Effects of climate change on the water cycle

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Effects_of_climate_change_on_the_water_cycle

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

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 when heavy rain events become even stronger. The effects of climate change on the water cycle have important negative effects on the availability of freshwater resources, as well as other water reservoirs such as oceans, ice sheets, the atmosphere and soil moisture. The water cycle is essential to life on Earth and plays a large role in the global climate system and ocean circulation. The warming of our planet is expected to be accompanied by changes in the water cycle for various reasons. For example, a warmer atmosphere can contain more water vapor which has effects on evaporation and rainfall.

The underlying cause of the intensifying water cycle is the increased amount of greenhouse gases in the atmosphere, which lead to a warmer atmosphere through the greenhouse effect. Fundamental laws of physics explain how the saturation vapor pressure in the atmosphere increases by 7% when temperature rises by 1 °C. This relationship is known as 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 hydrological cycle is a system whereby the evaporation of moisture in one place leads to precipitation (rain or snow) in another place. For example, evaporation always exceeds precipitation over the oceans. This allows moisture to be transported by the atmosphere from the oceans onto land where precipitation exceeds evapotranspiration. The runoff from the land flows into streams and rivers and discharges into the ocean, which completes the global 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.

The availability of water plays a major role in determining where the extra heat goes. It can go either into evaporation or into air temperature increases. If water is available (like over the oceans and the tropics), extra heat goes mostly into evaporation. If water is not available (like over dry areas on land), the extra heat goes into raising air temperature. Also, the water holding capacity of the atmosphere increases proportionally with temperature increase. For these reasons, the temperature increases dominate in the Arctic (polar amplification) and on land but not over the oceans and the tropics.

Several inherent characteristics have the potential to cause sudden (abrupt) changes in the water cycle. However, the likelihood that such changes will occur during the 21st century is currently regarded as low.

Overview

The water cycle

Heating of the Earth leads to more energy cycling within its climate system, causing changes to the global water cycle. These include first and foremost an increased water vapor pressure in the atmosphere. This causes changes in precipitation patterns with regards to frequency and intensity, as well as changes in groundwater and soil moisture. Taken together, these changes are often referred to as an "intensification and acceleration" of the water cycle. Key processes that will also be affected are droughts and floods, tropical cyclones, glacier retreat, snow cover, ice jam floods and extreme weather events.

The increasing amount of greenhouse gases in the atmosphere leads to extra heating of the lower atmosphere, also known as the troposphere. The saturation vapor pressure of air rises along with its temperature, which means that warmer air can contain more water vapor. Transfers of heat to land, ocean and ice surfaces additionally promote more evaporation. The greater amount of water in the troposphere then increases the chances for more intense rainfall events.

This relation between temperature and saturation vapor pressure is described in the Clausius–Clapeyron equation, which states that saturation pressure will increase by 7% when temperature rises by 1 °C. This is visible in measurements of the tropospheric water vapor, which are provided by satellites, radiosondes and surface stations. The IPCC AR5 concludes that tropospheric water vapor has increased by 3.5% over the last 40 years, which is consistent with the observed temperature increase of 0.5 °C.

The human influence on the water cycle can be observed by analyzing the ocean's surface salinity and the "precipitation minus evaporation (P–E)" patterns over the ocean. Both are elevated. Research published in 2012 based on surface ocean salinity over the period 1950 to 2000 confirm this projection of an intensified global water cycle with salty areas becoming more saline and fresher areas becoming more fresh over the period. IPCC indicates there is high confidence that heavy precipitation events associated with both tropical and extratropical cyclones, and atmospheric moisture transport and heavy precipitation events will intensify.

Intermittency in precipitation

Climate models do not simulate the water cycle very well. One reason is that precipitation is a difficult quantity to deal with because it is inherently intermittent. Often, only the average amount is considered. People tend to use the term "precipitation" as if it was the same as "precipitation amount". What actually matters when describing changes to Earth's precipitation patterns is more than just the total amount: it is also about the intensity (how hard it rains or snows), frequency (how often), duration (how long), and type (whether rain or snow). Scientists have researched the characteristics of precipitation and found that it is the frequency and intensity that matter for extremes, and those are difficult to calculate in climate models.

Observations and predictions

Predicted changes in precipitation event intensity and evapotranspiration under the SSP2-4.5 scenario.

Since the middle of the 20th century, human-caused climate change has included 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.

