Water cycle

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Water_cycle 

 

Diagram depicting the global water cycle.

The water cycle, also known as the hydrologic cycle or the hydrological cycle, is a biogeochemical cycle that describes the continuous movement of water on, above and below the surface of the Earth. The mass of water on Earth remains fairly constant over time but the partitioning of the water into the major reservoirs of ice, fresh water, saline water (salt water) and atmospheric water is variable depending on a wide range of climatic variables. The water moves from one reservoir to another, such as from river to ocean, or from the ocean to the atmosphere, by the physical processes of evaporation, transpiration, condensation, precipitation, infiltration, surface runoff, and subsurface flow. In doing so, the water goes through different forms: liquid, solid (ice) and vapor. The ocean plays a key role in the water cycle as it is the source of 86% of global evaporation.

The water cycle involves the exchange of energy, which leads to temperature changes. When water evaporates, it takes up energy from its surroundings and cools the environment. When it condenses, it releases energy and warms the environment. These heat exchanges influence climate.

The evaporative phase of the cycle purifies water which then replenishes the land with freshwater. The flow of liquid water and ice transports minerals across the globe. It is also involved in reshaping the geological features of the Earth, through processes including erosion and sedimentation. The water cycle is also essential for the maintenance of most life and ecosystems on the planet.

Description

Diagram of the water cycle

Overall process

Further information: Water distribution on Earth

The water cycle is powered from the energy emitted by the sun. This energy heats water in the ocean and seas. Water evaporates as water vapor into the air. Some ice and snow sublimates directly into water vapor. Evapotranspiration is water transpired from plants and evaporated from the soil. The water molecule H
₂O has smaller molecular mass than the major components of the atmosphere, nitrogen (N
₂) and oxygen (O
₂) and hence is less dense. Due to the significant difference in density, buoyancy drives humid air higher. As altitude increases, air pressure decreases and the temperature drops (see Gas laws). The lower temperature causes water vapor to condense into tiny liquid water droplets which are heavier than the air, and which fall unless supported by an updraft. A huge concentration of these droplets over a large area in the atmosphere become visible as cloud, while condensation near ground level is referred to as fog.

Atmospheric circulation moves water vapor around the globe; cloud particles collide, grow, and fall out of the upper atmospheric layers as precipitation. Some precipitation falls as snow, hail, or sleet, and can accumulate in ice caps and glaciers, which can store frozen water for thousands of years. Most water falls as rain back into the ocean or onto land, where the water flows over the ground as surface runoff. A portion of this runoff enters rivers, with streamflow moving water towards the oceans. Runoff and water emerging from the ground (groundwater) may be stored as freshwater in lakes. Not all runoff flows into rivers; much of it soaks into the ground as infiltration. Some water infiltrates deep into the ground and replenishes aquifers, which can store freshwater for long periods of time. Some infiltration stays close to the land surface and can seep back into surface-water bodies (and the ocean) as groundwater discharge or be taken up by plants and transferred back to the atmosphere as water vapor by transpiration. Some groundwater finds openings in the land surface and emerges as freshwater springs. In river valleys and floodplains, there is often continuous water exchange between surface water and ground water in the hyporheic zone. Over time, the water returns to the ocean, to continue the water cycle.

The ocean plays a key role in the water cycle. The ocean holds "97% of the total water on the planet; 78% of global precipitation occurs over the ocean, and it is the source of 86% of global evaporation".

Processes leading to movements and phase changes in water

Important physical processes within the water cycle include the following (in alphabetical order):

