NationMaster - Encyclopedia: Particle in a box

In physics, the particle in a box (also known as the infinite potential well or the infinite square well) is a very simple problem consisting of a single particle bouncing around inside of an immovable box, from which it cannot escape, and which loses no energy when it collides with the walls of the box. 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. When it reflects from a wall, it always reflects at an equal but opposite angle to its angle of approach, and its speed does not change.

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. Nevertheless, it remains a very simple and solvable problem. This article will only be concerned with the quantum mechanical solution. A simple introduction to this subject is provided in Basics of quantum mechanics. ...

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. 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. Potential energy is stored energy. ... In physics, the SchrÃ¶dinger equation, proposed by the Austrian physicist Erwin SchrÃ¶dinger in 1925, describes the time-dependence of quantum mechanical systems. ...

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. In quantum mechanics, when the Schrödinger equation is applied to the proposed system, the results are not intuitive. In the first place, the particle can only have certain specific energy levels, and the zero energy level is not one of them. Secondly, the chances of detecting the particle in the box at any specific energy level is not uniform - there are certain locations in the box where the particle might be found, but there are also places where it can never be found. Both of these results differ from the usual way we perceive the world, yet rest on principles that have been extensively experimentally verified. It has been suggested that this article or section be merged with Newtonian mechanics. ... Newtons First and Second laws, in Latin, from the original 1687 edition of the Principia Mathematica. ... In physics, the SchrÃ¶dinger equation, proposed by the Austrian physicist Erwin SchrÃ¶dinger in 1925, describes the time-dependence of quantum mechanical systems. ...

1 Formal Introduction
2 Solutions
- 2.1 The particle in a 1-dimensional box
- 2.2 The particle in a 2-dimensional or 3-dimensional rectangular box
3 Free propagation
4 References
5 See also
6 External links

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. It is the situation of a particle confined within a finite region of space (the box) by an infinite potential that exists at the walls of the box. The particle experiences no forces while inside the box, but is constrained by the walls to remain in the box. This is similar to the situation of a gas confined in a container. For simplicity we start with the 1-dimensional case, where all motion is constrained to a single dimension. Later we will extend the discussion to the 2 and 3 dimensional cases. See also the Particle in a spherically symmetric potential where the case is treated of a particle in a spherical box, or the particle in a ring which shows the case for a particle in a 1D ring. The statistical mechanics of many particles in a box is developed in the gas in a box article. A simple introduction to this subject is provided in Basics of quantum mechanics. ... Potential energy is stored energy. ... The particle in a spherically symmetric potential describes the dynamics of a particle in a central force field, i. ... In quantum mechanics, the case of a particle in a one-dimensional ring is similar to the particle in a box. ... 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 containing a large number particles which do not interact with each other except for instantaneous thermalizing collisions. ...

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. This is a particularly useful illustration because this behaviour is not "forced" on the system, it arises naturally from the initial 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, the SchrÃ¶dinger equation, proposed by the Austrian physicist Erwin SchrÃ¶dinger in 1925, describes the time-dependence of quantum mechanical systems. ... It has been suggested that this article or section be merged with Newtonian mechanics. ... This article or section does not cite its references or sources. ... Magic or sorcery are terms referring to the alleged influencing of events and physical phenomena by supernatural, mystical, or paranormal means. ...

The quantum behavior in the box includes:

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).
Zero-point energy - The lowest possible energy level of the particle, called the zero-point energy, is nonzero.
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 analytically the Schrödinger equation 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 trap spatially particles, typically electrons and holes. If one considers complex valued solutions, one describes conveniently a particle propagating freely in a constrained volume (like a solid). In physics, the zero-point energy refers to the lowest possible energy that quantum mechanical model of a physical system may posses; it is the energy of the ground state of the system. ... In physics, the zero-point energy refers to the lowest possible energy that quantum mechanical model of a physical system may posses; it is the energy of the ground state of the system. ... An area location represented by a point, or a line segment that is bound or unbound, or a plane surface bound or unbound, or a structure that can be represented by multi-plane surfaces that bounds the contained area. ... A quantum well is a potential well that confines particles in one dimension, forcing them to occupy a planar region. ... Properties The electron is a lightweight fundamental subatomic particle that carries a negative electric charge. ... In solid state physics, an electron hole (usually referred to simply as a hole) is the absence of an electron from the otherwise full valence band. ...

