Physics:Quantum Eigenstates and eigenvalues

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Quantum eigenstates and eigenvalues describe the states of a quantum system that yield definite results when a physical observable is measured. Each observable is represented by an operator, whose eigenvalues correspond to measurable quantities.

Eigenstates of a quantum system correspond to definite measurement outcomes, with eigenvalues representing observable quantities such as energy.

Mathematical formulation

In quantum mechanics, observables are represented by operators acting on wavefunctions. An eigenstate $\psi$ satisfies:

$\hat{A}\psi =a\psi$

where:

• $\hat{A}$ is a linear operator
• $a$ is the eigenvalue
• $\psi$ is the eigenfunction (eigenstate)

This equation means that applying the operator does not change the form of the state, only its magnitude.

Physical interpretation

Eigenstates correspond to states with definite measurement outcomes:

• Measuring observable $A$ in eigenstate $\psi$ yields $a$ with certainty
• After measurement, the system remains in that eigenstate
• General states can be expressed as superpositions of eigenstates

This is a central postulate of quantum mechanics.

Energy eigenstates

A key example is the Hamiltonian operator $\hat{H}$, which represents the total energy:

$\hat{H}{\psi }_n=E_n{\psi }_n$

where:

• $E_n$ are discrete energy levels
• ${\psi }_n$ are stationary states

These states evolve in time as:

${\psi }_n(x,t)={\psi }_n(x)e^{-i{E}_{n}t/\hslash }$

Orthogonality and completeness

Eigenstates of a Hermitian operator have important properties:

• Orthogonality: $\int {\psi }_m^{*}{\psi }_{n}dx=0(m\ne n)$

• Completeness: Any wavefunction can be expressed as a sum of eigenstates

These properties allow expansion of arbitrary quantum states in a basis of eigenfunctions.

Applications

Eigenstates and eigenvalues are fundamental in:

• Atomic and molecular spectra
• Quantum measurements
• Quantum computing (basis states)
• Solving Schrödinger equations

They provide the link between mathematical operators and physical observables.

Table of content (70 articles)

Core pathway

1. Physics:Quantum basics
2. Physics:Quantum mechanics
3. Physics:Quantum mechanics measurements
4. Physics:Quantum Interpretations of quantum mechanics
5. Physics:Quantum Mathematical Foundations of Quantum Theory
6. Physics:Quantum Atomic structure and spectroscopy
7. Physics:Quantum Density matrix
8. Physics:Quantum Open systems
9. Physics:Quantum Statistical mechanics
10. Physics:Quantum Kinetic theory
11. Physics:Plasma physics (fusion context)
12. Physics:Tokamak physics
13. Physics:Tokamak edge physics and recycling asymmetries

Full contents

Foundations

1. Physics:Quantum basics
2. Physics:Quantum mechanics
3. Physics:Quantum mechanics measurements
4. Physics:Quantum Mathematical Foundations of Quantum Theory

Conceptual and interpretations

5. Physics:Quantum Interpretations of quantum mechanics
6. Physics:Quantum A Spooky Action at a Distance
7. Physics:Quantum A Walk Through the Universe
8. Physics:Quantum: The Secret of Cohesion: How Waves Hold Matter Together

Mathematical structure and systems

9. Physics:Quantum Density matrix
10. Physics:Quantum Exactly solvable quantum systems
11. Physics:Quantum Formulas Collection
12. Physics:Quantum A Matter Of Size
13. Physics:Quantum Symmetry in quantum mechanics
14. Physics:Quantum Angular momentum operator
15. Physics:Runge–Lenz vector
16. Physics:Quantum Approximation Methods
17. Physics:Quantum Matter Elements and Particles

Atomic and spectroscopy

18. Physics:Quantum Atomic structure and spectroscopy
19. Physics:Quantum Hydrogen atom
20. Physics:Quantum Selection rules
21. Physics:Quantum Fermi's golden rule
22. Physics:Quantum Spectral lines and series

Wavefunctions and modes

23. Physics:Quantum Wavefunction
24. Physics:Quantum Superposition principle
25. Physics:Quantum Eigenstates and eigenvalues
26. Physics:Quantum Boundary conditions and quantization
27. Physics:Quantum Standing waves and modes
28. Physics:Quantum Normal modes and field quantization
29. Physics:Number of independent spatial modes in a spherical volume
30. Physics:Quantum Density of states

Quantum information and computing

31. Physics:Quantum information theory
32. Physics:Quantum Qubit
33. Physics:Quantum Entanglement
34. Physics:Quantum Gates and circuits
35. Physics:Quantum Computing Algorithms in the NISQ Era
36. Physics:Quantum Noisy Qubits

Quantum optics and experiments

37. Physics:Quantum Nonlinear King plot anomaly in calcium isotope spectroscopy
38. Physics:Quantum optics beam splitter experiments
39. Physics:Quantum Ultra fast lasers
40. Physics:Quantum Experimental quantum physics
41. Template:Quantum optics operators

Open quantum systems

42. Physics:Quantum Open systems
43. Physics:Quantum Master equation
44. Physics:Quantum Lindblad equation
45. Physics:Quantum Decoherence
46. Physics:Quantum Markovian dynamics
47. Physics:Quantum Non-Markovian dynamics
48. Physics:Quantum Trajectories

Quantum field theory

49. Physics:Quantum field theory (QFT) basics
50. Physics:Quantum field theory (QFT) core

Statistical mechanics and kinetic theory



51. Physics:Quantum Statistical mechanics
52. Physics:Quantum Partition function
53. Physics:Quantum Distribution functions
54. Physics:Quantum Liouville equation
55. Physics:Quantum Kinetic theory
56. Physics:Quantum Boltzmann equation
57. Physics:Quantum BBGKY hierarchy
58. Physics:Quantum Transport theory
59. Physics:Quantum Relaxation and thermalization

Plasma and fusion physics

60. Physics:Plasma physics (fusion context)
61. Physics:Tokamak physics
62. Physics:Tokamak edge physics and recycling asymmetries
• Hierarchy of modern physics models showing the progression from quantum statistical mechanics to kinetic theory and plasma physics, culminating in tokamak edge transport and recycling asymmetries.

Timeline

63. Physics:Quantum mechanics/Timeline
64. Physics:Quantum_mechanics/Timeline/Quiz/

Advanced and frontier topics

65. Physics:Quantum Supersymmetry
66. Physics:Quantum Black hole thermodynamics
67. Physics:Quantum Holographic principle
68. Physics:Quantum gravity
69. Physics:Quantum De Sitter invariant special relativity
70. Physics:Quantum Doubly special relativity

References

Author: Harold Foppele