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A stationary state is a
quantum state In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution i ...
with all
observable In physics, an observable is a physical quantity that can be measured. Examples include position and momentum. In systems governed by classical mechanics, it is a real-valued "function" on the set of all possible system states. In quantum phy ...
s independent of time. It is an
eigenvector In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted ...
of the energy operator (instead of a quantum superposition of different energies). It is also called energy eigenvector, energy eigenstate, energy eigenfunction, or energy eigenket. It is very similar to the concept of
atomic orbital In atomic theory and quantum mechanics, an atomic orbital is a function describing the location and wave-like behavior of an electron in an atom. This function can be used to calculate the probability of finding any electron of an atom in an ...
and
molecular orbital In chemistry, a molecular orbital is a mathematical function describing the location and wave-like behavior of an electron in a molecule. This function can be used to calculate chemical and physical properties such as the probability of findin ...
in chemistry, with some slight differences explained below.


Introduction

A stationary state is called ''stationary'' because the system remains in the same state as time elapses, in every observable way. For a single-particle Hamiltonian, this means that the particle has a constant probability distribution for its position, its velocity, its spin, etc. (This is true assuming the particle's environment is also static, i.e. the Hamiltonian is unchanging in time.) The
wavefunction A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements ...
itself is not stationary: It continually changes its overall complex
phase factor For any complex number written in polar form (such as ), the phase factor is the complex exponential factor (). As such, the term "phase factor" is related to the more general term phasor, which may have any magnitude (i.e. not necessarily on th ...
, so as to form a standing wave. The oscillation frequency of the standing wave, times Planck's constant, is the energy of the state according to the
Planck–Einstein relation The Planck relationFrench & Taylor (1978), pp. 24, 55.Cohen-Tannoudji, Diu & Laloë (1973/1977), pp. 10–11. (referred to as Planck's energy–frequency relation,Schwinger (2001), p. 203. the Planck relation, Planck equation, and Planck formula, ...
. Stationary states are
quantum state In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution i ...
s that are solutions to the time-independent
Schrödinger equation The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of th ...
: \hat H , \Psi\rangle = E_\Psi , \Psi\rangle, where : , \Psi\rangle is a
quantum state In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution i ...
, which is a stationary state if it satisfies this equation; : \hat H is the
Hamiltonian operator Hamiltonian may refer to: * Hamiltonian mechanics, a function that represents the total energy of a system * Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system ** Dyall Hamiltonian, a modified Hamiltonia ...
; : E_\Psi is a
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
, and corresponds to the energy eigenvalue of the state , \Psi\rangle. This is an
eigenvalue equation In linear algebra, linear transformations can be represented by matrices. If T is a linear transformation mapping \mathbb^n to \mathbb^m and \mathbf x is a column vector with n entries, then T( \mathbf x ) = A \mathbf x for some m \times n matrix ...
: \hat H is a linear operator on a vector space, , \Psi\rangle is an eigenvector of \hat H, and E_\Psi is its eigenvalue. If a stationary state , \Psi\rangle is plugged into the time-dependent Schrödinger equation, the result is i\hbar\frac , \Psi\rangle = E_\Psi, \Psi\rangle. Assuming that \hat H is time-independent (unchanging in time), this equation holds for any time . Therefore, this is a
differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
describing how , \Psi\rangle varies in time. Its solution is , \Psi(t)\rangle = e^ , \Psi(0)\rangle. Therefore, a stationary state is a standing wave that oscillates with an overall complex
phase factor For any complex number written in polar form (such as ), the phase factor is the complex exponential factor (). As such, the term "phase factor" is related to the more general term phasor, which may have any magnitude (i.e. not necessarily on th ...
, and its oscillation
angular frequency In physics, angular frequency "''ω''" (also referred to by the terms angular speed, circular frequency, orbital frequency, radian frequency, and pulsatance) is a scalar measure of rotation rate. It refers to the angular displacement per unit tim ...
is equal to its energy divided by \hbar.


