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{{Quantum book backlink|Quantum dynamics and evolution}}

A '''stationary state''' is a [[quantum state]] with all [[observable]]s independent of time. It is an [[eigenvector]] of the [[energy operator]] (rather than a [[quantum superposition]] of different energies). It is also called an '''energy eigenstate''', '''energy eigenfunction''', or '''energy [[Bra-ket notation|eigenket]]'''. Stationary states are fundamental in [[quantum mechanics]] and are closely related to concepts such as [[atomic orbital]]s and [[molecular orbital]]s.

[[File:Quantum_stationary_states_energy_levels_yellow.jpg|thumb|400px|Stationary states as energy eigenstates: discrete energy levels with time-independent probability densities.]]
[[File:QuantumHarmonicOscillatorAnimation.gif|thumb|400px|A [[harmonic oscillator]] in classical mechanics (A–B) and quantum mechanics (C–H). Some of the quantum solutions are stationary states, corresponding to standing waves with fixed probability distributions.]]

== Introduction ==
A stationary state is called ''stationary'' because the system remains unchanged in every observable way as time elapses. For a system with a time-independent [[Hamiltonian (quantum mechanics)|Hamiltonian]], this means that measurable quantities such as position probability, momentum, or [[Spin (physics)|spin]] remain constant in time.<ref>[[Claude Cohen-Tannoudji]], Bernard Diu, and [[Franck Laloë]]. ''Quantum Mechanics: Volume One''. Hermann, 1977. p. 32.</ref>

The [[wavefunction]] itself is not constant: it evolves by a global complex [[phase factor]], forming a [[standing wave]]. The oscillation frequency of this phase, multiplied by the [[Planck constant]], corresponds to the energy via the [[Planck–Einstein relation]].

== Definition ==
Stationary states are solutions to the time-independent [[Schrödinger equation]]:
<math display="block">\hat H |\Psi\rangle = E |\Psi\rangle</math>

where:
* <math>|\Psi\rangle</math> is the quantum state,
* <math>\hat H</math> is the Hamiltonian operator,
* <math>E</math> is the energy eigenvalue.

This is an [[Eigenvalues and eigenvectors|eigenvalue equation]]: the stationary states are eigenvectors of the Hamiltonian.

When inserted into the time-dependent Schrödinger equation, the evolution is:<ref>Quanta: A handbook of concepts, P. W. Atkins, Oxford University Press, 1974, {{ISBN|0-19-855493-1}}.</ref>
<math display="block">i\hbar \frac{\partial}{\partial t} |\Psi\rangle = E |\Psi\rangle</math>

with solution:
<math display="block">|\Psi(t)\rangle = e^{-iEt/\hbar} |\Psi(0)\rangle</math>

Thus, a stationary state evolves only by a phase factor, with angular frequency <math>\omega = E/\hbar</math>.

== Stationary state properties ==
[[File:StationaryStatesAnimation.gif|thumb|400px|Two stationary states (top) and a non-stationary superposition (bottom). Only stationary states have time-independent probability densities.]]

Although the wavefunction changes in time,
<math display="block">|\Psi(t)\rangle = e^{-iEt/\hbar}|\Psi(0)\rangle</math>

all observable quantities remain constant. For example, the probability density:
<math display="block">
|\Psi(x,t)|^2 = |\Psi(x,0)|^2
</math>

is time-independent.

In the [[Heisenberg picture]], stationary states are mathematically constant in time.

These results assume a time-independent Hamiltonian; if the system changes, the state will generally no longer be stationary.

== Superposition ==
A general [[quantum state]] need not be stationary. A [[superposition]] of stationary states with different energies leads to time-dependent interference effects, producing a changing probability distribution.

== Spontaneous decay ==
In ideal (nonrelativistic) quantum mechanics, systems such as the [[hydrogen atom]] have many stationary states. However, in reality, excited states are not perfectly stationary: an electron in a higher energy level can undergo [[spontaneous emission]], emitting a [[photon]] and decaying to a lower-energy state.<ref>Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles (2nd Edition), R. Eisberg, R. Resnick, John Wiley & Sons, 1985, {{ISBN|978-0-471-87373-0}}</ref>

This occurs because the usual Hamiltonian is an approximation; more complete descriptions from [[quantum field theory]] include effects such as [[vacuum fluctuations]], which break exact stationarity for excited states.

== Comparison to orbitals ==
{{main|Atomic orbital|Molecular orbital}}

An [[atomic orbital]] or [[molecular orbital]] can be interpreted as a stationary state (or approximation thereof) for a single electron.<ref>Physical chemistry, P. W. Atkins, Oxford University Press, 1978, {{ISBN|0-19-855148-7}}.</ref>

For single-electron systems (such as [[hydrogen]]), orbitals correspond directly to stationary states. For many-electron systems, however, the full stationary state is a many-particle state, often approximated using methods such as [[Slater determinant]]s.<ref name=lowdin55_1>{{cite journal |first1=Per-Olov |last1=Löwdin |journal=[[Physical Review]] |title=Quantum theory of many-particle systems |volume=97 |issue=6 |pages=1474–1489 |year=1955}}</ref>

In practice, orbitals are useful approximations based on treating electrons independently (the single-electron approximation), often combined with the [[Born–Oppenheimer approximation]].

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

== References ==
{{reflist|3}}

{{Author|Harold Foppele}}
{{Sourceattribution|Physics:Stationary state|1}}

Physics:Quantum Stationary states - Revision history

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