# Particle in a Box

## The particle in a box model provides one of the very few problems in quantum mechanics which can be solved analytically.

#### Key Points

• The model is mainly used as a hypothetical example to illustrate the differences between classical and quantum systems.

• The particle may only occupy certain positive energy levels.

• The particle is more likely to be found at certain positions than at others.

• The particle may never be detected at certain positions, known as spatial nodes.

#### Terms

• A number, between 0 and 1, expressing the precise likelihood of an event happening.

• A mathematical function that describes the propagation of the quantum mechanical wave associated with a particle (or system of particles), related to the probability of finding the particle in a particular region of space.

#### Figures

1. ##### The Potential Well

Energy/position relationships in the particle in a box.

2. ##### Solutions to Particle in a Box

The first four solutions to the one dimensional particle in a box.

## The Particle in a Box

Also called the infinite square well problem, the "particle in a box" is actually one of the very few problems in quantum mechanics that can be solved without approximations, or analytically.

Recall the Schrödinger equation:

$-\frac { { \hbar }^{ 2 } }{ 2m } \frac { { \partial }^{ 2 }\Psi }{ { \partial x }^{ 2 } } +V(x)\Psi \quad =\quad i\hbar \frac { \partial \Psi }{ \partial t }$

The potential function, V, is time independent, while the wave function itself is time dependent.

The infinite square well is defined by a potential function in which the value V(x) is 0 for values of x between 0 and L where L is the length of the box, and infinite in all other positions Figure 1. In this case, classical and quantum physics are essentially describing the same thing: the impossibility of a particle "leaping" over an infinitely high boundary.

Separating the variables reduces the problem to one of simply solving the spatial part of the equation:

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

E represents the possible energies that can describe the system.

Inside the box, no forces act on 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:

$\psi (x,t)\quad =\quad [A\sin { (kx) } +\quad B\cos { (kx)]{ e }^{ -i\omega t } }$

B and A are arbitrary complex numbers. The frequency of the oscillations through space and time are given by the wave number and the angular frequency, respectively. These are related to the energy of the particle by:

$E=\hbar \omega =\frac { { \hbar }^{ 2 }{ k }^{ 2 } }{ 2m }$

The size or amplitude of the wavefunction at any point is given by the probability of finding the particle there by: $P(x,t)={ \left\lfloor \psi (x,t) \right\rfloor }^{ 2 }$

Therefore, the wavefunction must vanish everywhere beyond the edges of the box. Also, the amplitude of the wavefunction may not "jump" abruptly from one point to the next. These two conditions are only satisfied by wave functions with the form:${ \psi }_{ n }(x,t)\quad =\quad A\sin { { (k }_{ n } } x){ e }^{ -i{ \omega }_{ n }t }$

where 0

${ k }_{ n }\quad =\quad \frac { n\pi }{ L }$  where n = {1,2,3,4...} and L is the size of the box Figure 2. Negative values are neglected, since they give wave functions identical to the positive solutions except for a physically unimportant sign change. Finally, the unknown constant may be found by normalizing the wavefunction so that the total probability density of finding the particle in the system is 1.

Normalizing, we get $\left| A \right| =\sqrt { \frac { 2 }{ L } }$

A may be any complex number with absolute value $\left| A \right| =\sqrt { \frac { 2 }{ L } }$

When treated as a probability function, the wavefunction describes the probability of finding the particle at a given point and at a given time. Four conditions, proposed by Max Born, must be met for this to be true:

1. The wavefunction must be single-valued
2. The wavefunction must be "square integrable"
3. The wave unction must be continuous everywhere
4. The first derivative of the wavefunction must be continuous.

#### Key Term Glossary

amplitude
the maximum absolute value of some quantity that varies, especially a wave
##### Appears in these related concepts:
constant
Consistently recurring over time; persistent
##### Appears in these related concepts:
density
a measure of the amount of matter contained by a given volume
##### Appears in these related concepts:
energy
a quantity that denotes the ability to do work and is measured in a unit dimensioned in mass × distance²/time² (ML²/T²) or the equivalent
##### Appears in these related concepts:
frequency
The number of occurrences of a repeating event per unit of time.
##### Appears in these related concepts:
probability
A number, between 0 and 1, expressing the precise likelihood of an event happening.
##### Appears in these related concepts:
quanta
discrete packets with energy stored inside
##### Appears in these related concepts:
quantum
the smallest possible, and therefore indivisible, unit of a given quantity or quantifiable phenomenon
##### Appears in these related concepts:
solution
A homogeneous mixture, which may be liquid, gas or solid, formed by dissolving one or more substances.
##### Appears in these related concepts:
Solution
A homogeneous mixture, which may be liquid, gas or solid, formed by dissolving one or more substances.
##### Appears in these related concepts:
system
the part of the universe being studied, arbitrarily defined to any size desired
##### Appears in these related concepts:
wave
A shape that alternatingly curves in opposite directions.
##### Appears in these related concepts:
wavefunction
A mathematical function that describes the propagation of the quantum mechanical wave associated with a particle (or system of particles), related to the probability of finding the particle in a particular region of space.