• Advection: The movement of water through the atmosphere. Without advection, water that evaporated over the oceans could not precipitate over land. Atmospheric rivers that move large volumes of water vapor over long distances are an example of advection.
• Condensation: The transformation of water vapor to liquid water droplets in the air, creating clouds and fog.
• Evaporation: The transformation of water from liquid to gas phases as it moves from the ground or bodies of water into the overlying atmosphere. The source of energy for evaporation is primarily solar radiation. Evaporation often implicitly includes transpiration from plants, though together they are specifically referred to as evapotranspiration. Total annual evapotranspiration amounts to approximately 505,000 km³ (121,000 cu mi) of water, 434,000 km³ (104,000 cu mi) of which evaporates from the oceans. 86% of global evaporation occurs over the ocean.
• Infiltration: The flow of water from the ground surface into the ground. Once infiltrated, the water becomes soil moisture or groundwater. A recent global study using water stable isotopes, however, shows that not all soil moisture is equally available for groundwater recharge or for plant transpiration.
• Percolation: Water flows vertically through the soil and rocks under the influence of gravity.
• Precipitation: Condensed water vapor that falls to the Earth's surface. Most precipitation occurs as rain, but also includes snow, hail, fog drip, graupel, and sleet. Approximately 505,000 km³ (121,000 cu mi) of water falls as precipitation each year, 398,000 km³ (95,000 cu mi) of it over the oceans. The rain on land contains 107,000 km³ (26,000 cu mi) of water per year and a snowing only 1,000 km³ (240 cu mi). 78% of global precipitation occurs over the ocean.
• Runoff: The variety of ways by which water moves across the land. This includes both surface runoff and channel runoff. As it flows, the water may seep into the ground, evaporate into the air, become stored in lakes or reservoirs, or be extracted for agricultural or other human uses.
• Subsurface flow: The flow of water underground, in the vadose zone and aquifers. Subsurface water may return to the surface (e.g. as a spring or by being pumped) or eventually seep into the oceans. Water returns to the land surface at lower elevation than where it infiltrated, under the force of gravity or gravity induced pressures. Groundwater tends to move slowly and is replenished slowly, so it can remain in aquifers for thousands of years.
• Transpiration: The release of water vapor from plants and soil into the air.

Residence times

Average reservoir residence times

Reservoir	Average residence time
Antarctica	20,000 years
Oceans	3,200 years
Glaciers	20 to 100 years
Seasonal snow cover	2 to 6 months
Soil moisture	1 to 2 months
Groundwater: shallow	100 to 200 years
Groundwater: deep	10,000 years
Lakes (see lake retention time)	50 to 100 years
Rivers	2 to 6 months
Atmosphere	9 days

The residence time of a reservoir within the hydrologic cycle is the average time a water molecule will spend in that reservoir (see adjacent table). It is a measure of the average age of the water in that reservoir.

Groundwater can spend over 10,000 years beneath Earth's surface before leaving. Particularly old groundwater is called fossil water. Water stored in the soil remains there very briefly, because it is spread thinly across the Earth, and is readily lost by evaporation, transpiration, stream flow, or groundwater recharge. After evaporating, the residence time in the atmosphere is about 9 days before condensing and falling to the Earth as precipitation.

The major ice sheets – Antarctica and Greenland – store ice for very long periods. Ice from Antarctica has been reliably dated to 800,000 years before present, though the average residence time is shorter.

In hydrology, residence times can be estimated in two ways. The more common method relies on the principle of conservation of mass (water balance) and assumes the amount of water in a given reservoir is roughly constant. With this method, residence times are estimated by dividing the volume of the reservoir by the rate by which water either enters or exits the reservoir. Conceptually, this is equivalent to timing how long it would take the reservoir to become filled from empty if no water were to leave (or how long it would take the reservoir to empty from full if no water were to enter).

An alternative method to estimate residence times, which is gaining in popularity for dating groundwater, is the use of isotopic techniques. This is done in the subfield of isotope hydrology.

Water in storage

Further information: Water resources and Water distribution on Earth

 

Water cycle showing human influences and major pools (storages) and fluxes.

The water cycle describes the processes that drive the movement of water throughout the hydrosphere. However, much more water is "in storage" (or in "pools") for long periods of time than is actually moving through the cycle. The storehouses for the vast majority of all water on Earth are the oceans. It is estimated that of the 1,386,000,000 km³ of the world's water supply, about 1,338,000,000 km³ is stored in oceans, or about 97%. It is also estimated that the oceans supply about 90% of the evaporated water that goes into the water cycle. The Earth's ice caps, glaciers, and permanent snowpack stores another 24,064,000 km³ accounting for only 1.7% of the planet's total water volume. However, this quantity of water is 68.7% of all freshwater on the planet.

Changes caused by humans

Extreme weather (heavy rains, droughts, heat waves) is one consequence of a changing water cycle due to global warming. These events will be progressively more common as the Earth warms more and more.

 

Predicted changes in average soil moisture for a scenario of 2°C global warming. This can disrupt agriculture and ecosystems. A reduction in soil moisture by one standard deviation means that average soil moisture will approximately match the ninth driest year between 1850 and 1900 at that location.

