Physics:Quantum Superposition principle

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Quantum superposition principle states that if a quantum system can be in one of two or more states, then any linear combination of those states is also a valid quantum state.

Superposition of two quantum states forming a new state as a linear combination of wavefunctions.

Mathematical formulation

If ${\psi }_1$ and ${\psi }_2$ are valid wavefunctions, then any linear combination

$\psi =c_1{\psi }_1+c_2{\psi }_2$

is also a valid wavefunction, where:

• $c_1$ and $c_2$ are complex coefficients

For a normalized two-state system,

$|c_1{|}^2+|c_2{|}^2=1$.

Physical interpretation

Superposition means that a system is described by a combination of possible states rather than a single definite classical state. In standard quantum mechanics, measurement is associated with probabilistic outcomes and state reduction.

Interference effects

Superposition gives rise to interference phenomena:

• Constructive interference — amplitudes reinforce
• Destructive interference — amplitudes cancel

This is observed in wave and quantum experiments such as interference and wave-packet formation.

Basis states and Hilbert space

Quantum states form a vector space (Hilbert space):

• States can be expressed in different bases
• Superposition depends on the chosen basis
• Eigenstates form a complete set

This vector-space structure is part of the standard postulates of quantum mechanics.

Applications

Superposition is central to modern quantum technologies:

• Quantum computing
• Quantum interference devices
• Atomic and optical physics

Britannica’s overview of quantum computing explicitly describes qubits as using superposition to hold multiple possible values at once.

Table of contents (118 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. Plasma and fusion physics
14. Timeline
15. 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 (9)

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:Runge–Lenz vector
25. Physics:Quantum Approximation Methods
26. Physics:Quantum Matter Elements and Particles

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4. Atomic and spectroscopy (11)

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

27. Physics:Quantum Atomic structure and spectroscopy
28. Physics:Quantum Hydrogen atom
29. Physics:Quantum Multi-electron atoms
30. Physics:Quantum Fine structure
31. Physics:Quantum Hyperfine structure
32. Physics:Quantum Isotopic shift
33. Physics:Quantum Zeeman effect
34. Physics:Quantum Stark effect
35. Physics:Quantum Spectral lines and series
36. Physics:Quantum Selection rules
37. 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.

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

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6. Quantum dynamics and evolution (7)

46. Physics:Quantum Time evolution
47. Physics:Quantum Schrödinger equation
48. Physics:Quantum Time-dependent Schrödinger equation
49. Physics:Quantum Stationary states
50. Physics:Quantum Perturbation theory
51. Physics:Quantum Time-dependent perturbation theory
52. Physics:Quantum Adiabatic theorem

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7. Measurement and information (5)

53. Physics:Quantum Measurement theory
54. Physics:Quantum Projective measurement
55. Physics:Quantum POVM
56. Physics:Quantum Entanglement
57. Physics:Quantum Information theory

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8. Quantum information and computing (6)

58. Physics:Quantum information theory
59. Physics:Quantum Qubit
60. Physics:Quantum Entanglement
61. Physics:Quantum Gates and circuits
62. Physics:Quantum Computing Algorithms in the NISQ Era
63. 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.

64. Physics:Quantum Nonlinear King plot anomaly in calcium isotope spectroscopy
65. Physics:Quantum optics beam splitter experiments
66. Physics:Quantum Ultra fast lasers
67. Physics:Quantum Experimental quantum physics
Template:Quantum optics operators

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10. Open quantum systems (7)

68. Physics:Quantum Open systems
69. Physics:Quantum Master equation
70. Physics:Quantum Lindblad equation
71. Physics:Quantum Decoherence
72. Physics:Quantum Markovian dynamics
73. Physics:Quantum Non-Markovian dynamics
74. Physics:Quantum Trajectories

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11. Quantum field theory (18)

Structural dependency map of quantum field theory.

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

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12. Statistical mechanics and kinetic theory (9)

93. Physics:Quantum Statistical mechanics
94. Physics:Quantum Partition function
95. Physics:Quantum Distribution functions
96. Physics:Quantum Liouville equation
97. Physics:Quantum Kinetic theory
98. Physics:Quantum Boltzmann equation
99. Physics:Quantum BBGKY hierarchy
100. Physics:Quantum Transport theory
101. Physics:Quantum Relaxation and thermalization

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13. 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.

102. Physics:Quantum Plasma (fusion context)
103. Physics:Quantum Fusion reactions and Lawson criterion
104. Physics:Quantum Magnetic confinement fusion
105. Physics:Quantum Inertial confinement fusion
106. Physics:Quantum Plasma instabilities and turbulence
107. Physics:Quantum Tokamak
108. Physics:Quantum Tokamak core plasma
109. Physics:Quantum Tokamak edge physics and recycling asymmetries
110. Physics:Quantum Stellarator

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14. Timeline (2)

111. Physics:Quantum mechanics/Timeline
112. Physics:Quantum_mechanics/Timeline/Quiz/

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15. Advanced and frontier topics (6)

113. Physics:Quantum Supersymmetry
114. Physics:Quantum Black hole thermodynamics
115. Physics:Quantum Holographic principle
116. Physics:Quantum gravity
117. Physics:Quantum De Sitter invariant special relativity
118. Physics:Quantum Doubly special relativity

References

Author: Harold Foppele