In physics, the particle in a box (also known as the infinite potential well or the infinite square well) is a problem consisting a single particle inside of an infinitely deep potential well, from which it cannot escape, and which loses no energy when it collides with the walls of the box. Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. In classical mechanics, the solution to the problem is trivial: The particle moves in a straight line, always at the same speed, until it reflects from a wall.

The problem becomes very interesting when one attempts a quantum-mechanical solution, since many fundamental quantum mechanical concepts need to be introduced in order to find the solution. Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons Nevertheless, it remains a very simple and solvable problem. This article will only be concerned with the quantum mechanical solution.

The problem may be expressed in any number of dimensions, but the simplest problem is one dimensional, while the most useful solution is the particle in the three dimensional box. In one dimension this amounts to the particle existing on a line segment, with the "walls" being the endpoints of the segment.

In physical terms, the particle in a box is defined as a single point particle, enclosed in a box inside of which it experiences no force whatsoever, i. e. it is at zero potential energy. Potential energy can be thought of as Energy stored within a physical system At the walls of the box, the potential rises to infinity, forming an impenetrable wall. Using this description in terms of potentials allows the Schrödinger equation to be used to determine the solution. In Physics, especially Quantum mechanics, the Schrödinger equation is an equation that describes how the Quantum state of a Physical system

As mentioned above, if we were studying this system under the rules of classical mechanics we would apply Newton's laws of motion to the initial conditions and the result would seem reasonable and intuitive. Classical mechanics is used for describing the motion of Macroscopic objects from Projectiles to parts of Machinery, as well as Astronomical objects Newton's laws of motion are three Physical laws which provide relationships between the Forces acting on a body and the motion of the In quantum mechanics, when the Schrödinger equation is applied to the proposed system, the results are not intuitive. The particle's confinement leads to a quantization of its allowed energy eigenvalues, where a vanishing energy eigenvalue is not allowed. In Mathematics, given a Linear transformation, an of that linear transformation is a nonzero vector which when that transformation is applied to it changes In other words, the least energy state of the system has a non-zero energy. In Physics, the zero-point energy is the lowest possible Energy that a Quantum mechanical Physical system may possess and is the energy of the Furthermore, the probability of measuring particle in the box at a given energy level is non-uniform and probabilistic -- upon measurement, certain locations in the "box" where the particle may be found have a higher probability than others of being the location where the particle is actually found when a measurement is made. There are also places within the box where the particle cannot ever be found. Both of these results differ from the classical description of classical mechanics and everyday experience, but yet the results rest on principles that have been extensively experimentally verified. Mechanics ( Greek) is the branch of Physics concerned with the behaviour of physical bodies when subjected to Forces or displacements

Formal introduction

The particle in a box (or the infinite potential well or infinite square well) is a simple idealized system that can be completely solved within quantum mechanics. Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons The infinite potential well is a finite sized region in space (the box) with an infinite potential at its boundaries (the walls). Potential energy can be thought of as Energy stored within a physical system A particle experiences no forces while inside the box, but as the walls are 'infinitely high', it is constrained to remain in the box. This is similar to the situation of a gas confined in a non-porous container. For simplicity we start with the 1-dimensional case, where the motion of the particle is considered only in a single dimension - the direction of the quantum confinement. Later we extend the discussion to the 2- and 3-dimensional cases.

As we shall see, the solution of the Schrödinger equation for the particle in a box problem reveals some decidedly quantum behavior of the particle that agrees with observation but contrasts sharply with the predictions of classical mechanics. In Physics, especially Quantum mechanics, the Schrödinger equation is an equation that describes how the Quantum state of a Physical system Classical mechanics is used for describing the motion of Macroscopic objects from Projectiles to parts of Machinery, as well as Astronomical objects This is a particularly useful illustration because this behaviour is not "forced" on the system, it arises naturally from the boundary conditions. It neatly demonstrates that quantum behaviour is a natural outcome of any wave-like system, contrary to the common concept of a "quantum leap" where the behavior is almost magical. In Physics, a quantum leap or quantum jump is a change of an Electron from one energy state to another within an Atom. Magic, sometimes known as sorcery, is a Conceptual system that asserts human ability to control the natural world (including events objects people and

