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{{Quantum book backlink|Mathematical structure and systems}}
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.<ref>{{cite book |last=Griffiths |first=David J. |title=Introduction to Quantum Mechanics |publisher=Pearson |year=2018}}</ref>

'''<div style="float:right; border:1px solid #ccc; padding:5px; background:#f9f9f9; width:400px; margin:0 0 10px 10px;">
[[File:Exactly solvable quantum systems2.jpg]]
<div style="font-size:90%; margin-top:5px;">
Exactly solvable quantum systems, including the particle in a box, harmonic oscillator, hydrogen atom, and central potentials.
</div>
</div>'''
=== Model ===
The potential is defined as

<math>
V(x) =
\begin{cases}
0 & 0 < x < L \\
\infty & \text{otherwise}
\end{cases}
</math>

The particle is therefore restricted to the interval <math>0 < x < L</math>.

=== Schrödinger equation ===

Inside the box, the time-independent Schrödinger equation is

<math>
-\frac{\hbar^2}{2m} \frac{d^2 \psi}{dx^2} = E \psi.
</math>

The boundary conditions require

<math>
\psi(0) = \psi(L) = 0.
</math>

=== Solutions ===

The normalized solutions are

<math>
\psi_n(x) = \sqrt{\frac{2}{L}} \sin\left(\frac{n\pi x}{L}\right),
\quad n = 1,2,3,\dots
</math>

The corresponding energy levels are

<math>
E_n = \frac{n^2 \pi^2 \hbar^2}{2mL^2}.
</math>

=== 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 (<math>n=1</math>) has nonzero energy.
* '''Wavefunction nodes''' — higher states have increasing numbers of nodes.

The probability density

<math>
|\psi_n(x)|^2
</math>

describes the likelihood of finding the particle at position <math>x</math>.

=== 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.<ref>{{cite book |last=Sakurai |first=J. J. |title=Modern Quantum Mechanics |publisher=Addison-Wesley |year=1994}}</ref>

=== 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

<math>
V(x) = \tfrac{1}{2} m \omega^2 x^2.
</math>

This model appears in many physical contexts, including molecular vibrations, lattice dynamics, and quantum field theory.<ref>{{cite book |last=Griffiths |first=David J. |title=Introduction to Quantum Mechanics |publisher=Pearson |year=2018}}</ref>

=== Schrödinger equation ===

The time-independent Schrödinger equation is

<math>
-\frac{\hbar^2}{2m}\frac{d^2 \psi}{dx^2} + \tfrac{1}{2} m \omega^2 x^2 \psi = E \psi.
</math>

Its solutions can be expressed in terms of Hermite polynomials.

=== Energy spectrum ===

The allowed energy levels are

<math>
E_n = \hbar \omega \left(n + \tfrac{1}{2}\right),
\quad n = 0,1,2,\dots
</math>

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

=== Ground state ===

The ground-state wavefunction is

<math>
\psi_0(x) = \left(\frac{m\omega}{\pi\hbar}\right)^{1/4} e^{-m\omega x^2/(2\hbar)},
</math>

with energy

<math>
E_0 = \tfrac{1}{2}\hbar\omega.
</math>

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''':

<math>
\hat{a} = \frac{1}{\sqrt{2\hbar m\omega}} (m\omega x + i \hat{p}),
\quad
\hat{a}^\dagger = \frac{1}{\sqrt{2\hbar m\omega}} (m\omega x - i \hat{p}).
</math>

They satisfy the commutation relation

<math>
[\hat{a}, \hat{a}^\dagger] = 1.
</math>

The Hamiltonian can be written as

<math>
\hat{H} = \hbar \omega \left(\hat{a}^\dagger \hat{a} + \tfrac{1}{2}\right).
</math>

These operators allow transitions between energy levels:

<math>
\hat{a}^\dagger \psi_n \propto \psi_{n+1}, \quad \hat{a} \psi_n \propto \psi_{n-1}.
</math>

=== 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.<ref>{{cite book |last=Sakurai |first=J. J. |title=Modern Quantum Mechanics |publisher=Addison-Wesley |year=1994}}</ref>

=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.<ref>{{cite book |last=Griffiths |first=David J. |title=Introduction to Quantum Mechanics |publisher=Pearson |year=2018}}</ref>

=== Potential ===

The electron moves in the Coulomb potential

<math>
V(r) = -\frac{e^2}{4\pi \varepsilon_0 r}.
</math>

Because the potential depends only on the radial distance <math>r</math>, the system has spherical symmetry.