The report also found that: Precipitation over land has increased since 1950, and the rate of increase has become faster since the 1980s and in higher latitudes. Water vapour in the atmosphere (in particular the troposphere) has increased since at least the 1980s. It is expected that over the course of the 21st century, the annual global precipitation over land will increase due to a higher global surface temperature.

A warming climate makes extremely wet and very dry occurrences more severe. There can also be changes in atmospheric circulation patterns. This will affect the regions and frequency for these extremes to occur. In most parts of the world and under all climate change scenarios, water cycle variability and accompanying extremes are anticipated to rise more quickly than the changes of average values.

In 2024 the World Meteorological Organization published a report saying that climate change had severely destabilized water cycle during the year 2023, causing both stronger rainfall and stronger drought. The world's rivers had their driest year in at least 30 years and many of the world's major river basins were drying up like the basins of Mississippi, Amazon, Ganges, Brahmaputra and Mekong. For 3 years in a row, more than 50% of global catchment areas had lower than normal river discharges. Glaciers lost more than 600 gigatons of water – the biggest water loss in the last 50 years. It was the second year in a row in which all glaciated regions had ice loss.

Changes to regional weather patterns

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.

Regional weather patterns across the globe are also changing due to tropical ocean warming. The Indo-Pacific warm pool has been warming rapidly and expanding during the recent decades, largely in response to increased carbon emissions from fossil fuel burning. The warm pool expanded to almost double its size, from an area of 22 million km² during 1900–1980, to an area of 40 million km² during 1981–2018. This expansion of the warm pool has altered global rainfall patterns, by changing the life cycle of the Madden Julian Oscillation (MJO), which is the most dominant mode of weather fluctuation originating in the tropics.

Potential for abrupt change

Several characteristics of the water cycle have the potential to cause sudden (abrupt) changes of the water cycle. The definition for "abrupt change" is: a regional to global scale change in the climate system that happens more quickly than it has in the past, indicating that the climate response is not linear. There may be "rapid transitions between wet and dry states" as a result of non-linear interactions between the ocean, atmosphere, and land surface.

For example, a collapse of the Atlantic meridional overturning circulation (AMOC), if it did occur, could have large regional impacts on the water cycle. The initiation or termination of solar radiation modification could also result in abrupt changes in the water cycle. There could also be abrupt water cycle responses to changes in the land surface: Amazon deforestation and drying, greening of the Sahara and the Sahel, amplification of drought by dust are all processes which could contribute.

The scientific understanding of the likelihood of such abrupt changes to the water cycle is not yet clear. Sudden changes in the water cycle due to human activity are a possibility that cannot be ruled out, with current scientific knowledge. However, the likelihood that such changes will occur during the 21st century is currently regarded as low.

Measurement and modelling techniques

Changes in ocean salinity

See also: Effects of climate change on oceans

The yearly average distribution of precipitation minus evaporation. The image shows how the region around the equator is dominated by precipitation, and the subtropics are mainly dominated by evaporation.

Due to global warming and increased glacier melt, thermohaline circulation patterns may be altered by increasing amounts of freshwater released into oceans and, therefore, changing ocean salinity. Thermohaline circulation is responsible for bringing up cold, nutrient-rich water from the depths of the ocean, a process known as upwelling.

Seawater consists of fresh water and salt, and the concentration of salt in seawater is called salinity. Salt does not evaporate, thus the precipitation and evaporation of freshwater influences salinity strongly. Changes in the water cycle are therefore strongly visible in surface salinity measurements, which has already been known since the 1930s.

The global pattern of the oceanic surface salinity. It can be seen how the by evaporation dominated subtropics are relatively saline. The tropics and higher latitudes are less saline. When comparing with the map above it can be seen how the high salinity regions match the by evaporation dominated areas, and the lower salinity regions match the by precipitation dominated areas.

The advantage of using surface salinity is that it is well documented in the last 50 years, for example with in-situ measurement systems as ARGO. Another advantage is that oceanic salinity is stable on very long time scales, which makes small changes due to anthropogenic forcing easier to track. The oceanic salinity is not homogeneously distributed over the globe, there are regional differences that show a clear pattern. The tropic regions are relatively fresh, since these regions are dominated by rainfall. The subtropics are more saline, since these are dominated by evaporation, these regions are also known as the 'desert latitudes'. The latitudes close to the polar regions are then again less saline, with the lowest salinity values found in these regions. This is because there is a low amount of evaporation in this region, and a high amount of fresh meltwater entering the Arctic Ocean.

