Particle in a Box and Potential Wells

The particle in a box and the square potential well are the simplest exactly solvable quantum systems: confining a particle forces its energy into discrete levels and its wavefunction into standing-wave patterns, illustrating quantization in its purest form.

Definition

The particle in a box is a model of a quantum particle confined to a region by an infinitely or finitely deep potential, whose stationary states are standing waves with quantized energies fixed by the boundary conditions.

Scope

The topic covers the infinite square well with its exact energy levels and sinusoidal standing waves, the finite square well with its limited number of bound states and exponential leakage into classically forbidden regions, matching conditions on the wavefunction and its derivative at boundaries, zero-point energy, and the extension to two- and three-dimensional boxes with degeneracies.

Core questions

• Why does confining a particle produce discrete energy levels?
• What boundary conditions must the wavefunction satisfy at the walls of a well?
• Why does a finite well support only a limited number of bound states?
• What is zero-point energy and why can it not be removed?

Key concepts

• infinite square well
• finite square well
• standing wave
• boundary conditions
• zero-point energy
• degeneracy

Key theories

Infinite square well

A particle confined between impenetrable walls has wavefunctions that vanish at the walls, forcing standing waves with an integer number of half-wavelengths and energies that grow as the square of that integer, the cleanest example of quantization.

Finite square well

When the walls are of finite height the wavefunction leaks exponentially into the forbidden region and the well supports only finitely many bound states determined by a transcendental matching condition, with at least one bound state always present in one dimension.

Clinical relevance

The box model is the foundation of nanoscience: quantum wells, wires, and dots in semiconductors behave like particles in engineered boxes, with their discrete levels tuning the colors of quantum-dot displays and the operation of quantum-well lasers and detectors.

History

The confined-particle model emerged immediately after Schrodinger's 1926 equation as the simplest illustration of quantization; it became central again in the late twentieth century when molecular-beam epitaxy made it possible to fabricate real semiconductor quantum wells matching the textbook idealization.

Key figures

• Erwin Schrodinger
• Arnold Sommerfeld
• Lev Landau

Seminal works

• griffiths2018
• cohentannoudji2019

Frequently asked questions

Why does the lowest energy in a box have to be greater than zero?

Confining a particle to a finite region forces its wavefunction to curve and gives it a nonzero spread in momentum by the uncertainty principle, so the kinetic energy cannot vanish; this irreducible minimum is the zero-point energy.

How does a finite well differ from an infinite well?

An infinite well has infinitely many bound states and wavefunctions that strictly vanish at the walls, whereas a finite well supports only a finite number of bound states whose wavefunctions extend a short distance into the classically forbidden region.