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Revision as of 10:47, 22 May 2026

Book I

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Eigenstates and eigenvalues 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. 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. In quantum mechanics, observables are represented by operators acting on wavefunctions. This equation means that applying the operator does not change the form of the state, only its magnitude. Eigenstates correspond to states with definite measurement outcomes:

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.[1]

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.[2]

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[3]

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.[4]

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.[5]

See also

Table of contents (217 articles)

Index

Full contents

References

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


Source attribution: Physics:Quantum Eigenstates and eigenvalues

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