The long-term observation records show a clear trend: the global salinity patterns are amplifying in this period. This means that the high saline regions have become more saline, and regions of low salinity have become less saline. The regions of high salinity are dominated by evaporation, and the increase in salinity shows that evaporation is increasing even more. The same goes for regions of low salinity that are become less saline, which indicates that precipitation is intensifying only more. This spatial pattern is similar to the spatial pattern of evaporation minus precipitation. The amplification of the salinity patterns is therefore indirect evidence for an intensifying water cycle.

To further investigate the relation between ocean salinity and the water cycle, models play a large role in current research. General Circulation Models (GCMs) and more recently Atmosphere-Ocean General Circulation Models (AOGCMs) simulate the global circulations and the effects of changes such as an intensifying water cycle. The outcome of multiple studies based on such models support the relationship between surface salinity changes and the amplifying precipitation minus evaporation patterns.

A metric to capture the difference in salinity between high and low salinity regions in the top 2000 meters of the ocean is captured in the SC2000 metric. The observed increase of this metric is 5.2% (±0.6%) from 1960 to 2017. But this trend is accelerating, as it increased 1.9% (±0.6%) from 1960 to 1990, and 3.3% (±0.4%) from 1991 to 2017. Amplification of the pattern is weaker below the surface. This is because ocean warming increases near-surface stratification, subsurface layer is still in equilibrium with the colder climate. This causes the surface amplification to be stronger than older models predicted.

An instrument carried by the SAC-D satellite Aquarius, launched in June 2011, measured global sea surface salinity.

Between 1994 and 2006, satellite observations showed an 18% increase in the flow of freshwater into the world's oceans, partly from melting ice sheets, especially Greenland and partly from increased precipitation driven by an increase in global ocean evaporation.

Salinity evidence for changes in the water cycle

Essential processes of the water cycle are precipitation and evaporation. The local amount of precipitation minus evaporation (often noted as P-E) shows the local influence of the water cycle. Changes in the magnitude of P-E are often used to show changes in the water cycle. But robust conclusions about changes in the amount of precipitation and evaporation are complex. About 85% of the earth's evaporation and 78% of the precipitation happens over the ocean surface, where measurements are difficult. Precipitation on the one hand, only has long term accurate observation records over land surfaces where the amount of rainfall can be measured locally (called in-situ). Evaporation on the other hand, has no long time accurate observation records at all. This prohibits confident conclusions about changes since the industrial revolution. The AR5 (Fifth Assessment Report) of the IPCC creates an overview of the available literature on a topic, and labels the topic then on scientific understanding. They assign only low confidence to precipitation changes before 1951, and medium confidence after 1951, because of the scarcity of data. These changes are attributed to human influence, but only with medium confidence as well. There have been limited changes in regional monsoon precipitation observed over the 20th century because increases caused by global warming have been neutralized by cooling effects of anthropogenic aerosols. Different regional climate models project changes in monsoon precipitation whereby more regions are projected with increases than those with decreases.

Convection-permitting models to predict weather extremes

The representation of convection in climate models has so far restricted the ability of scientists to accurately simulate African weather extremes, limiting climate change predictions. Convection-permitting models (CPMs) are able to better simulate the diurnal cycle of tropical convection, the vertical cloud structure and the coupling between moist convection and convergence and soil moisture-convection feedbacks in the Sahel. The benefits of CPMs have also been demonstrated in other regions, including a more realistic representation of the precipitation structure and extremes. A convection-permitting (4.5 km grid-spacing) model over an Africa-wide domain shows future increases in dry spell length during the wet season over western and central Africa. The scientists concludes that, with the more accurate representation of convection, projected changes in both wet and dry extremes over Africa may be more severe. In other words: "both ends of Africa's weather extremes will get more severe".

Impacts on water management aspects

See also: Effects of climate change

The human-caused changes to the water cycle will increase hydrologic variability and therefore have a profound impact on the water sector and investment decisions. They will affect water availability (water resources), water supply, water demand, water security and water allocation at regional, basin, and local levels.

