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Repair Quantum Collection B backlink template

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{{Quantum book backlink|Wavefunctions and modes}}
'''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>

[[File:Quantum_eigenstates.svg|thumb|400px|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 <math>\psi</math> satisfies:

<math>\hat{A}\psi = a \psi</math>

where:
* <math>\hat{A}</math> is a linear operator
* <math>a</math> is the eigenvalue
* <math>\psi</math> is the eigenfunction (eigenstate)

This equation means that applying the operator does not change the form of the state, only its magnitude.<ref>[https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Introductory_Quantum_Mechanics_%28Fitzpatrick%29/03%3A_Fundamentals_of_Quantum_Mechanics/3.08%3A_Eigenstates_and_Eigenvalues Eigenstates and Eigenvalues – LibreTexts]</ref>

== Physical interpretation ==

Eigenstates correspond to states with definite measurement outcomes:

* Measuring observable <math>A</math> in eigenstate <math>\psi</math> yields <math>a</math> 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.<ref>[https://plato.stanford.edu/entries/qm/ Quantum Mechanics – Stanford Encyclopedia of Philosophy]</ref>

== Energy eigenstates ==

A key example is the Hamiltonian operator <math>\hat{H}</math>, which represents the total energy:

<math>\hat{H}\psi_n = E_n \psi_n</math>

where:
* <math>E_n</math> are discrete energy levels
* <math>\psi_n</math> are stationary states

These states evolve in time as:

<math>\psi_n(x,t) = \psi_n(x)e^{-iE_n t / \hbar}</math><ref>[https://openstax.org/books/university-physics-volume-3/pages/7-4-the-quantum-particle-in-a-box The Quantum Particle in a Box – OpenStax]</ref>

== Orthogonality and completeness ==

Eigenstates of a Hermitian operator have important properties:

* Orthogonality: <math>\int \psi_m^* \psi_n dx = 0 \quad (m \neq n)</math>

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

These properties allow expansion of arbitrary quantum states in a basis of eigenfunctions.<ref>[https://mathworld.wolfram.com/OrthogonalFunctions.html Orthogonal Functions – Wolfram MathWorld]</ref>

== 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.<ref>[https://www.britannica.com/science/quantum-mechanics-physics Quantum mechanics – Britannica]</ref>

=See also=
{{#invoke:PhysicsQC|tocHeadingAndList|Physics:Quantum basics/See also}}

==References==
{{reflist|3}}
{{Author|Harold Foppele}}

{{Sourceattribution|Quantum Eigenstates and eigenvalues|1}}

Physics:Quantum Eigenstates and eigenvalues - Revision history

imported>WikiHarold: Repair Quantum Collection B backlink template

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