Stationary state
Encyclopedia
In quantum mechanics
, a stationary state is an eigenvector of the Hamiltonian
, implying the probability density
associated with the wavefunction
is independent of time . This corresponds to a quantum state with a single definite energy (instead of a probability distribution
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
and molecular orbital
in chemistry, with some slight differences explained below.
for its position, its velocity, its spin
, etc. (This is true assuming the rest of the system is also static, i.e. the Hamiltonian is unchanging in time.) The wavefunction
itself is not stationary: It continually changes its overall complex phase factor
, 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 de Broglie relation.
Stationary states are quantum states that are solutions to the time-independent Schrödinger Equation
:
,
where
This is an eigenvalue equation:
is a linear operator on a vector space, 
is an eigenvector of 
, and 
is its eigenvalue.
If a stationary state
is plugged into the time-dependent Schrödinger Equation
, the result is :
Assuming that
is time-independent (unchanging in time), this equation holds for any time t. Therefore this is a differential equation
describing how
varies in time. Its solution is:
Therefore a stationary state is a standing wave
that oscillates with an overall complex phase factor
, and its oscillation angular frequency
is equal to its energy divided by
.
However, all observable properties of the state are in fact constant. For example, if
represents a simple one-dimensional single-particle wavefunction 
, the probability that the particle is at location x is:
which is independent of the time t.
The Heisenberg picture
is an alternative mathematical 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, a 1s electron
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.
, 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
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
used in nonrelativistic quantum mechanics is only an approximation to the true Hamiltonian of the universe
. The higher-energy electron states (2s, 2p, 3s, etc.) are stationary states according to the approximate Hamiltonian, but not stationary according to the true Hamiltonian, because of vacuum fluctuations. On the other hand, the 1s state is truly a stationary state, according to both the approximate and the true Hamiltonian.
; more specifically, an atomic orbital
for an electron in an atom, or a molecular orbital
for an electron in a molecule. However, there are some differences between "orbital" and "stationary state". First, when there is no spin-orbit coupling, there will be pairs of stationary states with the same configuration in space, but with opposite electron spin
. These two states are considered to be just one orbital; therefore the Pauli exclusion principle
allows two electrons per orbital, but only one electron per stationary state. Second, an orbital is usually a wavefunction describing just one electron, even though the true stationary state is a many-particle state
requiring a more complicated description (such as a Slater determinant
of individual orbitals). In this case an orbital is only approximately a stationary state.
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
, a stationary state is an eigenvector of the Hamiltonian
Hamiltonian (quantum mechanics)
In quantum mechanics, the Hamiltonian H, also Ȟ or Ĥ, is the operator corresponding to the total energy of the system. Its spectrum is the set of possible outcomes when one measures the total energy of a system...
, implying the probability density
Probability density
Probability density may refer to:* Probability density function in probability theory* The product of the probability amplitude with its complex conjugate in quantum mechanics...
associated with the wavefunction
Wavefunction
Not to be confused with the related concept of the Wave equationA wave function or wavefunction is a probability amplitude in quantum mechanics describing the quantum state of a particle and how it behaves. Typically, its values are complex numbers and, for a single particle, it is a function of...
is independent of time . This corresponds to a quantum state with a single definite energy (instead of a probability distribution
Probability distribution
In probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values....
of different energies). It is also called energy eigenvector, energy eigenstate, energy eigenfunction, or energy eigenket
Bra-ket notation
Bra-ket notation is a standard notation for describing quantum states in the theory of quantum mechanics composed of angle brackets and vertical bars. It can also be used to denote abstract vectors and linear functionals in mathematics...
. It is very similar to the concept of atomic orbital
Atomic orbital
An atomic orbital is a mathematical function that describes the wave-like behavior of either one electron or a pair of electrons in an atom. This function can be used to calculate the probability of finding any electron of an atom in any specific region around the atom's nucleus...
and molecular orbital
Molecular orbital
In chemistry, a molecular orbital is a mathematical function describing the 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 finding an electron in any specific region. The term "orbital" was first...
in chemistry, with some slight differences explained below.
Introduction
A stationary state is called stationary because a particle remains in the same state as time elapses, in every observable way. It has a constant probability distributionProbability distribution
In probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values....
for its position, its velocity, its spin
Spin (physics)
In quantum mechanics and particle physics, spin is a fundamental characteristic property of elementary particles, composite particles , and atomic nuclei.It is worth noting that the intrinsic property of subatomic particles called spin and discussed in this article, is related in some small ways,...