The quantum behavior in the box includes:

1. Energy quantization - It is not possible for the particle to have any arbitrary definite energy. Instead only discrete definite energy levels are allowed (if the state is not a steady state, however, any energy past zero-point energy is allowed on average). In Physics, the zero-point energy is the lowest possible Energy that a Quantum mechanical Physical system may possess and is the energy of the
2. Zero-point energy - The lowest possible energy level of the particle, called the zero-point energy, is nonzero.
3. Spatial nodes - In contrast to classical mechanics the Schrödinger equation predicts that for some energy levels there are nodes, implying positions at which the particle can never be found.

One can solve the Schrödinger equation analytically for such a simple potential. However trivial, this case is both of great technical value for the insights it allows, and of paramount physical importance. Depending on the boundary conditions, one can use the solutions to describe two important systems. If one considers real valued solutions (of which detailed derivation is given below), one describes actual potentials of heterostructures called quantum wells which spatially confine particles - typically electrons and holes. A heterojunction is the interface that occurs between two layers or regions of dissimilar Crystalline Semiconductors These semiconducting materials have unequal A quantum well is a Potential well that confines particles which were originally free to move in three dimensions to two dimensions forcing them to occupy a planar region The electron is a fundamental Subatomic particle that was identified and assigned the negative charge in 1897 by J An electron hole is the conceptual and mathematical Opposite of an Electron, useful in the study of Physics and Chemistry. If one considers complex valued solutions, one describes conveniently a particle propagating freely in a constrained volume (like a solid).

1-dimensional box

Solution

The eigenfunction and eigenenergies of a particle of mass m in a one dimensional box of length L are

Derivation

The problem is defined as the case of a particle situated in a 1D infinite square well with momentum only in the direction of quantum confinement (the x direction). In Classical mechanics, momentum ( pl momenta SI unit kg · m/s, or equivalently N · s) is the product

The 1D time-independent Schrödinger equation can be written as:

where

is the Reduced Planck Constant

is the mass of the particle

is the complex-valued stationary time-independent wavefunction that we want to find

is the spatially varying potential and

is the energy, a real number. The Planck constant (denoted h\ is a Physical constant used to describe the sizes of quanta. Mass is a fundamental concept in Physics, roughly corresponding to the Intuitive idea of how much Matter there is in an object Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted A wave function or wavefunction is a mathematical tool used in Quantum mechanics to describe any physical system In Physics and other Sciences energy (from the Greek grc ἐνέργεια - Energeia, "activity operation" from grc ἐνεργός

For the case of the particle in a 1-dimensional box of length L, the potential is zero inside the box, but rises abruptly to infinity at x = 0 and x = L. Thus for the region inside the box V(x) = 0 and Equation 1 reduces to:

The potential is zero inside the box, and infinite elsewhere

This is a well studied differential equation and eigenvalue problem with a general solution of:

Here, A and B can be any complex numbers and the wavenumber, k can be any real number (k must be real because E is real). A differential equation is a mathematical Equation for an unknown function of one or several variables that relates the values of the In Mathematics, given a Linear transformation, an of that linear transformation is a nonzero vector which when that transformation is applied to it changes Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted

Now in order to find the specific solution for the problem at hand, we must specify the appropriate boundary conditions and find the values for A and B that satisfy those conditions. One usually resorts to one of the following two choices, describing two kinds of systems. The first case, with which we shall pursue our derivation, demands that ψ(x) equal zero at x = 0 and x = L. A hand-waving argument to motivate these boundary conditions is that the particle is unlikely to be found at a location with a high potential (the potential repulses the particle), thus the probability of finding the particle, |ψ(x)|², must be small in these regions and decreases with increasing potential. For the case of an infinite potential, |ψ(x)|² must be infinitesimally small or 0, thus ψ(x) must also be zero in this region. In summary,

