Physics:Quantum Exactly solvable quantum systems

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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.

Exactly solvable quantum systems, including the particle in a box, harmonic oscillator, hydrogen atom, and central potentials.

Model

The potential is defined as

$V(x)=\{\begin{array}{ll}0& 0<x<L\\ \mathrm{\infty }& \text{otherwise}\end{array}$

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

Schrödinger equation

Inside the box, the time-independent Schrödinger equation is

$-\frac{{\hslash }^2}{2m}\frac{d^2\psi }{d{x}^2}=E\psi .$

The boundary conditions require

$\psi (0)=\psi (L)=0.$

Solutions

The normalized solutions are

${\psi }_n(x)=\sqrt{\frac{2}{L}}\sin \left(\frac{n\pi x}{L}\right),n=1,2,3,\dots$

The corresponding energy levels are

$E_n=\frac{n^2{\pi }^2{\hslash }^2}{2m{L}^2}.$

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

$|{\psi }_n(x){|}^2$

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.

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

$V(x)=\frac{1}{2}m{\omega }^2{x}^2.$

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

Schrödinger equation

The time-independent Schrödinger equation is

$-\frac{{\hslash }^2}{2m}\frac{d^2\psi }{d{x}^2}+\frac{1}{2}m{\omega }^2{x}^2\psi =E\psi .$

Its solutions can be expressed in terms of Hermite polynomials.

Energy spectrum

The allowed energy levels are

$E_n=\hslash \omega (n+\frac{1}{2}),n=0,1,2,\dots$

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

Ground state

The ground-state wavefunction is

${\psi }_0(x)={\left(\frac{m\omega }{\pi \hslash }\right)}^{1/4}{e}^{-m\omega x^2/(2\hslash )},$

with energy

$E_0=\frac{1}{2}\hslash \omega .$

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:

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

They satisfy the commutation relation

$[\hat{a},{\hat{a}}^{†}]=1.$

The Hamiltonian can be written as

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

These operators allow transitions between energy levels:

${\hat{a}}^{†}{\psi }_n\propto {\psi }_{n+1},\hat{a}{\psi }_n\propto {\psi }_{n-1}.$

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.

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.

Potential

The electron moves in the Coulomb potential

$V(r)=-\frac{e^2}{4\pi {\epsilon }_{0}r}.$

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

$-\frac{{\hslash }^2}{2m}{\mathrm{\nabla }}^2\psi +V(r)\psi =E\psi .$

Using spherical coordinates, the wavefunction separates as

$\psi (r,\theta ,\varphi )=R_{nl}(r)Y_{lm}(\theta ,\varphi ),$

where $Y_{lm}$ are spherical harmonics.

Energy levels

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

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

This degeneracy reflects the high symmetry of the Coulomb potential.

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

The probability density

$|\psi (r,\theta ,\varphi ){|}^2$

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

${\hat{L}}^2{Y}_{lm}={\hslash }^{2}l(l+1)Y_{lm},{\hat{L}}_z{Y}_{lm}=\hslash m{Y}_{lm}.$

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.

Central potentials

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

$V(\mathbf{𝐫})=V(r).$

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

Schrödinger equation

The time-independent Schrödinger equation is

$-\frac{{\hslash }^2}{2m}{\mathrm{\nabla }}^2\psi +V(r)\psi =E\psi .$

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

$\psi (r,\theta ,\varphi )=R(r)Y_{lm}(\theta ,\varphi ),$

where $Y_{lm}$ are spherical harmonics.

Radial equation

The radial function satisfies

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

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

The term

$\frac{{\hslash }^{2}l(l+1)}{2m{r}^2}$

is called the centrifugal potential and arises from angular momentum.

Angular momentum

Central potentials conserve angular momentum. The operators satisfy

${\hat{L}}^2{Y}_{lm}={\hslash }^{2}l(l+1)Y_{lm},{\hat{L}}_z{Y}_{lm}=\hslash m{Y}_{lm}.$

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.