, etc. (This is true assuming the rest of the system is also static, i.e. the Hamiltonian is unchanging in time.) The wavefunction
Wavefunction
Not to be confused with the related concept of the Wave equationA wave function or wavefunction is a probability amplitude in quantum mechanics describing the quantum state of a particle and how it behaves. Typically, its values are complex numbers and, for a single particle, it is a function of...
itself is not stationary: It continually changes its overall complex phase factor
Phase factor
For any complex number written in polar form , the phase factor is the exponential part, i.e. eiθ. As such, the term "phase factor" is similar to the term phasor, although the former term is more common in quantum mechanics. This phase factor is itself a complex number of absolute value 1...
, so as to form a standing wave
Standing wave
In physics, a standing wave – also known as a stationary wave – is a wave that remains in a constant position.This phenomenon can occur because the medium is moving in the opposite direction to the wave, or it can arise in a stationary medium as a result of interference between two waves traveling...
. The oscillation frequency of the standing wave, times Planck's constant, is the energy of the state according to the de Broglie relation.
Stationary states are quantum states that are solutions to the time-independent Schrödinger Equation
Schrödinger equation
The Schrödinger equation was formulated in 1926 by Austrian physicist Erwin Schrödinger. Used in physics , it is an equation that describes how the quantum state of a physical system changes in time....
:
,where
is a quantum state, which is a stationary state if it satisfies this equation;
is the Hamiltonian operator;
is a real number, and corresponds to the energy eigenvalue of the state 
.
This is an eigenvalue equation:
is a linear operator on a vector space, is an eigenvector of , and is its eigenvalue.If a stationary state
is plugged into the time-dependent Schrödinger EquationSchrödinger equation
The Schrödinger equation was formulated in 1926 by Austrian physicist Erwin Schrödinger. Used in physics , it is an equation that describes how the quantum state of a physical system changes in time....
, the result is :
Assuming that
is time-independent (unchanging in time), this equation holds for any time t. Therefore this is a differential equationDifferential equation
A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...
describing how
varies in time. Its solution is:Therefore a stationary state is a standing wave
Standing wave
In physics, a standing wave – also known as a stationary wave – is a wave that remains in a constant position.This phenomenon can occur because the medium is moving in the opposite direction to the wave, or it can arise in a stationary medium as a result of interference between two waves traveling...
that oscillates with an overall complex phase factor
Phase factor
For any complex number written in polar form , the phase factor is the exponential part, i.e. eiθ. As such, the term "phase factor" is similar to the term phasor, although the former term is more common in quantum mechanics. This phase factor is itself a complex number of absolute value 1...
, and its oscillation angular frequency
Angular frequency
In physics, angular frequency ω is a scalar measure of rotation rate. Angular frequency is the magnitude of the vector quantity angular velocity...
is equal to its energy divided by
.Stationary state properties
As shown above, a stationary state is not mathematically constant:However, all observable properties of the state are in fact constant. For example, if
represents a simple one-dimensional single-particle wavefunction , the probability that the particle is at location x is:which is independent of the time t.
The Heisenberg picture
Heisenberg picture
In physics, the Heisenberg picture is a formulation of quantum mechanics in which the operators incorporate a dependency on time, but the state vectors are time-independent. It stands in contrast to the Schrödinger picture in which the operators are constant and the states evolve in time...
is an alternative mathematical formulation of quantum mechanics
Mathematical formulation of quantum mechanics
The mathematical formulations of quantum mechanics are those mathematical formalisms that permit a rigorous description of quantum mechanics. Such are distinguished from mathematical formalisms for theories developed prior to the early 1900s by the use of abstract mathematical structures, such as...
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
Atomic orbital
An atomic orbital is a mathematical function that describes the wave-like behavior of either one electron or a pair of electrons in an atom. This function can be used to calculate the probability of finding any electron of an atom in any specific region around the atom's nucleus...
in a hydrogen atom
Hydrogen atom
A hydrogen atom is an atom of the chemical element hydrogen. The electrically neutral atom contains a single positively-charged proton and a single negatively-charged electron bound to the nucleus by the Coulomb force...
