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

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.^[1] 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.^[2] 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

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) (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 wavefunction.^[3] The wavefunction ${\displaystyle \psi (x,t)}$ can be found by solving the Schrödinger equation for the system

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:^[1][4]

$${\displaystyle \psi (x,t)=\lefte^{-i\omega t},}$$

(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

which is known as the dispersion relation for a free particle.^[1] 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.

The 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.^[1][4] Also, the amplitude of the wavefunction may not "jump" abruptly from one point to the next.^[1] These two conditions are only satisfied by wavefunctions with the form

where^[5] and