Table of contents (138 articles)

Index

Core theory

1. Foundations
2. Conceptual and interpretations
3. Mathematical structure and systems
4. Atomic and spectroscopy
5. Wavefunctions and modes
6. Quantum dynamics and evolution
7. Measurement and information
8. Quantum information and computing

Applications and extensions

9. Quantum optics and experiments
10. Open quantum systems
11. Quantum field theory
12. Statistical mechanics and kinetic theory
13. Condensed matter and solid-state physics
14. Plasma and fusion physics
15. Timeline
16. Advanced and frontier topics

Full contents

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1. Foundations (7)

1. Physics:Quantum basics
2. Physics:Quantum Postulates
3. Physics:Quantum Hilbert space
4. Physics:Quantum Observables and operators
5. Physics:Quantum mechanics
6. Physics:Quantum mechanics measurements
7. Physics:Quantum Mathematical Foundations of Quantum Theory

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2. Conceptual and interpretations (10)

8. Physics:Quantum Interpretations of quantum mechanics
9. Physics:Quantum Wave–particle duality
10. Physics:Quantum Complementarity principle
11. Physics:Quantum Uncertainty principle
12. Physics:Quantum Measurement problem
13. Physics:Quantum Bell's theorem
14. Physics:Quantum Hidden variable theory
15. Physics:Quantum A Spooky Action at a Distance
16. Physics:Quantum A Walk Through the Universe
17. Physics:Quantum The Secret of Cohesion and How Waves Hold Matter Together

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3. Mathematical structure and systems (11)

18. Physics:Quantum Density matrix
19. Physics:Quantum Exactly solvable quantum systems
20. Physics:Quantum Formulas Collection
21. Physics:Quantum A Matter Of Size
22. Physics:Quantum Symmetry in quantum mechanics
23. Physics:Quantum Angular momentum operator
24. Physics:Quantum Runge–Lenz vector
25. Physics:Quantum Approximation Methods
26. Physics:Quantum Matter Elements and Particles
27. Physics:Quantum Dirac equation
28. Physics:Quantum Klein–Gordon equation

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4. Atomic and spectroscopy (11)

Quantum atomic structure and spectroscopy: orbitals, energy levels, and emission and absorption spectra.

29. Physics:Quantum Atomic structure and spectroscopy
30. Physics:Quantum Hydrogen atom
31. Physics:Quantum Multi-electron atoms
32. Physics:Quantum Fine structure
33. Physics:Quantum Hyperfine structure
34. Physics:Quantum Isotopic shift
35. Physics:Quantum Zeeman effect
36. Physics:Quantum Stark effect
37. Physics:Quantum Spectral lines and series
38. Physics:Quantum Selection rules
39. Physics:Quantum Fermi's golden rule

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5. Wavefunctions and modes (8)

A quantum wavefunction showing probability amplitude in space; the square of its magnitude gives the probability density.

40. Physics:Quantum Wavefunction
41. Physics:Quantum Superposition principle
42. Physics:Quantum Eigenstates and eigenvalues
43. Physics:Quantum Boundary conditions and quantization
44. Physics:Quantum Standing waves and modes
45. Physics:Quantum Normal modes and field quantization
46. Physics:Number of independent spatial modes in a spherical volume
47. Physics:Quantum Density of states

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6. Quantum dynamics and evolution (9)

48. Physics:Quantum Time evolution
49. Physics:Quantum Schrödinger equation
50. Physics:Quantum Time-dependent Schrödinger equation
51. Physics:Quantum Stationary states
52. Physics:Quantum Perturbation theory
53. Physics:Quantum Time-dependent perturbation theory
54. Physics:Quantum Adiabatic theorem
55. Physics:Quantum Scattering theory
56. Physics:Quantum S-matrix

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7. Measurement and information (6)

57. Physics:Quantum Measurement theory
58. Physics:Quantum Measurement operators
59. Physics:Quantum Projective measurement
60. Physics:Quantum POVM
61. Physics:Quantum Weak measurement
62. Physics:Quantum Measurement collapse

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8. Quantum information and computing (6)

63. Physics:Quantum information theory
64. Physics:Quantum Qubit
65. Physics:Quantum Entanglement
66. Physics:Quantum Gates and circuits
67. Physics:Quantum Computing Algorithms in the NISQ Era
68. Physics:Quantum Noisy Qubits

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9. Quantum optics and experiments (4)

Experimental quantum physics: qubits, dilution refrigerators, quantum communication, and laboratory systems.