Stationary state properties

As shown above, a stationary state is not mathematically constant: , \Psi(t)\rangle = e^, \Psi(0)\rangle. However, all observable properties of the state are in fact constant in time. For example, if , \Psi(t)\rangle represents a simple one-dimensional single-particle wavefunction \Psi(x, t), the probability that the particle is at location is , \Psi(x, t), ^2 = \left, e^ \Psi(x, 0)\^2 = \left, e^\^2 \left, \Psi(x, 0)\^2 = \left, \Psi(x, 0)\^2, which is independent of the time . The
Heisenberg picture In physics, the Heisenberg picture (also called the Heisenberg representation) is a formulation (largely due to Werner Heisenberg in 1925) of quantum mechanics in which the operators (observables and others) incorporate a dependency on time, but ...
is an alternative
mathematical formulation of quantum mechanics The mathematical formulations of quantum mechanics are those mathematical formalisms that permit a rigorous description of quantum mechanics. This mathematical formalism uses mainly a part of functional analysis, especially Hilbert spaces, which ...
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, a
1s electron In atomic theory and quantum mechanics, an atomic orbital is a function describing the location and wave-like behavior of an electron in an atom. This function can be used to calculate the probability of finding any electron of an atom in any spe ...
in a hydrogen 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

Spontaneous decay complicates the question of stationary states. For example, according to simple (
nonrelativistic The theory of relativity usually encompasses two interrelated theories by Albert Einstein: special relativity and general relativity, proposed and published in 1905 and 1915, respectively. Special relativity applies to all physical phenomena in ...
)
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistr ...
, the hydrogen 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 will spontaneously emit one or more
photon A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless, so they a ...
s to decay into the ground state. This seems to contradict the idea that stationary states should have unchanging properties. The explanation is that the
Hamiltonian Hamiltonian may refer to: * Hamiltonian mechanics, a function that represents the total energy of a system * Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system ** Dyall Hamiltonian, a modified Hamiltonian ...
used in nonrelativistic quantum mechanics is only an approximation to the Hamiltonian from quantum field theory. The higher-energy electron states (2s, 2p, 3s, etc.) are stationary states according to the approximate Hamiltonian, but stationary according to the true Hamiltonian, because of
vacuum fluctuations In quantum physics, a quantum fluctuation (also known as a vacuum state fluctuation or vacuum fluctuation) is the temporary random change in the amount of energy in a point in space, as prescribed by Werner Heisenberg's uncertainty principle. ...
. On the other hand, the 1s state is truly a stationary state, according to both the approximate and the true Hamiltonian.


Comparison to "orbital" in chemistry

An orbital is a stationary state (or approximation thereof) of a one-electron atom or molecule; more specifically, an
atomic orbital In atomic theory and quantum mechanics, an atomic orbital is a function describing the location and wave-like behavior of an electron in an atom. This function can be used to calculate the probability of finding any electron of an atom in an ...
for an electron in an atom, or a
molecular orbital In chemistry, a molecular orbital is a mathematical function describing the location and wave-like behavior of an electron in a molecule. This function can be used to calculate chemical and physical properties such as the probability of findin ...
for an electron in a molecule.Physical chemistry, P. W. Atkins, Oxford University Press, 1978, . For a molecule that contains only a single electron (e.g. atomic
hydrogen Hydrogen is the chemical element with the symbol H and atomic number 1. Hydrogen is the lightest element. At standard conditions hydrogen is a gas of diatomic molecules having the formula . It is colorless, odorless, tasteless, non-toxic ...
or H2+), 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 a many-particle state requiring a more complicated description (such as a
Slater determinant In quantum mechanics, a Slater determinant is an expression that describes the wave function of a multi-fermionic system. It satisfies anti-symmetry requirements, and consequently the Pauli principle, by changing sign upon exchange of two elect ...
). 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 the
Born–Oppenheimer approximation In quantum chemistry and molecular physics, the Born–Oppenheimer (BO) approximation is the best-known mathematical approximation in molecular dynamics. Specifically, it is the assumption that the wave functions of atomic nuclei and elect ...
.


See also

* Transition of state * Quantum number * Quantum mechanic vacuum or
vacuum state In quantum field theory, the quantum vacuum state (also called the quantum vacuum or vacuum state) is the quantum state with the lowest possible energy. Generally, it contains no physical particles. The word zero-point field is sometimes used as ...
*
Virtual particle A virtual particle is a theoretical transient particle that exhibits some of the characteristics of an ordinary particle, while having its existence limited by the uncertainty principle. The concept of virtual particles arises in the perturba ...
*
Steady state In systems theory, a system or a process is in a steady state if the variables (called state variables) which define the behavior of the system or the process are unchanging in time. In continuous time, this means that for those properties ''p' ...


References


Further reading

*''Stationary states'', Alan Holden, Oxford University Press, 1971, {{DEFAULTSORT:Stationary State Quantum mechanics ja:基底状態 zh:定态