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Quantum mechanics
${\displaystyle i\hslash \frac{d}{dt}|\mathrm{\Psi }⟩=\hat{H}|\mathrm{\Psi }⟩}$ Schrödinger equation
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Astationary state is aquantum state with allobservables independent of time. It is aneigenvector of theenergy operator (instead of aquantum superposition of different energies). It is also calledenergy eigenvector,energy eigenstate,energy eigenfunction, orenergyeigenket. It is very similar to the concept ofatomic orbital andmolecular orbital in chemistry, with some slight differences explainedbelow.

A stationary state is calledstationary because the system remains in the same state as time elapses, in every observable way. For a single-particleHamiltonian, this means that the particle has a constantprobability distribution for its position, its velocity, itsspin, etc.^[1] (This is true assuming the particle's environment is also static, i.e. the Hamiltonian is unchanging in time.) Thewavefunction itself is not stationary: It continually changes its overall complexphase factor, so as to form astanding wave. The oscillation frequency of the standing wave, multiplied by thePlanck constant, is the energy of the state according to thePlanck–Einstein relation.

Stationary states arequantum states that are solutions to the time-independentSchrödinger equation:$${\displaystyle \hat{H}|\mathrm{\Psi }⟩=E_{\mathrm{\Psi }}|\mathrm{\Psi }⟩,}$$where

This is aneigenvalue equation:${\displaystyle \hat{H}}$ is alinear operator on a vector space,${\displaystyle |\mathrm{\Psi }⟩}$ is an eigenvector of${\displaystyle \hat{H}}$, and${\displaystyle E_{\mathrm{\Psi }}}$ is its eigenvalue.

If a stationary state${\displaystyle |\mathrm{\Psi }⟩}$ is plugged into the time-dependent Schrödinger equation, the result is^[2]$${\displaystyle i\hslash \frac{\mathrm{\partial }}{\mathrm{\partial }t}|\mathrm{\Psi }⟩=E_{\mathrm{\Psi }}|\mathrm{\Psi }⟩.}$$

Assuming that${\displaystyle \hat{H}}$ is time-independent (unchanging in time), this equation holds for any timet. Therefore, this is adifferential equation describing how${\displaystyle |\mathrm{\Psi }⟩}$ varies in time. Its solution is$${\displaystyle |\mathrm{\Psi }(t)⟩=e^{-i{E}_{\mathrm{\Psi }}t/\hslash }|\mathrm{\Psi }(0)⟩.}$$

Therefore, a stationary state is astanding wave that oscillates with an overall complexphase factor, and its oscillationangular frequency is equal to its energy divided by${\displaystyle \hslash }$.

As shown above, a stationary state is not mathematically constant:$${\displaystyle |\mathrm{\Psi }(t)⟩=e^{-i{E}_{\mathrm{\Psi }}t/\hslash }|\mathrm{\Psi }(0)⟩.}$$

However, all observable properties of the state are in fact constant in time. For example, if${\displaystyle |\mathrm{\Psi }(t)⟩}$ represents a simple one-dimensional single-particle wavefunction${\displaystyle \mathrm{\Psi }(x,t)}$, the probability that the particle is at locationx is$${\displaystyle |\mathrm{\Psi }(x,t){|}^2={\left|e^{-i{E}_{\mathrm{\Psi }}t/\hslash }\mathrm{\Psi }(x,0)\right|}^2={\left|e^{-i{E}_{\mathrm{\Psi }}t/\hslash }\right|}^2{\left|\mathrm{\Psi }(x,0)\right|}^2={\left|\mathrm{\Psi }(x,0)\right|}^2,}$$which is independent of the timet.

TheHeisenberg picture is an alternativemathematical formulation of quantum mechanics where stationary states are truly mathematically constant in time.

As mentioned above, these equations assume that the Hamiltonian is time-independent. This means simply that stationary states are only stationary when the rest of the system is fixed and stationary as well. For example, a1s electron in ahydrogen atom is in a stationary state, but if the hydrogen atom reacts with another atom, then the electron will of course be disturbed.

Spontaneous decay complicates the question of stationary states. For example, according to simple (nonrelativistic)quantum mechanics, thehydrogen atom has many stationary states:1s, 2s, 2p, and so on, are all stationary states. But in reality, only the ground state 1s is truly "stationary": An electron in a higher energy level willspontaneously emit one or morephotons to decay into the ground state.^[3] This seems to contradict the idea that stationary states should have unchanging properties.

The explanation is that theHamiltonian used in nonrelativistic quantum mechanics is only an approximation to the Hamiltonian fromquantum field theory. The higher-energy electron states (2s, 2p, 3s, etc.) are stationary states according to the approximate Hamiltonian, butnot stationary according to the true Hamiltonian, because ofvacuum fluctuations. On the other hand, the 1s state is truly a stationary state, according to both the approximate and the true Hamiltonian.

An orbital is a stationary state (or approximation thereof) of a one-electron atom or molecule; more specifically, anatomic orbital for an electron in an atom, or amolecular orbital for an electron in a molecule.^[4]

For a molecule that contains only a single electron (e.g. atomichydrogen orH₂⁺), an orbital is exactly the same as a total stationary state of the molecule. However, for a many-electron molecule, an orbital is completely different from a total stationary state, which is amany-particle state requiring a more complicated description (such as aSlater determinant).^[5] In particular, in a many-electron molecule, an orbital is not the total stationary state of the molecule, but rather the stationary state of a single electron within the molecule. This concept of an orbital is only meaningful under the approximation that if we ignore the electron–electron instantaneous repulsion terms in the Hamiltonian as a simplifying assumption, we can decompose the total eigenvector of a many-electron molecule into separate contributions from individual electron stationary states (orbitals), each of which are obtained under the one-electron approximation. (Luckily, chemists and physicists can often (but not always) use this "single-electron approximation".) In this sense, in a many-electron system, an orbital can be considered as the stationary state of an individual electron in the system.

In chemistry, calculation of molecular orbitals typically also assume theBorn–Oppenheimer approximation.

• Transition of state
• Quantum number
• Quantum mechanic vacuum orvacuum state
• Virtual particle
• Steady state

• Stationary states, Alan Holden, Oxford University Press, 1971,ISBN 0-19-851121-3

• Quantum mechanics

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• Short description is different from Wikidata