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Stationary states 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). A stationary state is called stationary because the system remains unchanged in every observable way as time elapses.

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

Related topic: Atomic orbital, Molecular orbital

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}}
+{{Quantum article nav|previous=Physics:Quantum Time-dependent Schrödinger equation|previous label=Time-dependent Schrödinger equation|next=Physics:Quantum Perturbation theory|next label=Perturbation theory}}
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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).
+'''Stationary states''' 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). A stationary state is called stationary because the system remains unchanged in every observable way as time elapses.
 </div>
@@ Line 22: / Line 21: @@
 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 ==
@@ Line 61: / Line 60: @@
 == 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.
+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>
+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.
+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}}
+''Related topic:'' 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.&nbsp;W. Atkins, Oxford University Press, 1978, {{ISBN|0-19-855148-7}}.</ref>
+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 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>
+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.

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Physics:Quantum Stationary states: Difference between revisions

Latest revision as of 11:32, 22 May 2026

Introduction

Definition

Stationary state properties

Superposition

Spontaneous decay

Comparison to orbitals

See also

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