Water security

Impacts of climate change that are tied to water, affect people's water security on a daily basis. They include more frequent and intense heavy precipitation which affects the frequency, size and timing of floods. Also droughts can alter the total amount of freshwater and cause a decline in groundwater storage, and reduction in groundwater recharge. Reduction in water quality due to extreme events can also occur. Faster melting of glaciers can also occur.

Global climate change will probably make it more complex and expensive to ensure water security. It creates new threats and adaptation challenges. This is because climate change leads to increased hydrological variability and extremes. Climate change has many impacts on the water cycle. These result in higher climatic and hydrological variability, which can threaten water security. Changes in the water cycle threaten existing and future water infrastructure. It will be harder to plan investments for future water infrastructure as there are so many uncertainties about future variability for the water cycle. This makes societies more exposed to risks of extreme events linked to water and therefore reduces water security.

Water scarcity

Climate change could have a big impact on water resources around the world because of the close connections between the climate and hydrological cycle. Rising temperatures will increase evaporation and lead to increases in precipitation. However there will be regional variations in rainfall. Both droughts and floods may become more frequent and more severe in different regions at different times. There will be generally less snowfall and more rainfall in a warmer climate. Changes in snowfall and snow melt in mountainous areas will also take place. Higher temperatures will also affect water quality in ways that scientists do not fully understand. Possible impacts include increased eutrophication. Climate change could also boost demand for irrigation systems in agriculture. There is now ample evidence that greater hydrologic variability and climate change have had a profound impact on the water sector, and will continue to do so. This will show up in the hydrologic cycle, water availability, water demand, and water allocation at the global, regional, basin, and local levels.

The United Nations' FAO states that by 2025 1.9 billion people will live in countries or regions with absolute water scarcity. It says two thirds of the world's population could be under stress conditions. The World Bank says that climate change could profoundly alter future patterns of water availability and use. This will make water stress and insecurity worse, at the global level and in sectors that depend on water.

Droughts

Climate change affects many factors associated with droughts. These include how much rain falls and how fast the rain evaporates again. Warming over land increases the severity and frequency of droughts around much of the world. In some tropical and subtropical regions of the world, there will probably be less rain due to global warming. This will make them more prone to drought. Droughts are set to worsen in many regions of the world. These include Central America, the Amazon and south-western South America. They also include West and Southern Africa. The Mediterranean and south-western Australia are also some of these regions.

Higher temperatures increase evaporation. This dries the soil and increases plant stress. Agriculture suffers as a result. This means even regions where overall rainfall is expected to remain relatively stable will experience these impacts. These regions include central and northern Europe. Without climate change mitigation, around one third of land areas are likely to experience moderate or more severe drought by 2100. Due to global warming droughts are more frequent and intense than in the past.

Several impacts make their impacts worse. These are increased water demand, population growth and urban expansion in many areas. Land restoration can help reduce the impact of droughts. One example of this is agroforestry.

Desertification

Research into desertification is complex, and there is no single metric which can define all aspects. However, more intense climate change is still expected to increase the current extent of drylands on the Earth's continents: from 38% in late 20th century to 50% or 56% by the end of the century, under the "moderate" and high-warming Representative Concentration Pathways 4.5 and 8.5. Most of the expansion will be seen over regions such as "southwest North America, the northern fringe of Africa, southern Africa, and Australia".

Drylands cover 41% of the earth's land surface and include 45% of the world's agricultural land. These regions are among the most vulnerable ecosystems to anthropogenic climate and land use change and are under threat of desertification. An observation-based attribution study of desertification was carried out in 2020 which accounted for climate change, climate variability, CO₂ fertilization as well as both the gradual and rapid ecosystem changes caused by land use. The study found that, between 1982 and 2015, 6% of the world's drylands underwent desertification driven by unsustainable land use practices compounded by anthropogenic climate change. Despite an average global greening, anthropogenic climate change has degraded 12.6% (5.43 million km²) of drylands, contributing to desertification and affecting 213 million people, 93% of who live in developing economies.

Floods

Due to an increase in heavy rainfall events, floods are likely to become more severe when they do occur. The interactions between rainfall and flooding are complex. There are some regions in which flooding is expected to become rarer. This depends on several factors. These include changes in rain and snowmelt, but also soil moisture. Climate change leaves soils drier in some areas, so they may absorb rainfall more quickly. This leads to less flooding. Dry soils can also become harder. In this case heavy rainfall runs off into rivers and lakes. This increases risks of flooding.