69. Physics:Quantum Nonlinear King plot anomaly in calcium isotope spectroscopy
70. Physics:Quantum optics beam splitter experiments
71. Physics:Quantum Ultra fast lasers
72. Physics:Quantum Experimental quantum physics
Template:Quantum optics operators

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10. Open quantum systems (7)

73. Physics:Quantum Open systems
74. Physics:Quantum Master equation
75. Physics:Quantum Lindblad equation
76. Physics:Quantum Decoherence
77. Physics:Quantum Markovian dynamics
78. Physics:Quantum Non-Markovian dynamics
79. Physics:Quantum Trajectories

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11. Quantum field theory (18)

Structural dependency map of quantum field theory.

80. Physics:Quantum field theory (QFT) basics
81. Physics:Quantum field theory (QFT) core
82. Physics:Quantum Fields and Particles
83. Physics:Quantum Second quantization
84. Physics:Quantum Harmonic Oscillator field modes
85. Physics:Quantum Creation and annihilation operators
86. Physics:Quantum vacuum fluctuations
87. Physics:Quantum Propagators in quantum field theory
88. Physics:Quantum Feynman diagrams
89. Physics:Quantum Path integral formulation
90. Physics:Quantum Renormalization in field theory
91. Physics:Quantum Renormalization group
92. Physics:Quantum Field Theory Gauge symmetry
93. Physics:Quantum Non-Abelian gauge theory
94. Physics:Quantum Electrodynamics (QED)
95. Physics:Quantum chromodynamics (QCD)
96. Physics:Quantum Electroweak theory
97. Physics:Quantum Standard Model

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12. Statistical mechanics and kinetic theory (10)

98. Physics:Quantum Statistical mechanics
99. Physics:Quantum Partition function
100. Physics:Quantum Distribution functions
101. Physics:Quantum Liouville equation
102. Physics:Quantum Kinetic theory
103. Physics:Quantum Boltzmann equation
104. Physics:Quantum BBGKY hierarchy
105. Physics:Quantum Transport theory
106. Physics:Quantum Relaxation and thermalization
107. Physics:Quantum Thermodynamics

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13. Condensed matter and solid-state physics (7)

108. Physics:Quantum Band structure
109. Physics:Quantum Fermi surfaces
110. Physics:Quantum Semiconductor physics
111. Physics:Quantum Phonons
112. Physics:Quantum Electron-phonon interaction
113. Physics:Quantum Superconductivity
114. Physics:Quantum Topological phases of matter

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14. Plasma and fusion physics (9)

Conceptual illustration of plasma physics in a fusion context, showing magnetically confined ionized gas in a tokamak and the collective behavior governed by electromagnetic fields and transport processes.

115. Physics:Quantum Plasma (fusion context)
116. Physics:Quantum Fusion reactions and Lawson criterion
117. Physics:Quantum Magnetic confinement fusion
118. Physics:Quantum Inertial confinement fusion
119. Physics:Quantum Plasma instabilities and turbulence
120. Physics:Quantum Tokamak
121. Physics:Quantum Tokamak core plasma
122. Physics:Quantum Tokamak edge physics and recycling asymmetries
123. Physics:Quantum Stellarator

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15. Timeline (8)

124. Physics:Quantum mechanics/Timeline
125. Physics:Quantum mechanics/Timeline/Pre-quantum era
126. Physics:Quantum mechanics/Timeline/Old quantum theory
127. Physics:Quantum mechanics/Timeline/Modern quantum mechanics
128. Physics:Quantum mechanics/Timeline/Quantum field theory era
129. Physics:Quantum mechanics/Timeline/Quantum information era
130. Physics:Quantum mechanics/Timeline/Quantum technology era
131. Physics:Quantum mechanics/Timeline/Quiz/

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16. Advanced and frontier topics (6)

132. Physics:Quantum Supersymmetry
133. Physics:Quantum Black hole thermodynamics
134. Physics:Quantum Holographic principle
135. Physics:Quantum gravity
136. Physics:Quantum De Sitter invariant special relativity
137. Physics:Quantum Doubly special relativity

References

Author: Harold Foppele