted well, with <math display="inline">x_c = L/2</math>

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

<math display="block">V(x) = \begin{cases}

0, & x_c-\tfrac{L}{2} < x <x_c+\tfrac{L}{2},\\

\infty, & \text{otherwise,}

\end{cases},</math>

where L is the length of the box, x<sub>c</sub> 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<sub>c</sub> = 0) and the shifted box (x<sub>c</sub> = L/2) (pictured).

Position wave function

In quantum mechanics, the wave function 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 wave function.

The wave function <math>\psi(x,t)</math> can be found by solving the Schrödinger equation for the system

<math display="block">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>

where <math>\hbar</math> is the reduced Planck constant, <math>m</math> is the mass of the particle, <math>i</math> is the imaginary unit and <math>t</math> is time.

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

where <math>A</math> and <math>B</math> are arbitrary complex numbers. The frequency of the oscillations through space and time is given by the wave number <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

<math display="block">E = \hbar\omega = \frac{\hbar^2 k^2}{2m},</math>

which is known as the dispersion relation for a free particle. Here one sees that only a discrete set of energy values and wave numbers k are allowed for the particle. Usually in quantum mechanics it is also demanded that the derivative of the wave function in addition to the wave function 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 wave function is not a differentiable function at the boundary of the box, and thus it can be said that the wave function does not solve the Schrödinger equation at the boundary points <math> x = 0 </math> and <math> x = L </math> (but does solve it everywhere else).

Finally, the unknown constant <math>A</math> may be found by normalizing the wave function. That is, it follows from

<math display="block">\int_0^L \left\vert \psi(x) \right\vert^2 dx = 1,</math>

that any complex number <math>A</math> whose absolute value is

<math display="block">\left| A \right| = \sqrt{\frac{2 }{L,</math>

yields the same normalized state.

It is 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. Notice that <math>x_c - \tfrac{L}{2}</math> 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 spatial part of the wave function succinctly as:

<math display="block">\psi_n (x) = \begin{cases}

\sqrt{\frac{2}{L \sin(k_nx) \quad{} \text{for } n \text{ even} \\

\sqrt{\frac{2}{L \cos(k_nx) \quad{} \text{for } n \text{ odd}.

\end{cases}</math>

Momentum wave function

The momentum wave function is proportional to the Fourier transform of the position wave function. With <math>k = p / \hbar</math> (note that the parameter describing the momentum wave function below is not exactly the special above, linked to the energy eigenvalues), the momentum wave function is given by

<math display="block">\begin{align}

\phi_n(p,t)

&= \frac{1}{\sqrt{2\pi\hbar\int_{-\infty}^\infty \psi_n(x,t)e^{-ikx}\,dx \\[1ex]

&= \sqrt{\frac{L}{\pi \hbar \left(\frac{n\pi}{n\pi+k L}\right)\,\operatorname{sinc}\left(\tfrac{1}{2}(n\pi-k L)\right)e^{-i k x_c}e^{i (n-1) \tfrac{\pi}{2e^{-i\omega_n t} ,

\end{align}

</math>

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 wave number , the momentum can, when measured, also attain other values beyond <math> p = \pm \hbar k_n </math>.

Hence, it also appears that, since the energy is <math display="inline"> E_n = \frac{\hbar^2 k_n^2}{2m} </math> for the nth eigenstate, the relation <math display="inline"> E = \frac{p^2}{2m} </math> does not strictly hold for the measured momentum ; the energy eigenstate <math> \psi_n </math> 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 wave function 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

<math display="block">P_n(x,t) = \begin{cases}

\frac{2}{L} \sin^2\left(k_n \left(x-x_c+\tfrac{L}{2}\right)\right), & x_c-\frac{L}{2} < x < x_c+\frac{L}{2},\\

0, & \text{otherwise,}

\end{cases}</math>

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. If relativistic wave equations are considered, however, the probability density does not go to zero at the nodes (apart from the trivial case <math>n=0</math>).

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

<math display="block">\langle x \rangle = \int_{-\infty}^{\infty} x P_n(x)\,\mathrm{d}x.</math>

For the steady state particle in a box, it can be shown that the average position is always <math>\langle x \rangle =x_c</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>.

