Particle in a box

The particle in a box (or infinite potential well) is one of the simplest exactly solvable models in quantum mechanics. It describes a particle confined to a finite region of space with infinitely high potential barriers at the boundaries.^[1]

Model

The potential is defined as

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

The particle is therefore restricted to the interval $0 < x < L$ .

Schrödinger equation

Inside the box, the time-independent Schrödinger equation is

$- \frac{ℏ^{2}}{2 m} \frac{d^{2} ψ}{d x^{2}} = E ψ .$

The boundary conditions require

$ψ (0) = ψ (L) = 0 .$

Solutions

The normalized solutions are

$ψ_{n} (x) = \sqrt{\frac{2}{L}} \sin (\frac{n π x}{L}), n = 1, 2, 3, \dots$

The corresponding energy levels are

$E_{n} = \frac{n^{2} π^{2} ℏ^{2}}{2 m L^{2}} .$

Physical features

The particle in a box illustrates several key quantum phenomena:

Energy quantization — only discrete energy values are allowed.
Zero-point energy — the ground state ( $n = 1$ ) has nonzero energy.
Wavefunction nodes — higher states have increasing numbers of nodes.

The probability density

$| ψ_{n} (x) |^{2}$

describes the likelihood of finding the particle at position $x$ .

Higher dimensions

The model can be extended to two or three dimensions, where the solutions become products of one-dimensional eigenfunctions and the energy spectrum depends on multiple quantum numbers.^[2]

Importance

The particle in a box serves as a fundamental example of:

boundary conditions in quantum systems,
quantization arising from confinement,
the connection between wavefunctions and measurable probabilities.

It is widely used as an introductory model and as a building block for more complex systems.

Harmonic oscillator

The quantum harmonic oscillator is one of the most important exactly solvable systems in quantum mechanics. It describes a particle subject to a restoring force proportional to its displacement, with potential

$V (x) = \frac{1}{2} m ω^{2} x^{2} .$

This model appears in many physical contexts, including molecular vibrations, lattice dynamics, and quantum field theory.^[3]

Schrödinger equation

The time-independent Schrödinger equation is

$- \frac{ℏ^{2}}{2 m} \frac{d^{2} ψ}{d x^{2}} + \frac{1}{2} m ω^{2} x^{2} ψ = E ψ .$

Its solutions can be expressed in terms of Hermite polynomials.

Energy spectrum

The allowed energy levels are

$E_{n} = ℏ ω (n + \frac{1}{2}), n = 0, 1, 2, \dots$

Unlike the particle in a box, the energy levels are equally spaced.

Ground state

The ground-state wavefunction is

$ψ_{0} (x) = {(\frac{m ω}{π ℏ})}^{1 / 4} e^{- m ω x^{2} / (2 ℏ)},$

with energy

$E_{0} = \frac{1}{2} ℏ ω .$

This nonzero energy is known as the zero-point energy.

Ladder operators

A powerful method for solving the harmonic oscillator uses creation and annihilation operators:

$\hat{a} = \frac{1}{\sqrt{2 ℏ m ω}} (m ω x + i \hat{p}), {\hat{a}}^{†} = \frac{1}{\sqrt{2 ℏ m ω}} (m ω x - i \hat{p}) .$

They satisfy the commutation relation

$[\hat{a}, {\hat{a}}^{†}] = 1 .$

The Hamiltonian can be written as

$\hat{H} = ℏ ω ({\hat{a}}^{†} \hat{a} + \frac{1}{2}) .$

These operators allow transitions between energy levels:

${\hat{a}}^{†} ψ_{n} \propto ψ_{n + 1}, \hat{a} ψ_{n} \propto ψ_{n - 1} .$

Physical significance

The harmonic oscillator is fundamental because:

many systems can be approximated as harmonic near equilibrium,
it provides a basis for quantizing fields (each mode behaves like an oscillator),
it introduces operator methods used throughout quantum theory.

It is one of the most widely used models in both quantum mechanics and quantum field theory.^[4]

Hydrogen atom

The hydrogen atom is the most important exactly solvable system in quantum mechanics involving a central potential. It describes an electron bound to a proton via the Coulomb interaction and provides the foundation for atomic physics.^[5]

Potential

The electron moves in the Coulomb potential

$V (r) = - \frac{e^{2}}{4 π ε_{0} r} .$

Because the potential depends only on the radial distance $r$ , the system has spherical symmetry.

Schrödinger equation

The time-independent Schrödinger equation in three dimensions is

$- \frac{ℏ^{2}}{2 m} \nabla^{2} ψ + V (r) ψ = E ψ .$

Using spherical coordinates, the wavefunction separates as

$ψ (r, θ, ϕ) = R_{n l} (r) Y_{l m} (θ, ϕ),$

where $Y_{l m}$ are spherical harmonics.

Energy levels

The allowed energy levels depend only on the principal quantum number $n$ :

$E_{n} = - \frac{13.6 eV}{n^{2}}, n = 1, 2, 3, \dots$

This degeneracy reflects the high symmetry of the Coulomb potential.^[6]

Quantum numbers

The solutions are labeled by three quantum numbers:

$n$ — principal quantum number
$l = 0, 1, \dots, n - 1$ — orbital angular momentum
$m = - l, \dots, l$ — magnetic quantum number

These arise from the separation of variables and the rotational symmetry of the system.