Water cycle intensification due to climate change

Main articles: Effects of climate change on the water cycle and Effects of climate change on oceans

Since the middle of the 20th century, human-caused climate change has resulted in observable changes in the global water cycle. The IPCC Sixth Assessment Report in 2021 predicted that these changes will continue to grow significantly at the global and regional level. These findings are a continuation of scientific consensus expressed in the IPCC Fifth Assessment Report from 2007 and other special reports by the Intergovernmental Panel on Climate Change which had already stated that the water cycle will continue to intensify throughout the 21st century.

The effects of climate change on the water cycle are profound and have been described as an intensification or a strengthening of the water cycle (also called hydrologic cycle). This effect has been observed since at least 1980. One example is the intensification of heavy precipitation events. This has important negative effects on the availability of freshwater resources, as well as other water reservoirs such as oceans, ice sheets, atmosphere and land surface. The water cycle is essential to life on Earth and plays a large role in the global climate and the ocean circulation. The warming of our planet is expected to cause changes in the water cycle for various reasons. For example, warmer atmosphere can contain more water vapor which has effects on evaporation and rainfall. Oceans play a large role as well, since they absorb 93% of heat. The increase in ocean heat content since 1971 has a big effect on the ocean as well as the water cycle.

The underlying cause of the intensifying water cycle is the increased amount of greenhouse gases, which lead to a warmer atmosphere through the greenhouse effect. Physics dictates that saturation vapor pressure increases by 7% when temperature rises by 1 °C (as described in the Clausius-Clapeyron equation).

The strength of the water cycle and its changes over time are of considerable interest, especially as the climate changes. The essence of the overall hydrological cycle is the evaporation of moisture in one place and the precipitation in other places. In particular, evaporation exceeds precipitation over the oceans, which allows moisture to be transported by the atmosphere from the oceans onto land where precipitation exceeds evapotranspiration, and the runoff flows into streams and rivers and discharges into the ocean, completing the cycle. The water cycle is a key part of Earth's energy cycle through the evaporative cooling at the surface which provides latent heat to the atmosphere, as atmospheric systems play a primary role in moving heat upward.

Changes due to other human activities

Relationship between impervious surfaces and surface runoff

Human activities, other than those that lead to global warming from greenhouse gas emissions, can also alter the water cycle. The IPCC Sixth Assessment Report stated that there is "abundant evidence that changes in land use and land cover alter the water cycle globally, regionally and locally, by changing precipitation, evaporation, flooding, groundwater, and the availability of freshwater for a variety of uses".

Examples for such land use changes are converting fields to urban areas or clearing forests. Such changes can affect the ability of soils to soak up surface water. Deforestation has local as well as regional effects. For example it reduces soil moisture, evaporation and rainfall at the local level. Furthermore, deforestation causes regional temperature changes that can affect rainfall patterns.

Aquifer drawdown or overdrafting and the pumping of fossil water increase the total amount of water in the hydrosphere. This is because the water that was originally in the ground has now become available for evaporation as it is now in contact with the atmosphere.

Related processes

Biogeochemical cycling

While the water cycle is itself a biogeochemical cycle, flow of water over and beneath the Earth is a key component of the cycling of other biogeochemicals. Runoff is responsible for almost all of the transport of eroded sediment and phosphorus from land to waterbodies. The salinity of the oceans is derived from erosion and transport of dissolved salts from the land. Cultural eutrophication of lakes is primarily due to phosphorus, applied in excess to agricultural fields in fertilizers, and then transported overland and down rivers. Both runoff and groundwater flow play significant roles in transporting nitrogen from the land to waterbodies. The dead zone at the outlet of the Mississippi River is a consequence of nitrates from fertilizer being carried off agricultural fields and funnelled down the river system to the Gulf of Mexico. Runoff also plays a part in the carbon cycle, again through the transport of eroded rock and soil.

Slow loss over geologic time

Main article: Atmospheric escape

The hydrodynamic wind within the upper portion of a planet's atmosphere allows light chemical elements such as Hydrogen to move up to the exobase, the lower limit of the exosphere, where the gases can then reach escape velocity, entering outer space without impacting other particles of gas. This type of gas loss from a planet into space is known as planetary wind. Planets with hot lower atmospheres could result in humid upper atmospheres that accelerate the loss of hydrogen.

Historical interpretations

Floating land mass

In ancient times, it was widely thought that the land mass floated on a body of water, and that most of the water in rivers has its origin under the earth. Examples of this belief can be found in the works of Homer (circa 800 BCE).

