Infinite Square Well

The infinite square well is a fundamental one-dimensional model in quantum mechanics that describes a particle confined to a potential energy well with infinitely high walls. This simple model is used to introduce and study basic quantum mechanical concepts such as energy quantization and wavefunctions. The infinite square well provides a valuable pedagogical tool for understanding the behavior of particles in more complex potential wells.

For the 3D case, view the page on Particle in a Box.

Table of Contents

Potential and System Description

The infinite square well potential is defined as:

$V(x) = \begin{cases}0, & 0 < x < L \\\infty , & \text{otherwise}\end{cases}$

where $L$ is the width of the well. A particle of mass $m$ is confined to this potential well and can only exist in the region between $0$ and $L$ .

Time-Independent Schrödinger Equation

The time-independent Schrödinger equation for the infinite square well potential is given by:

$-\dfrac{\hbar^2}{2m} \dfrac{d^2\psi(x)}{dx^2} = E\psi(x)$

where

$\hbar$ is the reduced Planck constant,
$m$ is the mass of the particle,
$E$ is the energy of the particle,
$\psi(x)$ is the particle’s wavefunction.

Wavefunction and Energy Quantization

Solving the time-independent Schrödinger equation for the infinite square well yields the following wavefunction and energy quantization:

$\psi_n(x) = \sqrt{\dfrac{2}{L}}\sin\left(\dfrac{n\pi x}{L}\right)$

$E_n = \dfrac{n^2 \pi^2 \hbar^2}{2mL^2}$

where $n$ is a positive integer called the quantum number.

The wavefunctions represent the stationary states of the particle, and the quantized energy levels demonstrate that the particle can only have discrete energy values.

Probability Distribution and Expectation Values

The probability distribution for a particle in an infinite square well is given by the square of the wavefunction:

$P_n(x) = \left|\psi_n(x)\right|^2 = \dfrac{2}{L}\sin^2\left(\dfrac{n\pi x}{L}\right)$

This distribution describes the likelihood of finding the particle at a particular location within the well. Expectation values for position, momentum, and other observables can be calculated using the wavefunctions.

Applications and Extensions

The infinite square well model is often extended to include finite square wells, periodic potentials, and other more realistic scenarios. These extensions allow for the study of more complex quantum systems, such as quantum wells in semiconductor devices, and provide a foundation for understanding the behavior of particles in more complicated potential landscapes.

Summary

The infinite square well is a fundamental model in quantum mechanics that describes a particle confined to a potential well with infinitely high walls. It is used to introduce and study basic quantum mechanical concepts, such as energy quantization and wavefunctions. The model provides valuable insights into the behavior of particles in more complex potential wells and serves as a foundation for understanding more advanced quantum systems.

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Potential and System Description

Time-Independent Schrödinger Equation

Wavefunction and Energy Quantization

Probability Distribution and Expectation Values

Applications and Extensions

Summary

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Infinite Square Well

Potential and System Description

Time-Independent Schrödinger Equation

Wavefunction and Energy Quantization

Probability Distribution and Expectation Values

Applications and Extensions

Summary

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