Solutions

The particle in a 1-dimensional box

For the 1-dimensional case in the $x$ direction, the time-independent Schrödinger equation can be written as:

$-frac{hbar^2}{2 m} frac{d^2 psi}{d x^2} + V(x) psi = E psi quad (1)$

where

$hbar = frac{h}{2 pi}$

$h ,$ is Planck's constant

$m ,$ is the mass of the particle

$psi,$ is the (complex valued) wavefunction that we want to find

$Vleft(xright),$ is a function describing the potential at each point x and

$E,$ is the energy, a real number.

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: A commemoration plaque for Max Planck on his discovery of Plancks constant, in front of Humboldt University, Berlin. ... Mass is a property of a physical object that quantifies the amount of matter it contains. ... This article discusses the concept of a wavefunction as it relates to quantum mechanics. ...

The Potential is 0 inside the box, and infinite elsewhere

$-frac{hbar^2}{2 m} frac{d^2 psi}{d x^2} = E psi quad (2)$

This is a well studied differential equation and eigenvalue problem with a general solution of: Image File history File links 1-D_Box. ... Image File history File links 1-D_Box. ... Graph of a differential equation In mathematics, a differential equation is an equation in which the derivatives of a function appear as variables. ... In linear algebra, the eigenvectors (from the German eigen meaning inherent, characteristic) of a linear operator are non-zero vectors which, when operated on by the operator, result in a scalar multiple of themselves. ...

$psi = A sin(kx) + B cos(kx)quad$

$E = frac{k^2 hbar^2}{2m} quad (3)$

Here, A and B can be any complex numbers, and k can be any real number (k must be real because E is real). Wikibooks Algebra has more about this subject: Complex numbers In mathematics, a complex number is an expression of the form where a and b are real numbers, and i is a specific imaginary number, called the imaginary unit, with the property i 2 = âˆ’1. ...

Now in order 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 equal zero at x = 0 and x = L. A handwaving 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|², must be small in these regions and decreases with increasing potential. For the case of an infinite potential, |ψ|² must infinitesimally small or 0, thus ψ must also be zero in this region. In summary,

$psi(0)=psi(L)=0 quad (4)$

The second case, to which solutions are given in section free propagation at the end of this article, does not compel the wavefunction to vanish at the boundary. This means that when the particle reaches one border of the well, it instantaneously disappears from this side to reappear on the opposite side, as if the well was some kind of torus. The value of the solutions are discussed in the appropriate section. We now resume derivation with vanishing boundary conditions. A torus. ...

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:

$psi = A sin(kx) quad (5)$

and at $x = L$ we find:

$psi = A sin(kL) = 0 quad (6)$

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 $A neq 0$ then $sin(k L) = 0$ , which is only true when:

$kL = n pi quad mathrm{where} quad n = 1,2,ldots$

$mathrm{or} quad k = frac{n pi}{L} quad (7)$

(note that $n = 0$ is ruled out because then $ψ = 0$ 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 $sin(n x))$ . Now in order to find $A$ we must undertake a process called normalising the wavefunction. We recognize that the particle must exist somewhere in space. $| ψ | 2$ 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: In quantum mechanics, wave functions which describe real particles must be normalisable: the probability of the particle to occupy any place must equal 1. ...

$1 = int_{-infty}^{infty} left| psi right|^2 , dx = left| A right|^2 int_0^L sin^2 kx , dx = left| A right|^2 frac{L}{2}$

$left| A right| = sqrt{frac{2}{L}} quad (8)$

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 modulus1) of a real number is its numerical value without regard to its sign. ...

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:

$psi_n = sqrt{frac{2}{L}} sinleft(frac{n pi x}{L}right) quad (9)$

$E_n = frac{n^2hbar^2 pi ^2}{2mL^2} = frac{n^2 h^2}{8mL^2} quad (10)$

with

$n=1,2,3,ldots$

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. 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. In quantum physics, the Heisenberg uncertainty principle states that one cannot assign, with full precision, values for certain pairs of observable variables, including the position and momentum, of a single elementary particle at the same time even in theory. ...

Also, since ψ consists of sine waves, for any value of n greater than one, there are regions within the box for which ψ and thus ψ² both equal zero, indicating that for these energy levels, nodes exist in the box where the probability of finding the particle is zero.

The particle in a 2-dimensional or 3-dimensional rectangular box

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:

$-frac{hbar^2}{2m} left( frac{partial^2psi}{partial x^2}+frac{partial^2 psi}{partial y^2} right) =Epsi quad(11)$

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. 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.: In mathematics, separation of variables is any of several methods of solving ordinary and partial differential equations. ...