Hebrew Bible

In the ancient Near East, Hebrew scholars observed that even though the rivers ran into the sea, the sea never became full. Some scholars conclude that the water cycle was described completely during this time in this passage: "The wind goeth toward the south, and turneth about unto the north; it whirleth about continually, and the wind returneth again according to its circuits. All the rivers run into the sea, yet the sea is not full; unto the place from whence the rivers come, thither they return again" (Ecclesiastes 1:6-7). Scholars are not in agreement as to the date of Ecclesiastes, though most scholars point to a date during the time of King Solomon, son of David and Bathsheba, "three thousand years ago, there is some agreement that the time period is 962–922 BCE. Furthermore, it was also observed that when the clouds were full, they emptied rain on the earth (Ecclesiastes 11:3). In addition, during 793–740 BCE a Hebrew prophet, Amos, stated that water comes from the sea and is poured out on the earth (Amos 5:8).

In the Biblical Book of Job, dated between 7th and 2nd centuries BCE, there is a description of precipitation in the hydrologic cycle, "For he maketh small the drops of water: they pour down rain according to the vapour thereof; which the clouds do drop and distil upon man abundantly" (Job 36:27-28).

Understanding of precipitation and percolation

In the Adityahridayam (a devotional hymn to the Sun God) of Ramayana, a Hindu epic dated to the 4th century BCE, it is mentioned in the 22nd verse that the Sun heats up water and sends it down as rain. By roughly 500 BCE, Greek scholars were speculating that much of the water in rivers can be attributed to rain. The origin of rain was also known by then. These scholars maintained the belief, however, that water rising up through the earth contributed a great deal to rivers. Examples of this thinking included Anaximander (570 BCE) (who also speculated about the evolution of land animals from fish) and Xenophanes of Colophon (530 BCE). Chinese scholars such as Chi Ni Tzu (320 BCE) and Lu Shih Ch'un Ch'iu (239 BCE) had similar thoughts.

The idea that the water cycle is a closed cycle can be found in the works of Anaxagoras of Clazomenae (460 BCE) and Diogenes of Apollonia (460 BCE). Both Plato (390 BCE) and Aristotle (350 BCE) speculated about percolation as part of the water cycle. Aristotle correctly hypothesized that the sun played a role in the Earth's hydraulic cycle in his book Meteorology, writing "By it [the sun's] agency the finest and sweetest water is everyday carried up and is dissolved into vapor and rises to the upper regions, where it is condensed again by the cold and so returns to the earth.", and believed that clouds were composed of cooled and condensed water vapor.

Up to the time of the Renaissance, it was wrongly assumed that precipitation alone was insufficient to feed rivers, for a complete water cycle, and that underground water pushing upwards from the oceans were the main contributors to river water. Bartholomew of England held this view (1240 CE), as did Leonardo da Vinci (1500 CE) and Athanasius Kircher (1644 CE).

Discovery of the correct theory

The first published thinker to assert that rainfall alone was sufficient for the maintenance of rivers was Bernard Palissy (1580 CE), who is often credited as the discoverer of the modern theory of the water cycle. Palissy's theories were not tested scientifically until 1674, in a study commonly attributed to Pierre Perrault. Even then, these beliefs were not accepted in mainstream science until the early nineteenth century.

Operator (physics)

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Operator_(physics)

In physics, an operator is a function over a space of physical states onto another space of physical states. The simplest example of the utility of operators is the study of symmetry (which makes the concept of a group useful in this context). Because of this, they are useful tools in classical mechanics. Operators are even more important in quantum mechanics, where they form an intrinsic part of the formulation of the theory.

Operators in classical mechanics

In classical mechanics, the movement of a particle (or system of particles) is completely determined by the Lagrangian ${\displaystyle L(q,\dot{q},t)}$ or equivalently the Hamiltonian ${\displaystyle H(q,p,t)}$, a function of the generalized coordinates q, generalized velocities ${\displaystyle \dot{q}=\mathrm{d}q/\mathrm{d}t}$ and its conjugate momenta:

${\displaystyle p=\frac{\mathrm{\partial }L}{\mathrm{\partial }\dot{q}}}$

If either L or H is independent of a generalized coordinate q, meaning the L and H do not change when q is changed, which in turn means the dynamics of the particle are still the same even when q changes, the corresponding momenta conjugate to those coordinates will be conserved (this is part of Noether's theorem, and the invariance of motion with respect to the coordinate q is a symmetry). Operators in classical mechanics are related to these symmetries.

More technically, when H is invariant under the action of a certain group of transformations G:

${\displaystyle S\in G,H(S(q,p))=H(q,p)}$.

