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{{Short description|Quantum Collection topic on 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.<ref>[https://www.britannica.com/science/eigenvalue Eigenvalue – Britannica]</ref>
'''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.<ref>[https://www.britannica.com/science/eigenvalue Eigenvalue – Britannica]</ref>
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[[File:Quantum_eigenstates.svg|thumb|280px|Quantum Eigenstates and eigenvalues.]]
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[[File:Quantum_eigenstates.svg|thumb|400px|Eigenstates of a quantum system correspond to definite measurement outcomes, with eigenvalues representing observable quantities such as energy.]]
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== Mathematical formulation ==
== Mathematical formulation ==

Revision as of 14:00, 17 May 2026


← Back to Wavefunctions and modes Book I

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.[1]

Quantum Eigenstates and eigenvalues.

Mathematical formulation

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

A^ψ=aψ

where:

  • A^ is a linear operator
  • a is the eigenvalue
  • ψ is the eigenfunction (eigenstate)

This equation means that applying the operator does not change the form of the state, only its magnitude.[2]

Physical interpretation

Eigenstates correspond to states with definite measurement outcomes:

  • Measuring observable A in eigenstate ψ 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.[3]

Energy eigenstates

A key example is the Hamiltonian operator H^, which represents the total energy:

H^ψn=Enψn

where:

  • En are discrete energy levels
  • ψn are stationary states

These states evolve in time as:

ψn(x,t)=ψn(x)e−iEnt/ℏ[4]

Orthogonality and completeness

Eigenstates of a Hermitian operator have important properties:

  • Orthogonality: ∫ψm*ψndx=0(m≠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.[5]

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.[6]

See also

Table of contents (217 articles)

Index

Full contents

References

  1. ↑ Eigenvalue – Britannica
  2. ↑ Eigenstates and Eigenvalues – LibreTexts
  3. ↑ Quantum Mechanics – Stanford Encyclopedia of Philosophy
  4. ↑ The Quantum Particle in a Box – OpenStax
  5. ↑ Orthogonal Functions – Wolfram MathWorld
  6. ↑ Quantum mechanics – Britannica
Author: Harold Foppele


Source attribution: Quantum Eigenstates and eigenvalues

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