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

<math display="block">\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>

The probability density for finding a particle with a given momentum is derived from the wave function 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

<math display="block">P_n(p)=\frac{L}{\pi \hbar} \left(\frac{n\pi}{n\pi+k L}\right)^2\,\textrm{sinc}^2\left(\tfrac{1}{2}(n\pi-k L)\right)</math>

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:

<math display="block">\mathrm{Var}(p)=\left(\frac{\hbar n\pi}{L}\right)^2</math>

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:

<math display="block">\Delta x \Delta p = \frac{\hbar}{2} \sqrt{\frac{n^2\pi^2}{3}-2}</math>

This product increases with increasing n, having a minimum 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>.

Another measure of uncertainty in position is the information entropy of the probability distribution H<sub>x</sub>:

<math display="block">H_x=\int_{-\infty}^\infty P_n(x) \log(P_n(x) x_0)\,dx =\log\left(\frac{2 L}{e \,x_0}\right)</math>

where x<sub>0</sub> is an arbitrary reference length.

Another measure of uncertainty in momentum is the information entropy of the probability distribution H<sub>p</sub>:

<math display="block">H_p(n)=\int_{-\infty}^\infty P_n(p) \log(P_n(p) p_0)\,dp</math>

<math display="block">\lim_{n\to\infty} H_p(n) = \log\left(\frac{4 \pi \hbar\, e^{2(1-\gamma){ L\, p_0}\right)</math>

where γ is Euler's constant. The quantum mechanical entropic uncertainty principle states that for <math>x_0\,p_0 = \hbar</math>

<math display="block">H_x+H_p(n) \ge \log(e\,\pi) \approx 2.14473...</math> (nats)

For <math>x_0\,p_0=\hbar</math>, the sum of the position and momentum entropies yields:

<math display="block">H_x+H_p(\infty) = \log\left(8\pi\, e^{1-2\gamma}\right) \approx 3.06974...</math>

where the unit is nat, and which satisfies the quantum entropic uncertainty principle.

Energy levels

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.

The energies that correspond with each of the permitted wave numbers may be written as

<math display="block">E_n = \frac{n^2\hbar^2 \pi ^2}{2mL^2} = \frac{n^2 h^2}{8mL^2}.</math>

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

<math display="block">E_1 = \frac{\hbar^2\pi^2}{2mL^2} = \frac{h^2}{8mL^2}.</math>

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

<math display="block">\Delta x\Delta p \geq \frac{\hbar}{2}</math>

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 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<sub>f</sub> and n<sub>i</sub> is:

<math display="block">\Delta E = \frac{\left(n_\text{f}^2 - n_\text{i}^2\right) h^2}{8mL^2}</math>

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:

<math display="block">\lambda = \frac{hc}{\Delta E}</math>

A common example of this phenomenon is in β-carotene. β-carotene (C<sub>40</sub>H<sub>56</sub>) is a conjugated polyene with an orange color and a molecular length of approximately 3.8&nbsp;nm (though its chain length is only approximately 2.4&nbsp;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;):

<math display="block">\Delta E = \frac{\left(n_\text{f}^2 - n_\text{i}^2\right) h^2}{8 m L^2}= \frac{\left(12^2 - 11^2\right) h^2}{8 m L^2}= 2.3658\times10^{-19} \text{ J}</math>

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

<math display="block">\lambda = \frac{ hc }{ \Delta E }= {0.000\,000\,84} \text{ m} = 840 \text{ nm}</math>

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&nbsp;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&nbsp;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 <math>(1/m) d\phi/dz</math>, which is consistent with experiment. The solution to the finite well particle in a box must be solved numerically, resulting in wave functions that are sine functions inside the quantum well and exponentially decaying functions in the barriers. 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. The laser is powered by a single electron that passes through two quantum dots; a double quantum dot. The electron moves from a state of higher energy, to a state of lower energy whilst emitting photons in the microwave region. These photons bounce off mirrors to create a beam of light; the laser. 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.

Different semiconducting materials are used to synthesize quantum dots of different sizes and therefore emit different wavelengths of light.

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.