Radial solutions

The radial functions $R_{n l} (r)$ are expressed in terms of associated Laguerre polynomials. The full solutions form a complete orthonormal set in Hilbert space.

The probability density

$| ψ (r, θ, ϕ) |^{2}$

describes the spatial distribution of the electron, leading to the familiar atomic orbitals.

Angular momentum

The hydrogen atom provides a natural setting for angular momentum in quantum mechanics. The operators satisfy

${\hat{L}}^{2} Y_{l m} = ℏ^{2} l (l + 1) Y_{l m}, {\hat{L}}_{z} Y_{l m} = ℏ m Y_{l m} .$

This connects the system to the representation theory of rotations.

Importance

The hydrogen atom is fundamental because it:

provides exact analytical solutions in three dimensions,
introduces quantum numbers and orbital structure,
explains atomic spectra,
serves as a basis for more complex atoms and quantum systems.

It is one of the cornerstone models linking quantum mechanics to experimental observations in spectroscopy.^[7]

Central potentials

A central potential is a potential that depends only on the radial distance from a fixed point,

$V (𝐫) = V (r) .$

Such systems are important because they possess spherical symmetry and can be solved by separating variables in spherical coordinates.^[8]

Schrödinger equation

The time-independent Schrödinger equation is

$- \frac{ℏ^{2}}{2 m} \nabla^{2} ψ + V (r) ψ = E ψ .$

Using spherical coordinates, the wavefunction separates into radial and angular parts:

$ψ (r, θ, ϕ) = R (r) Y_{l m} (θ, ϕ),$

where $Y_{l m}$ are spherical harmonics.

Radial equation

The radial function satisfies

$- \frac{ℏ^{2}}{2 m} \frac{d^{2} u}{d r^{2}} + [V (r) + \frac{ℏ^{2} l (l + 1)}{2 m r^{2}}] u = E u,$

where $u (r) = r R (r)$ .

The term

$\frac{ℏ^{2} l (l + 1)}{2 m r^{2}}$

is called the centrifugal potential and arises from angular momentum.

Angular momentum

Central potentials conserve angular momentum. The operators satisfy

${\hat{L}}^{2} Y_{l m} = ℏ^{2} l (l + 1) Y_{l m}, {\hat{L}}_{z} Y_{l m} = ℏ m Y_{l m} .$

This symmetry simplifies the problem and leads to quantum numbers $l$ and $m$ .

Examples

Important exactly solvable central potentials include:

Coulomb potential — hydrogen atom
Isotropic harmonic oscillator — quadratic potential in three dimensions
Free particle — $V (r) = 0$

Each case yields a discrete or continuous spectrum depending on the form of $V (r)$ .

Physical significance

Central potentials are fundamental because they:

describe atomic and molecular systems,
illustrate the role of symmetry in quantum mechanics,
provide a framework for solving three-dimensional Schrödinger equations.

They unify many exactly solvable systems under a common mathematical structure.^[9]

References

↑ Griffiths, David J. (2018). Introduction to Quantum Mechanics. Pearson.
↑ Sakurai, J. J. (1994). Modern Quantum Mechanics. Addison-Wesley.
↑ Griffiths, David J. (2018). Introduction to Quantum Mechanics. Pearson.
↑ Sakurai, J. J. (1994). Modern Quantum Mechanics. Addison-Wesley.
↑ Griffiths, David J. (2018). Introduction to Quantum Mechanics. Pearson.
↑ Sakurai, J. J. (1994). Modern Quantum Mechanics. Addison-Wesley.
↑ Bransden, B. H.; Joachain, C. J. (2003). Physics of Atoms and Molecules. Pearson.
↑ Griffiths, David J. (2018). Introduction to Quantum Mechanics. Pearson.
↑ Sakurai, J. J. (1994). Modern Quantum Mechanics. Addison-Wesley.

Author: Harold Foppele

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[1] Griffiths, David J. (2018). Introduction to Quantum Mechanics. Pearson.

[2] Sakurai, J. J. (1994). Modern Quantum Mechanics. Addison-Wesley.

[3] Griffiths, David J. (2018). Introduction to Quantum Mechanics. Pearson.

[4] Sakurai, J. J. (1994). Modern Quantum Mechanics. Addison-Wesley.

[5] Griffiths, David J. (2018). Introduction to Quantum Mechanics. Pearson.

[6] Sakurai, J. J. (1994). Modern Quantum Mechanics. Addison-Wesley.

[7] Bransden, B. H.; Joachain, C. J. (2003). Physics of Atoms and Molecules. Pearson.

[8] Griffiths, David J. (2018). Introduction to Quantum Mechanics. Pearson.

[9] Sakurai, J. J. (1994). Modern Quantum Mechanics. Addison-Wesley.

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Physics:Quantum Exactly solvable quantum systems

Particle in a box

Model

Schrödinger equation

Solutions

Physical features

Higher dimensions

Importance

Harmonic oscillator

Schrödinger equation

Energy spectrum

Ground state

Ladder operators

Physical significance

Hydrogen atom

Potential

Schrödinger equation

Energy levels

Quantum numbers

Radial solutions

Angular momentum

Importance

Central potentials

Schrödinger equation

Radial equation

Angular momentum

Examples

Physical significance

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Conceptual and interpretations

Mathematical and solvable systems

Symmetry and structure

Atomic and spectroscopy

Quantum wavefunctions and modes

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