Groundwater quantity and quality

The impacts of climate change on groundwater may be greatest through its indirect effects on irrigation water demand via increased evapotranspiration. There is an observed declined in groundwater storage in many parts of the world. This is due to more groundwater being used for irrigation activities in agriculture, particularly in drylands. Some of this increase in irrigation can be due to water scarcity issues made worse by effects of climate change on the water cycle. Direct redistribution of water by human activities amounting to ~24,000 km³ per year is about double the global groundwater recharge each year.

Climate change causes changes to the water cycle which in turn affect groundwater in several ways: There can be a decline in groundwater storage, and reduction in groundwater recharge and water quality deterioration due to extreme weather events. In the tropics intense precipitation and flooding events appear to lead to more groundwater recharge.

However, the exact impacts of climate change on groundwater are still under investigation. This is because scientific data derived from groundwater monitoring is still missing, such as changes in space and time, abstraction data and "numerical representations of groundwater recharge processes".

Effects of climate change could have different impacts on groundwater storage: The expected more intense (but fewer) major rainfall events could lead to increased groundwater recharge in many environments. But more intense drought periods could result in soil drying-out and compaction which would reduce infiltration to groundwater.

Photon polarization

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Photon_polarization

Photon polarization is the quantum mechanical description of the classical polarized sinusoidal plane electromagnetic wave. An individual photon can be described as having right or left circular polarization, or a superposition of the two. Equivalently, a photon can be described as having horizontal or vertical linear polarization, or a superposition of the two.

The description of photon polarization contains many of the physical concepts and much of the mathematical machinery of more involved quantum descriptions, such as the quantum mechanics of an electron in a potential well. Polarization is an example of a qubit degree of freedom, which forms a fundamental basis for an understanding of more complicated quantum phenomena. Much of the mathematical machinery of quantum mechanics, such as state vectors, probability amplitudes, unitary operators, and Hermitian operators, emerge naturally from the classical Maxwell's equations in the description. The quantum polarization state vector for the photon, for instance, is identical with the Jones vector, usually used to describe the polarization of a classical wave. Unitary operators emerge from the classical requirement of the conservation of energy of a classical wave propagating through lossless media that alter the polarization state of the wave. Hermitian operators then follow for infinitesimal transformations of a classical polarization state.

Many of the implications of the mathematical machinery are easily verified experimentally. In fact, many of the experiments can be performed with polaroid sunglass lenses.

The connection with quantum mechanics is made through the identification of a minimum packet size, called a photon, for energy in the electromagnetic field. The identification is based on the theories of Planck and the interpretation of those theories by Einstein. The correspondence principle then allows the identification of momentum and angular momentum (called spin), as well as energy, with the photon.

Polarization of classical electromagnetic waves

Main article: Polarization (waves)

Polarization states

Linear polarization

Main article: Linear polarization

Effect of a polarizer on reflection from mud flats. In the first picture, the polarizer is rotated to minimize the effect; in the second it is rotated 90° to maximize it: almost all reflected sunlight is eliminated.

The wave is linearly polarized (or plane polarized) when the phase angles ${\displaystyle {\alpha }_x,{\alpha }_y}$ are equal, $${\displaystyle {\alpha }_x={\alpha }_y\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}\alpha .}$$

This represents a wave with phase ${\displaystyle \alpha }$ polarized at an angle ${\displaystyle \theta }$ with respect to the x axis. In this case the Jones vector $${\displaystyle |\psi ⟩=\left(\begin{array}{c}\cos \theta \exp \left(i{\alpha }_x\right)\\ \sin \theta \exp \left(i{\alpha }_y\right)\end{array}\right)}$$ can be written with a single phase: $${\displaystyle |\psi ⟩=\left(\begin{array}{c}\cos \theta \\ \sin \theta \end{array}\right)\exp \left(i\alpha \right).}$$

The state vectors for linear polarization in x or y are special cases of this state vector.