The elements of G are physical operators, which map physical states among themselves.

Table of classical mechanics operators

Transformation	Operator	Position	Momentum
Translational symmetry	${\displaystyle X(\mathbf{a})}$	${\displaystyle \mathbf{r}\to \mathbf{r}+\mathbf{a}}$	${\displaystyle \mathbf{p}\to \mathbf{p}}$
Time translation symmetry	${\displaystyle U(t_0)}$	${\displaystyle \mathbf{r}(t)\to \mathbf{r}(t+t_0)}$	${\displaystyle \mathbf{p}(t)\to \mathbf{p}(t+t_0)}$
Rotational invariance	${\displaystyle R(\hat{\mathbf{n}},\theta )}$	${\displaystyle \mathbf{r}\to R(\hat{\mathbf{n}},\theta )\mathbf{r}}$	${\displaystyle \mathbf{p}\to R(\hat{\mathbf{n}},\theta )\mathbf{p}}$
Galilean transformations	${\displaystyle G(\mathbf{v})}$	${\displaystyle \mathbf{r}\to \mathbf{r}+\mathbf{v}t}$	${\displaystyle \mathbf{p}\to \mathbf{p}+m\mathbf{v}}$
Parity	${\displaystyle P}$	${\displaystyle \mathbf{r}\to -\mathbf{r}}$	${\displaystyle \mathbf{p}\to -\mathbf{p}}$
T-symmetry	${\displaystyle T}$	${\displaystyle \mathbf{r}\to \mathbf{r}(-t)}$	${\displaystyle \mathbf{p}\to -\mathbf{p}(-t)}$

where ${\displaystyle R(\hat{\mathit{n}},\theta )}$ is the rotation matrix about an axis defined by the unit vector ${\displaystyle \hat{\mathit{n}}}$ and angle θ.

Generators

If the transformation is infinitesimal, the operator action should be of the form

${\displaystyle I+ϵA,}$

where ${\displaystyle I}$ is the identity operator, ${\displaystyle ϵ}$ is a parameter with a small value, and ${\displaystyle A}$ will depend on the transformation at hand, and is called a generator of the group. Again, as a simple example, we will derive the generator of the space translations on 1D functions.

As it was stated, ${\displaystyle T_{a}f(x)=f(x-a)}$. If ${\displaystyle a=ϵ}$ is infinitesimal, then we may write

${\displaystyle T_{ϵ}f(x)=f(x-ϵ)\approx f(x)-ϵf^{\prime }(x).}$

This formula may be rewritten as

${\displaystyle T_{ϵ}f(x)=(I-ϵD)f(x)}$

where ${\displaystyle D}$ is the generator of the translation group, which in this case happens to be the derivative operator. Thus, it is said that the generator of translations is the derivative.

The exponential map

The whole group may be recovered, under normal circumstances, from the generators, via the exponential map. In the case of the translations the idea works like this.

The translation for a finite value of ${\displaystyle a}$ may be obtained by repeated application of the infinitesimal translation:

${\displaystyle T_{a}f(x)=\underset{N\to \mathrm{\infty }}{lim}{T}_{a/N}\cdots T_{a/N}f(x)}$

with the ${\displaystyle \cdots }$ standing for the application ${\displaystyle N}$ times. If ${\displaystyle N}$ is large, each of the factors may be considered to be infinitesimal:

${\displaystyle T_{a}f(x)=\underset{N\to \mathrm{\infty }}{lim}{\left(I-\frac{a}{N}D\right)}^{N}f(x).}$

But this limit may be rewritten as an exponential:

${\displaystyle T_{a}f(x)=\exp (-aD)f(x).}$

To be convinced of the validity of this formal expression, we may expand the exponential in a power series:

${\displaystyle T_{a}f(x)=\left(I-aD+\frac{a^2{D}^2}{2!}-\frac{a^3{D}^3}{3!}+\cdots \right)f(x).}$

The right-hand side may be rewritten as

${\displaystyle f(x)-a{f}^{\prime }(x)+\frac{a^2}{2!}{f}^{″}(x)-\frac{a^3}{3!}{f}^{(3)}(x)+\cdots }$

which is just the Taylor expansion of ${\displaystyle f(x-a)}$, which was our original value for ${\displaystyle T_{a}f(x)}$.

The mathematical properties of physical operators are a topic of great importance in itself. For further information, see C*-algebra and Gelfand–Naimark theorem.