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) quantum mechanicsQuantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
, the hydrogen atom
Hydrogen atom
A hydrogen atom is an atom of the chemical element hydrogen. The electrically neutral atom contains a single positively-charged proton and a single negatively-charged electron bound to the nucleus by the Coulomb force...
has many stationary states: 1s, 2s, 2p
Atomic orbital
An atomic orbital is a mathematical function that describes the wave-like behavior of either one electron or a pair of electrons in an atom. This function can be used to calculate the probability of finding any electron of an atom in any specific region around the atom's nucleus...
, 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
Spontaneous emission
Spontaneous emission is the process by which a light source such as an atom, molecule, nanocrystal or nucleus in an excited state undergoes a transition to a state with a lower energy, e.g., the ground state and emits a photon...
one or more photon
Photon
In physics, a photon is an elementary particle, the quantum of the electromagnetic interaction and the basic unit of light and all other forms of electromagnetic radiation. It is also the force carrier for the electromagnetic force...
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
Hamiltonian may refer toIn mathematics :* Hamiltonian system* Hamiltonian path, in graph theory** Hamiltonian cycle, a special case of a Hamiltonian path* Hamiltonian group, in group theory* Hamiltonian...
used in nonrelativistic quantum mechanics is only an approximation to the true Hamiltonian of the universe
Quantum field theory
Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
. The higher-energy electron states (2s, 2p, 3s, etc.) are stationary states according to the approximate Hamiltonian, but not stationary according to the true Hamiltonian, because of vacuum fluctuations. 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
In chemistry, a stationary state of an electron is called an orbitalOrbital
Orbital may refer to:In chemistry and physics:* Atomic orbital* Molecular orbitalIn astronomy and space flight:* Orbit* Orbital resonance* Orbital period* Orbital plane * Orbital elements* Orbital speed...
; more specifically, an atomic orbital
Atomic orbital
An atomic orbital is a mathematical function that describes the wave-like behavior of either one electron or a pair of electrons in an atom. This function can be used to calculate the probability of finding any electron of an atom in any specific region around the atom's nucleus...
for an electron in an atom, or a molecular orbital
Molecular orbital
In chemistry, a molecular orbital is a mathematical function describing the 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 finding an electron in any specific region. The term "orbital" was first...
for an electron in a molecule. However, there are some differences between "orbital" and "stationary state". First, when there is no spin-orbit coupling, there will be pairs of stationary states with the same configuration in space, but with opposite electron spin
Spin (physics)
In quantum mechanics and particle physics, spin is a fundamental characteristic property of elementary particles, composite particles , and atomic nuclei.It is worth noting that the intrinsic property of subatomic particles called spin and discussed in this article, is related in some small ways,...
. These two states are considered to be just one orbital; therefore the Pauli exclusion principle
Pauli exclusion principle
The Pauli exclusion principle is the quantum mechanical principle that no two identical fermions may occupy the same quantum state simultaneously. A more rigorous statement is that the total wave function for two identical fermions is anti-symmetric with respect to exchange of the particles...
allows two electrons per orbital, but only one electron per stationary state. Second, an orbital is usually a wavefunction describing just one electron, even though the true stationary state is a many-particle state
Identical particles
Identical particles, or indistinguishable particles, are particles that cannot be distinguished from one another, even in principle. Species of identical particles include elementary particles such as electrons, and, with some clauses, composite particles such as atoms and molecules.There are two...
requiring a more complicated description (such as a Slater determinant
Slater determinant
In quantum mechanics, a Slater determinant is an expression that describes the wavefunction of a multi-fermionic system that satisfies anti-symmetry requirements and consequently the Pauli exclusion principle by changing sign upon exchange of fermions . It is named for its discoverer, John C...
of individual orbitals). In this case an orbital is only approximately a stationary state.
See also
- Quantum numberQuantum numberQuantum numbers describe values of conserved quantities in the dynamics of the quantum system. Perhaps the most peculiar aspect of quantum mechanics is the quantization of observable quantities. This is distinguished from classical mechanics where the values can range continuously...
- Quantum mechanic vacuum or vacuum stateVacuum stateIn quantum field theory, the vacuum state is the quantum state with the lowest possible energy. Generally, it contains no physical particles...
- Virtual particleVirtual particleIn physics, a virtual particle is a particle that exists for a limited time and space. The energy and momentum of a virtual particle are uncertain according to the uncertainty principle...
- Steady StateSteady stateA system in a steady state has numerous properties that are unchanging in time. This implies that for any property p of the system, the partial derivative with respect to time is zero:...