If unit vectors are defined such that $${\displaystyle |x⟩\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}\left(\begin{array}{c}1\\ 0\end{array}\right)}$$ and $${\displaystyle |y⟩\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}\left(\begin{array}{c}0\\ 1\end{array}\right)}$$ then the linearly polarized polarization state can be written in the "x–y basis" as $${\displaystyle |\psi ⟩=\cos \theta \exp \left(i\alpha \right)|x⟩+\sin \theta \exp \left(i\alpha \right)|y⟩={\psi }_x|x⟩+{\psi }_y|y⟩.}$$

Circular polarization

Main article: Circular polarization

If the phase angles ${\displaystyle {\alpha }_x}$ and ${\displaystyle {\alpha }_y}$ differ by exactly ${\displaystyle \pi /2}$ and the x amplitude equals the y amplitude the wave is circularly polarized. The Jones vector then becomes $${\displaystyle |\psi ⟩=\frac{1}{\sqrt{2}}\left(\begin{array}{c}1\\ \pm i\end{array}\right)\exp \left(i{\alpha }_x\right)}$$ where the plus sign indicates left circular polarization and the minus sign indicates right circular polarization. In the case of circular polarization, the electric field vector of constant magnitude rotates in the x–y plane.

If unit vectors are defined such that $${\displaystyle |\mathrm{R}⟩\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}\frac{1}{\sqrt{2}}\left(\begin{array}{c}1\\ i\end{array}\right)}$$ and $${\displaystyle |\mathrm{L}⟩\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}\frac{1}{\sqrt{2}}\left(\begin{array}{c}1\\ -i\end{array}\right)}$$ then an arbitrary polarization state can be written in the "R–L basis" as $${\displaystyle |\psi ⟩={\psi }_{\mathrm{R}}|\mathrm{R}⟩+{\psi }_{\mathrm{L}}|\mathrm{L}⟩}$$ where $${\displaystyle {\psi }_{\mathrm{R}}=⟨\mathrm{R}|\psi ⟩=\frac{1}{\sqrt{2}}\left(\cos \theta \exp (i{\alpha }_x)-i\sin \theta \exp (i{\alpha }_y)\right)}$$ and $${\displaystyle {\psi }_{\mathrm{L}}=⟨\mathrm{L}|\psi ⟩=\frac{1}{\sqrt{2}}\left(\cos \theta \exp (i{\alpha }_x)+i\sin \theta \exp (i{\alpha }_y)\right).}$$

We can see that $${\displaystyle 1=|{\psi }_{\mathrm{R}}{|}^2+|{\psi }_{\mathrm{L}}{|}^2.}$$

Elliptical polarization

Main article: Elliptical polarization

The general case in which the electric field rotates in the x–y plane and has variable magnitude is called elliptical polarization. The state vector is given by $${\displaystyle |\psi ⟩\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}\left(\begin{array}{c}{\psi }_x\\ {\psi }_y\end{array}\right)=\left(\begin{array}{c}\cos \theta \exp \left(i{\alpha }_x\right)\\ \sin \theta \exp \left(i{\alpha }_y\right)\end{array}\right).}$$

Geometric visualization of an arbitrary polarization state

To get an understanding of what a polarization state looks like, one can observe the orbit that is made if the polarization state is multiplied by a phase factor of ${\displaystyle e^{i\omega t}}$ and then having the real parts of its components interpreted as x and y coordinates respectively. That is: $${\displaystyle \left(\begin{array}{c}x(t)\\ y(t)\end{array}\right)=\left(\begin{array}{c}\mathrm{\Re }(e^{i\omega t}{\psi }_x)\\ \mathrm{\Re }(e^{i\omega t}{\psi }_y)\end{array}\right)=\mathrm{\Re }\left[e^{i\omega t}\left(\begin{array}{c}{\psi }_x\\ {\psi }_y\end{array}\right)\right]=\mathrm{\Re }\left(e^{i\omega t}|\psi ⟩\right).}$$

If only the traced out shape and the direction of the rotation of (x(t), y(t)) is considered when interpreting the polarization state, i.e. only $${\displaystyle M(|\psi ⟩)=\left\{(x(t),y(t))\right|\mathrm{\forall }t\}}$$ (where x(t) and y(t) are defined as above) and whether it is overall more right circularly or left circularly polarized (i.e. whether |ψ_R| > |ψ_L| or vice versa), it can be seen that the physical interpretation will be the same even if the state is multiplied by an arbitrary phase factor, since $${\displaystyle M(e^{i\alpha }|\psi ⟩)=M(|\psi ⟩),\alpha \in \mathbb{R}}$$ and the direction of rotation will remain the same. In other words, there is no physical difference between two polarization states ${\displaystyle |\psi ⟩}$ and ${\displaystyle e^{i\alpha }|\psi ⟩}$, between which only a phase factor differs.