Operators in quantum mechanics

The mathematical formulation of quantum mechanics (QM) is built upon the concept of an operator.

Physical pure states in quantum mechanics are represented as unit-norm vectors (probabilities are normalized to one) in a special complex Hilbert space. Time evolution in this vector space is given by the application of the evolution operator.

Any observable, i.e., any quantity which can be measured in a physical experiment, should be associated with a self-adjoint linear operator. The operators must yield real eigenvalues, since they are values which may come up as the result of the experiment. Mathematically this means the operators must be Hermitian. The probability of each eigenvalue is related to the projection of the physical state on the subspace related to that eigenvalue. See below for mathematical details about Hermitian operators.

In the wave mechanics formulation of QM, the wavefunction varies with space and time, or equivalently momentum and time (see position and momentum space for details), so observables are differential operators.

In the matrix mechanics formulation, the norm of the physical state should stay fixed, so the evolution operator should be unitary, and the operators can be represented as matrices. Any other symmetry, mapping a physical state into another, should keep this restriction.

Wavefunction

Main article: wavefunction

The wavefunction must be square-integrable (see L^p spaces), meaning:

${\displaystyle {\iiint }_{{\mathbb{R}}^3}|\psi (\mathbf{r}){|}^2{d}^3\mathbf{r}={\iiint }_{{\mathbb{R}}^3}\psi (\mathbf{r}{)}^{\ast }\psi (\mathbf{r})d^3\mathbf{r}<\mathrm{\infty }}$

and normalizable, so that:

${\displaystyle {\iiint }_{{\mathbb{R}}^3}|\psi (\mathbf{r}){|}^2{d}^3\mathbf{r}=1}$

Two cases of eigenstates (and eigenvalues) are:

• for discrete eigenstates ${\displaystyle |{\psi }_i⟩}$ forming a discrete basis, so any state is a sum
  $${\displaystyle |\psi ⟩=\sum _i{c}_i|{\varphi }_i⟩}$$

  
  where c_i are complex numbers such that |c_i|² = c_i^*c_i is the probability of measuring the state ${\displaystyle |{\varphi }_i⟩}$, and the corresponding set of eigenvalues a_i is also discrete - either finite or countably infinite. In this case, the inner product of two eigenstates is given by ${\displaystyle ⟨{\varphi }_i|{\varphi }_j⟩={\delta }_{ij}}$, where ${\displaystyle {\delta }_{mn}}$ denotes the Kronecker Delta. However,
• for a continuum of eigenstates ${\displaystyle |{\psi }_i⟩}$ forming a continuous basis, any state is an integral
  $${\displaystyle |\psi ⟩=\int c(\varphi )d\varphi |\varphi ⟩}$$

  
  where c(φ) is a complex function such that |c(φ)|² = c(φ)^*c(φ) is the probability of measuring the state ${\displaystyle |\varphi ⟩}$, and there is an uncountably infinite set of eigenvalues a. In this case, the inner product of two eigenstates is defined as ${\displaystyle ⟨{\varphi }^{\prime }|\varphi ⟩=\delta (\varphi -{\varphi }^{\prime })}$, where here ${\displaystyle \delta (x-y)}$ denotes the Dirac Delta.

Linear operators in wave mechanics

Main articles: Wave function and Bra–ket notation

Let ψ be the wavefunction for a quantum system, and ${\displaystyle \hat{A}}$ be any linear operator for some observable A (such as position, momentum, energy, angular momentum etc.). If ψ is an eigenfunction of the operator ${\displaystyle \hat{A}}$, then

${\displaystyle \hat{A}\psi =a\psi ,}$

where a is the eigenvalue of the operator, corresponding to the measured value of the observable, i.e. observable A has a measured value a.

If ψ is an eigenfunction of a given operator ${\displaystyle \hat{A}}$, then a definite quantity (the eigenvalue a) will be observed if a measurement of the observable A is made on the state ψ. Conversely, if ψ is not an eigenfunction of ${\displaystyle \hat{A}}$, then it has no eigenvalue for ${\displaystyle \hat{A}}$, and the observable does not have a single definite value in that case. Instead, measurements of the observable A will yield each eigenvalue with a certain probability (related to the decomposition of ψ relative to the orthonormal eigenbasis of ${\displaystyle \hat{A}}$).

