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Stationary states is a Book I topic in the Quantum Collection. A stationary state is a quantum state with all observables 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 eigenket. Stationary states are fundamental in quantum mechanics and are closely related to concepts such as atomic orbitals and molecular orbitals. 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.]] A stationary state is a quantum state with all observables independent of time. It is an eigenvector of the energy operator (rather than a quantum superposition of different energies).

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, this means that measurable quantities such as position probability, momentum, or spin remain constant in time.^[1]

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: $\hat{H} | Ψ ⟩ = E | Ψ ⟩$

where:

$| Ψ ⟩$ is the quantum state,
$\hat{H}$ is the Hamiltonian operator,
$E$ is the energy eigenvalue.

This is an eigenvalue equation: the stationary states are eigenvectors of the Hamiltonian.

When inserted into the time-dependent Schrödinger equation, the evolution is:^[2] $i ℏ \frac{\partial}{\partial t} | Ψ ⟩ = E | Ψ ⟩$

with solution: $| Ψ (t) ⟩ = e^{- i E t / ℏ} | Ψ (0) ⟩$

Thus, a stationary state evolves only by a phase factor, with angular frequency $ω = E / ℏ$ .

Stationary state properties

Two stationary states (top) and a non-stationary superposition (bottom). Only stationary states have time-independent probability densities.

Although the wavefunction changes in time, $| Ψ (t) ⟩ = e^{- i E t / ℏ} | Ψ (0) ⟩$

all observable quantities remain constant. For example, the probability density: $| Ψ (x, t) |^{2} = | Ψ (x, 0) |^{2}$

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.^[3]

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

An atomic orbital or molecular orbital can be interpreted as a stationary state (or approximation thereof) for a single electron.^[4]

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 determinantss.^[5]

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

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 {{Short description|Quantum Collection topic on Quantum Stationary states}}
 {{Quantum book backlink|Quantum dynamics and evolution}}
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 <div style="flex:1; line-height:1.45; color:#006b45; column-count:2; column-gap:32px; column-rule:1px solid #b8d8c8;">
-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.
+'''Stationary states''' is a Book I topic in the Quantum Collection. A stationary state is a quantum state with all observables 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 eigenket. Stationary states are fundamental in quantum mechanics and are closely related to concepts such as atomic orbitals and molecular orbitals. 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.]] A stationary state is a quantum state with all observables independent of time. It is an eigenvector of the energy operator (rather than a quantum superposition of different energies).
-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.]]
 </div>
@@ Line 22: / Line 20: @@
 == 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.&nbsp;32.</ref>
+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, this means that measurable quantities such as position probability, momentum, or spin remain constant in time.<ref>Claude Cohen-Tannoudji, Bernard Diu, and Franck Laloë. ''Quantum Mechanics: Volume One''. Hermann, 1977. p.&nbsp;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]].
+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]]:
+Stationary states are solutions to the time-independent Schrödinger equation:
 <math display="block">\hat H |\Psi\rangle = E |\Psi\rangle</math>
@@ Line 35: / Line 33: @@
 * <math>E</math> is the energy eigenvalue.
-This is an [[Eigenvalues and eigenvectors|eigenvalue equation]]: the stationary states are eigenvectors of the Hamiltonian.
+This is an 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.&nbsp;W. Atkins, Oxford University Press, 1974, {{ISBN|0-19-855493-1}}.</ref>
@@ Line 58: / Line 56: @@
 is time-independent.
-In the [[Heisenberg picture]], stationary states are mathematically constant in time.
+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.
@@ Line 75: / Line 73: @@
 An [[atomic orbital]] or [[molecular orbital]] can be interpreted as a stationary state (or approximation thereof) for a single electron.<ref>Physical chemistry, P.&nbsp;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>
+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 determinantss.<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]].
+In practice, orbitals are useful approximations based on treating electrons independently (the single-electron approximation), often combined with the Born–Oppenheimer approximation.
 =See also=

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Physics:Quantum Stationary states: Difference between revisions

Revision as of 08:13, 20 May 2026

Introduction

Definition

Stationary state properties

Superposition

Spontaneous decay

Comparison to orbitals

See also

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