Substituting the general solution from Equation 3 into Equation 2 and evaluating at x = 0 (ψ = 0), we find that B = 0 (since sin(0) = 0 and cos(0) = 1). It follows that the wavefunction must be of the form:

and at x = L we find:

One solution for Equation 6 is A = 0, however, this "trivial solution" would imply that ψ = 0 everywhere (i. e. the particle isn't in the box. ) and can be thrown out. If then sin(kL) = 0 only when:

Note that n = 0 is ruled out because in this case, ψ = 0 everywhere, corresponding to the case where the particle is not in the box. Negative values of n are also neglected, since they merely change the sign of sin(nx), giving the same probability when squared. Now in order to find A we must normalize the wavefunction. In Quantum mechanics, Wave functions which describe real particles must be normalisable: the probability of the particle to occupy any place must We recognize that the particle must exist somewhere in space. | ψ(x) | ² is the probability of finding the particle at a particular point in space, so the integral of this value over all x must be equal to 1:

Probability density functions of finding a quantum particle in an infinite one dimensional potential well of width L for some energy levels m.

or

Thus, A may be any complex number with absolute value √(2/L); these different values of A yield the same physical state, so we choose A = √(2/L) to simplify. In Mathematics, the absolute value (or modulus) of a Real number is its numerical value without regard to its sign.

Energy Levels (dashed lines) and Wavefunctions (solid lines) of the one-dimensional particle in a box.

Finally, substituting the results from Equations 7 and 8 into Equation 3, the complete set of energy eigenfunctions for the 1-dimensional particle in a box problem is:

with n being a positive integer.

Note, that as mentioned previously, only "quantized" energy levels are possible. Also, since n cannot be zero, the lowest energy from Equation 10 is also non-zero. This zero-point energy, as it is called, can be explained in terms of the uncertainty principle. In Quantum physics, the Heisenberg uncertainty principle states that locating a particle in a small region of space makes the Momentum of the particle uncertain Because the particle is constrained within a finite region, the variance in its position is upper-bounded. Thus due to the uncertainty principle the variance in the particle's momentum cannot be zero, so the particle must contain some amount of energy that increases as the length of the box, L, decreases.

Also, since ψ consists of sine waves, for any value of n greater than one, there are regions within the box for which ψ(x) and thus ψ(x)² both equal zero, indicating that for these energy levels, nodes exist in the box where the probability of finding the particle is zero. A distance between the particle and any point of the box or a distance between two particles in the box may be calculated using the probability metric. A probability metric is a function defining a distance between Random variables or vectors.

Heuristic derivation of eigenenergies

The formula for the energy eigenvalues may be derived heuristically by considering that the wavefunction must have nodes at the boundary of the box. A node is a point along a Standing wave where the wave has minimal Amplitude. This is, in other words, a restatement of the continuity condition of the wavefunction at the boundaries of the box, which implies that a half-integer number of wavelengths must fit into the box:

Using the de Broglie relation, we may write the momentum of the particle as

The classical formula relating energy to momentum may then be used to give the energy eigenvalues of the system. In Physics wavelength is the distance between repeating units of a propagating Wave of a given Frequency. In Physics, the de Broglie hypothesis (pronounced /brœj/ as French breuil close to "broy" is the statement that all Matter (any object has a Wave

2-dimensional box

Solution

Derivation

For the 2-dimensional case the particle is confined to a rectangular surface of length L_x in the x-direction and L_y in the y-direction. Again the potential is zero inside the "box" and infinite at the walls. For the region inside the box, where the potential is zero, the two dimensional analogue of Equation 2 applies:

In this case ψ is a function of both x and y, so ψ=ψ(x,y). In order to solve Equation 11, we use the method of separation of variables. In Mathematics, separation of variables is any of several methods for solving ordinary and partial Differential equations in which algebra allows one to re-write an First, we assume that ψ can be expressed as the product of two independent functions, the first depending only on x and the second depending only on y; i. e. :

Substituting Equation 12 into Equation 11 and evaluating the partial derivatives gives:

which upon dividing by XY and rewriting d²X/dx² as X" and d²Y/dy² as Y" becomes:

Now we note that since X"/X is independent of y, varying y can only change the Y"/Y term. However, from Equation 14 we see that changing Y"/Y without varying X"/X, would also change E, but E is a constant, so Y"/Y must also be a constant, independent of y. The same argument can be applied to show that X"/X is independent of x. Since X"/X and Y"/Y are constants, we can write:

where E_x + E_y = E. Expanding X" and Y" in terms of the derivatives and rearranging gives:

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

each of which are of the same form as the 1-dimensional Schrödinger equation (Equation 2) we solved in the previous section. Thus, adapting the results from the previous section gives:

Finally, since ψ=XY and E = E_x + E_y, we obtain the solutions:

3-dimensional box and higher

The same separation of variables technique can be applied to the three dimensional case to give the energy eigenfunctions:

with

An interesting feature of the above solutions is that when two or more of the lengths are the same (e. In Mathematics, separation of variables is any of several methods for solving ordinary and partial Differential equations in which algebra allows one to re-write an g. L_x = L_y), there are multiple wavefunctions corresponding to the same total energy. For example the wavefunction with n_x = 2, n_y = 1 has the same energy as the wavefunction with 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.

Free propagation

If the potential is zero (or constant) everywhere, one describes a free particle. In Physics, a free particle is a particle that in some sense is not bound This leads to some difficulties of normalization of the momentum or energy eigenfunctions. One way around is to constrain the particle in a finite volume V of arbitrary (large) extension, in which it is free to propagate. It is expected that in the limit of V→ ∞ we recover the free particle while allowing in the intermediate calculations the use of properly normalized states. Also, when describing for instance a particle propagating in a solid, one does not expect spatially localized states but instead completely delocalized states (within the solid), meaning that the particles propagates inside it (since it can be everywhere with the same probability, conversely to the sine solutions we encountered where the particle has favored locations). This understanding follows from the solutions of the Schrödinger equation for zero potential following from the so-called Born-von Karman boundary conditions; i. The Born-von Karman boundary condition is a set of Boundary conditions which impose the restriction that a Wave function must be periodic on a certain Bravais e. , the wavefunction assumes same values on opposite sides of the box but it is not required to be zero here. One can then check that the following solutions obey eq. 1:

The energy remains (cf. eq. 3) but interestingly, now the k are twice as before (cf. eq. 7). This is because in the previous case, n was strictly positive whereas now it can be negative or zero (the ground state). The solutions where the sine does not superpose to itself after a translation of L can not be recovered with exponentials, since in this propagating particle interpretation, the derivative is discontinuous at the border, meaning that the particle acquires infinite velocity here. This shows how the two interpretations bear intrinsically differing behaviours.

References

• Griffiths, David J. (2004). Introduction to Quantum Mechanics (2nd ed. ). Prentice Hall. ISBN 0-13-111892-7.  

See also

• finite potential well
• Delta function potential
• gas in a box
• particle in a ring
• Particle in a spherically symmetric potential
• Delta potential well (QM)
• Semicircle potential well
• Configuration integral (statistical mechanics)

External links

• Scienceworld (Infinite Potential Well)
• Scienceworld (Finite Potential Well)
• 1-D quantum mechanics java applet simulates particle in a box, as well as other 1-dimensional cases. The finite potential well (also known as the finite square well) is a simple problem from Quantum mechanics. The Delta potential well is a common theoretical problem of Quantum mechanics. In Quantum mechanics, the results of the quantum Particle in a box can be used to look at the equilibrium situation for a quantum ideal gas in a box which is a box In Quantum mechanics, the case of a particle in a one-dimensional ring is similar to the Particle in a box. In Quantum mechanics, the particle in a spherically symmetric potential describes the dynamics of a particle in a Potential which has spherical symmetry The Delta potential well is a common theoretical problem of Quantum mechanics. In Quantum mechanics, the case of a particle in a one-dimensional ring is similar to the Particle in a box.
• 2-D particle in a box applet