In bra–ket notation the above can be written;

${\displaystyle \begin{array}{rl}\hat{A}\psi & =\hat{A}\psi (\mathbf{r})=\hat{A}⟨\mathbf{r}\mid \psi ⟩=⟨\mathbf{r}\left|\hat{A}\right|\psi ⟩\\ a\psi & =a\psi (\mathbf{r})=a⟨\mathbf{r}\mid \psi ⟩=⟨\mathbf{r}\mid a\mid \psi ⟩\end{array}}$

that are equal if ${\displaystyle |\psi ⟩}$ is an eigenvector, or eigenket of the observable A.

Due to linearity, vectors can be defined in any number of dimensions, as each component of the vector acts on the function separately. One mathematical example is the del operator, which is itself a vector (useful in momentum-related quantum operators, in the table below).

An operator in n-dimensional space can be written:

${\displaystyle \hat{\mathbf{A}}=\sum _{j=1}^n{\mathbf{e}}_j{\hat{A}}_j}$

where e_j are basis vectors corresponding to each component operator A_j. Each component will yield a corresponding eigenvalue ${\displaystyle a_j}$. Acting this on the wave function ψ:

${\displaystyle \hat{\mathbf{A}}\psi =\left(\sum _{j=1}^n{\mathbf{e}}_j{\hat{A}}_j\right)\psi =\sum _{j=1}^n\left({\mathbf{e}}_j{\hat{A}}_j\psi \right)=\sum _{j=1}^n\left({\mathbf{e}}_j{a}_j\psi \right)}$

in which we have used ${\displaystyle {\hat{A}}_j\psi =a_j\psi .}$

In bra–ket notation:

${\displaystyle \begin{array}{rl}\hat{\mathbf{A}}\psi =\hat{\mathbf{A}}\psi (\mathbf{r})=\hat{\mathbf{A}}⟨\mathbf{r}\mid \psi ⟩& =⟨\mathbf{r}\left|\hat{\mathbf{A}}\right|\psi ⟩\\ \left(\sum _{j=1}^n{\mathbf{e}}_j{\hat{A}}_j\right)\psi =\left(\sum _{j=1}^n{\mathbf{e}}_j{\hat{A}}_j\right)\psi (\mathbf{r})=\left(\sum _{j=1}^n{\mathbf{e}}_j{\hat{A}}_j\right)⟨\mathbf{r}\mid \psi ⟩& =⟨\mathbf{r}\left|\sum _{j=1}^n{\mathbf{e}}_j{\hat{A}}_j\right|\psi ⟩\end{array}}$

Commutation of operators on Ψ

Main article: Commutator

If two observables A and B have linear operators ${\displaystyle \hat{A}}$ and ${\displaystyle \hat{B}}$, the commutator is defined by,

${\displaystyle \left[\hat{A},\hat{B}\right]=\hat{A}\hat{B}-\hat{B}\hat{A}}$

The commutator is itself a (composite) operator. Acting the commutator on ψ gives:

${\displaystyle \left[\hat{A},\hat{B}\right]\psi =\hat{A}\hat{B}\psi -\hat{B}\hat{A}\psi .}$

If ψ is an eigenfunction with eigenvalues a and b for observables A and B respectively, and if the operators commute:

${\displaystyle \left[\hat{A},\hat{B}\right]\psi =0,}$

then the observables A and B can be measured simultaneously with infinite precision, i.e., uncertainties ${\displaystyle \mathrm{\Delta }A=0}$, ${\displaystyle \mathrm{\Delta }B=0}$ simultaneously. ψ is then said to be the simultaneous eigenfunction of A and B. To illustrate this:

${\displaystyle \begin{array}{rl}\left[\hat{A},\hat{B}\right]\psi & =\hat{A}\hat{B}\psi -\hat{B}\hat{A}\psi \\ & =a(b\psi )-b(a\psi )\\ & =0.\end{array}}$

It shows that measurement of A and B does not cause any shift of state, i.e., initial and final states are same (no disturbance due to measurement). Suppose we measure A to get value a. We then measure B to get the value b. We measure A again. We still get the same value a. Clearly the state (ψ) of the system is not destroyed and so we are able to measure A and B simultaneously with infinite precision.

If the operators do not commute:

${\displaystyle \left[\hat{A},\hat{B}\right]\psi \ne 0,}$

they cannot be prepared simultaneously to arbitrary precision, and there is an uncertainty relation between the observables

${\displaystyle \mathrm{\Delta }A\mathrm{\Delta }B\ge \left|\frac{1}{2}⟨[A,B]⟩\right|}$

even if ψ is an eigenfunction the above relation holds. Notable pairs are position-and-momentum and energy-and-time uncertainty relations, and the angular momenta (spin, orbital and total) about any two orthogonal axes (such as L_x and L_y, or s_y and s_z, etc.).