=== Schrödinger equation ===

The time-independent Schrödinger equation in three dimensions is

<math>
-\frac{\hbar^2}{2m} \nabla^2 \psi + V(r)\psi = E\psi.
</math>

Using spherical coordinates, the wavefunction separates as

<math>
\psi(r,\theta,\phi) = R_{nl}(r)\,Y_{lm}(\theta,\phi),
</math>

where <math>Y_{lm}</math> are [[Wikipedia:Spherical harmonics|spherical harmonics]].

=== Energy levels ===

The allowed energy levels depend only on the principal quantum number <math>n</math>:

<math>
E_n = -\frac{13.6\,\text{eV}}{n^2}, \quad n = 1,2,3,\dots
</math>

This degeneracy reflects the high symmetry of the Coulomb potential.<ref>{{cite book |last=Sakurai |first=J. J. |title=Modern Quantum Mechanics |publisher=Addison-Wesley |year=1994}}</ref>

=== Quantum numbers ===

The solutions are labeled by three quantum numbers:

* <math>n</math> — principal quantum number
* <math>l = 0,1,\dots,n-1</math> — orbital angular momentum
* <math>m = -l,\dots,l</math> — magnetic quantum number

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

=== Radial solutions ===

The radial functions <math>R_{nl}(r)</math> are expressed in terms of associated Laguerre polynomials. The full solutions form a complete orthonormal set in Hilbert space.

The probability density

<math>
|\psi(r,\theta,\phi)|^2
</math>

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

<math>
\hat{L}^2 Y_{lm} = \hbar^2 l(l+1) Y_{lm}, \quad
\hat{L}_z Y_{lm} = \hbar m Y_{lm}.
</math>

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.<ref>{{cite book |last=Bransden |first=B. H. |last2=Joachain |first2=C. J. |title=Physics of Atoms and Molecules |publisher=Pearson |year=2003}}</ref>

=Central potentials=

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

<math>
V(\mathbf{r}) = V(r).
</math>

Such systems are important because they possess spherical symmetry and can be solved by separating variables in spherical coordinates.<ref>{{cite book |last=Griffiths |first=David J. |title=Introduction to Quantum Mechanics |publisher=Pearson |year=2018}}</ref>

=== Schrödinger equation ===

The time-independent Schrödinger equation is

<math>
-\frac{\hbar^2}{2m} \nabla^2 \psi + V(r)\psi = E\psi.
</math>

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

<math>
\psi(r,\theta,\phi) = R(r)\,Y_{lm}(\theta,\phi),
</math>

where <math>Y_{lm}</math> are spherical harmonics.

=== Radial equation ===

The radial function satisfies

<math>
-\frac{\hbar^2}{2m} \frac{d^2 u}{dr^2} + \left[ V(r) + \frac{\hbar^2 l(l+1)}{2mr^2} \right] u = E u,
</math>

where <math>u(r) = r R(r)</math>.

The term

<math>
\frac{\hbar^2 l(l+1)}{2mr^2}
</math>

is called the '''centrifugal potential''' and arises from angular momentum.

=== Angular momentum ===

Central potentials conserve angular momentum. The operators satisfy

<math>
\hat{L}^2 Y_{lm} = \hbar^2 l(l+1) Y_{lm}, \quad
\hat{L}_z Y_{lm} = \hbar m Y_{lm}.
</math>

This symmetry simplifies the problem and leads to quantum numbers <math>l</math> and <math>m</math>.

=== Examples ===

Important exactly solvable central potentials include:

* '''Coulomb potential''' — hydrogen atom
* '''Isotropic harmonic oscillator''' — quadratic potential in three dimensions
* '''Free particle''' — <math>V(r)=0</math>

Each case yields a discrete or continuous spectrum depending on the form of <math>V(r)</math>.

=== 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.<ref>{{cite book |last=Sakurai |first=J. J. |title=Modern Quantum Mechanics |publisher=Addison-Wesley |year=1994}}</ref>

=See also=
{{#invoke:PhysicsQC|tocHeadingAndList|Physics:Quantum basics/See also}}

=References=
{{reflist|3}}

{{Author|Harold Foppele}}

{{Sourceattribution|Quantum Exactly solvable quantum systems|1}}

Physics:Quantum Exactly solvable quantum systems - Revision history

imported>WikiHarold: Repair Quantum Collection B backlink template

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