$psi(x,y) = X(x) Y(y) quad (12)$

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

$-frac{hbar^2}{2m} left( Yfrac{partial^2X}{partial x^2}+Xfrac{partial^2 Y}{partial y^2} right) =E X Y quad(13)$

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

$-frac{hbar^2}{2m} left( frac{X''}{X}+frac{Y''}{Y} right) =E quad(14)$

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:

$-frac{hbar^2}{2m}frac{X''}{X} = E_x quad and quad -frac{hbar^2}{2m}frac{Y''}{Y} = E_y quad (15)$

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

$-frac{hbar^2}{2m}frac{partial^2X}{partial x^2} = E_x Xquad (16)$

$-frac{hbar^2}{2m}frac{partial^2Y}{partial y^2} = E_y Yquad (17)$

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:

$X_{n_x}=sqrt{frac{2}{L_x}} sin left( frac{n_x pi x}{L_x} right) quad (18)$

$Y_{n_y}=sqrt{frac{2}{L_y}} sin left( frac{n_y pi y}{L_y} right) quad (19)$

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

$psi_{n_x,n_y} = sqrt{frac{4}{L_x L_y}} sin left( frac{n_x pi x}{L_x} right) sin left( frac{n_y pi y}{L_y} right) quad (20)$

$E_{n_x,n_y} = frac{h^2}{8m} left[ left( frac{n_x}{L_x} right)^2 + left( frac{n_y}{L_y} right)^2 right] quad (21)$

The same separation of variables technique can be applied to the three dimensional case to give the energy eigenfunctions: In mathematics, separation of variables is any of several methods of solving ordinary and partial differential equations. ...

$psi_{n_x,n_y,n_z} = sqrt{frac{8}{L_x L_y L_z}} sin left( frac{n_x pi x}{L_x} right) sin left( frac{n_y pi y}{L_y} right) sin left( frac{n_z pi z}{L_z} right) quad (22)$

$E_{n_x,n_y,n_z} = frac{hbar^2pi^2}{2m} left[ left( frac{n_x}{L_x} right)^2 + left( frac{n_y}{L_y} right)^2 + left( frac{n_z}{L_z} right)^2 right] quad (23)$

with

$n_i=1,2,3,ldots$

An interesting feature of the above solutions is that when two or more of the lengths are the same (e.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. 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 Von-Karman boundary conditions; i.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: In physics a free particle is a particle that is never under the influence of an external force Classical Free Particle The classical free particle is characterized simply by a fixed velocity. ...

$textrm{in~1D}: psi_k(x)={1oversqrt L}e^{ikx}; k={2npiover L}; quad ninmathbb{Z}$

$textrm{in~3D}: psi_{mathbf{k}}(x)={1oversqrt{L^3}}e^{imathbf{k}cdotmathbf{r}}; k_x={2n_xpiover L}; k_y={2n_ypiover L}; k_z={2n_zpiover L}; quad n_x, n_y, n_zinmathbb{Z}$

The energy remains $hbar^2 k^2/2m$ (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 0131118927.

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.
2-D particle in a box applet

Category: Quantum models

Results from FactBites:

Schrodinger equation (214 words)
	Assume the potential U(x) in the time-independent Schrodinger equation to be zero inside a one-dimensional box of length L and infinite outside the box.
	For a particle inside the box a free particle wavefunction is appropriate, but since the probability of finding the particle outside the box is zero, the wavefunction must go to zero at the walls.
	For the particle in a box with infinite walls, the probability must be equal to one for finding it within the box.

Particle in a Box (0 words)
	In this problem a particle of mass m is constrained to move along a line of length a.
	At the ends of the line are large potential energy barriers that keep the particle within the "box." In the quantum-mechanical treatment of the problem the Schrödinger wave equation is used to calculate the allowed energies of the particle.
	, in the "Box" is indicated by the presence of a plot of

More results at FactBites »

COMMENTARY

Share your thoughts, questions and commentary here

Want to know more?
Search encyclopedia, statistics and forums:

Feeds | Contact
The Wikipedia article included on this page is licensed under the GFDL.
Images may be subject to relevant owners' copyright.
All other elements are (c) copyright NationMaster.com 2003-5. All Rights Reserved.
Usage implies agreement with terms.

Contents