Expectation values of operators on Ψ

The expectation value (equivalently the average or mean value) is the average measurement of an observable, for particle in region R. The expectation value ${\displaystyle ⟨\hat{A}⟩}$ of the operator ${\displaystyle \hat{A}}$ is calculated from:

${\displaystyle ⟨\hat{A}⟩={\int }_R{\psi }^{\ast }\left(\mathbf{r}\right)\hat{A}\psi \left(\mathbf{r}\right){\mathrm{d}}^3\mathbf{r}=⟨\psi \left|\hat{A}\right|\psi ⟩.}$

This can be generalized to any function F of an operator:

${\displaystyle ⟨F\left(\hat{A}\right)⟩={\int }_R\psi (\mathbf{r}{)}^{\ast }\left[F\left(\hat{A}\right)\psi (\mathbf{r})\right]{\mathrm{d}}^3\mathbf{r}=⟨\psi \left|F\left(\hat{A}\right)\right|\psi ⟩,}$

An example of F is the 2-fold action of A on ψ, i.e. squaring an operator or doing it twice:

${\displaystyle \begin{array}{rl}F\left(\hat{A}\right)& ={\hat{A}}^2\\ \Rightarrow ⟨{\hat{A}}^2⟩& ={\int }_R{\psi }^{\ast }\left(\mathbf{r}\right){\hat{A}}^2\psi \left(\mathbf{r}\right){\mathrm{d}}^3\mathbf{r}=⟨\psi \left|{\hat{A}}^2\right|\psi ⟩\end{array}}$

Hermitian operators

Main article: Self-adjoint operator

The definition of a Hermitian operator is:

${\displaystyle \hat{A}={\hat{A}}^{†}}$

Following from this, in bra–ket notation:

${\displaystyle ⟨{\varphi }_i\left|\hat{A}\right|{\varphi }_j⟩={⟨{\varphi }_j\left|\hat{A}\right|{\varphi }_i⟩}^{\ast }.}$

Important properties of Hermitian operators include:

• real eigenvalues,
• eigenvectors with different eigenvalues are orthogonal,
• eigenvectors can be chosen to be a complete orthonormal basis,

Operators in matrix mechanics

An operator can be written in matrix form to map one basis vector to another. Since the operators are linear, the matrix is a linear transformation (aka transition matrix) between bases. Each basis element ${\displaystyle {\varphi }_j}$ can be connected to another, by the expression:

${\displaystyle A_{ij}=⟨{\varphi }_i\left|\hat{A}\right|{\varphi }_j⟩,}$

which is a matrix element:

${\displaystyle \hat{A}=\left(\begin{array}{cccc}{A}_{11}& A_{12}& \cdots & A_{1n}\\ A_{21}& A_{22}& \cdots & A_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ A_{n1}& A_{n2}& \cdots & A_{nn}\end{array}\right)}$

A further property of a Hermitian operator is that eigenfunctions corresponding to different eigenvalues are orthogonal. In matrix form, operators allow real eigenvalues to be found, corresponding to measurements. Orthogonality allows a suitable basis set of vectors to represent the state of the quantum system. The eigenvalues of the operator are also evaluated in the same way as for the square matrix, by solving the characteristic polynomial:

${\displaystyle det\left(\hat{A}-a\hat{I}\right)=0,}$

where I is the n × n identity matrix, as an operator it corresponds to the identity operator. For a discrete basis:

${\displaystyle \hat{I}=\sum _i|{\varphi }_i⟩⟨{\varphi }_i|}$

while for a continuous basis:

${\displaystyle \hat{I}=\int |\varphi ⟩⟨\varphi |\mathrm{d}\varphi }$

Inverse of an operator

A non-singular operator ${\displaystyle \hat{A}}$ has an inverse ${\displaystyle {\hat{A}}^{-1}}$ defined by:

${\displaystyle \hat{A}{\hat{A}}^{-1}={\hat{A}}^{-1}\hat{A}=\hat{I}}$

If an operator has no inverse, it is a singular operator. In a finite-dimensional space, an operator is non-singular if and only if its determinant is nonzero:

${\displaystyle det\left(\hat{A}\right)\ne 0}$

and hence the determinant is zero for a singular operator.