It can be seen that for a linearly polarized state, M will be a line in the xy plane, with length 2 and its middle in the origin, and whose slope equals to tan(θ). For a circularly polarized state, M will be a circle with radius 1/√2 and with the middle in the origin.

Energy, momentum, and angular momentum of a classical electromagnetic wave

Energy density of classical electromagnetic waves

Energy in a plane wave

Main article: Energy density

The energy per unit volume in classical electromagnetic fields is (cgs units) and also Planck units: $${\displaystyle {\mathcal{E}}_c=\frac{1}{8\pi }\left[{\mathbf{E}}^2(\mathbf{r},t)+{\mathbf{B}}^2(\mathbf{r},t)\right].}$$

For a plane wave, this becomes: $${\displaystyle {\mathcal{E}}_c=\frac{\mid \mathbf{E}{\mid }^2}{8\pi }}$$ where the energy has been averaged over a wavelength of the wave.

Fraction of energy in each component

The fraction of energy in the x component of the plane wave is $${\displaystyle f_x=\frac{|\mathbf{E}{|}^2{\cos }^2\theta }{|\mathbf{E}{|}^2}={\psi }_x^{\ast }{\psi }_x={\cos }^2\theta }$$ with a similar expression for the y component resulting in ${\displaystyle f_y={\sin }^2\theta }$.

The fraction in both components is $${\displaystyle {\psi }_x^{\ast }{\psi }_x+{\psi }_y^{\ast }{\psi }_y=⟨\psi |\psi ⟩=1.}$$

Momentum density of classical electromagnetic waves

The momentum density is given by the Poynting vector $${\displaystyle \mathcal{P}=\frac{1}{4\pi c}\mathbf{E}(\mathbf{r},t)\times \mathbf{B}(\mathbf{r},t).}$$

For a sinusoidal plane wave traveling in the z direction, the momentum is in the z direction and is related to the energy density: $${\displaystyle {\mathcal{P}}_{z}c={\mathcal{E}}_c.}$$

The momentum density has been averaged over a wavelength.

Angular momentum density of classical electromagnetic waves

Main article: Angular momentum of light

Electromagnetic waves can have both orbital and spin angular momentum.^[1] The total angular momentum density is $${\displaystyle \mathcal{L}=\mathbf{r}\times \mathcal{P}=\frac{1}{4\pi c}\mathbf{r}\times \left[\mathbf{E}(\mathbf{r},t)\times \mathbf{B}(\mathbf{r},t)\right].}$$

For a sinusoidal plane wave propagating along ${\displaystyle z}$ axis the orbital angular momentum density vanishes. The spin angular momentum density is in the ${\displaystyle z}$ direction and is given by $${\displaystyle \mathcal{L}=\frac{|\mathbf{E}{|}^2}{8\pi \omega }\left({\left|⟨\mathrm{R}|\psi ⟩\right|}^2-{\left|⟨\mathrm{L}|\psi ⟩\right|}^2\right)=\frac{1}{\omega }{\mathcal{E}}_c\left(|{\psi }_{\mathrm{R}}{|}^2-|{\psi }_{\mathrm{L}}{|}^2\right)}$$ where again the density is averaged over a wavelength.

Optical filters and crystals

Passage of a classical wave through a polaroid filter

Linear polarization

A linear filter transmits one component of a plane wave and absorbs the perpendicular component. In that case, if the filter is polarized in the x direction, the fraction of energy passing through the filter is $${\displaystyle f_x={\psi }_x^{\ast }{\psi }_x={\cos }^2\theta .}$$

Example of energy conservation: Passage of a classical wave through a birefringent crystal

An ideal birefringent crystal transforms the polarization state of an electromagnetic wave without loss of wave energy. Birefringent crystals therefore provide an ideal test bed for examining the conservative transformation of polarization states. Even though this treatment is still purely classical, standard quantum tools such as unitary and Hermitian operators that evolve the state in time naturally emerge.

Initial and final states

A birefringent crystal is a material that has an optic axis with the property that the light has a different index of refraction for light polarized parallel to the axis than it has for light polarized perpendicular to the axis. Light polarized parallel to the axis are called "extraordinary rays" or "extraordinary photons", while light polarized perpendicular to the axis are called "ordinary rays" or "ordinary photons". If a linearly polarized wave impinges on the crystal, the extraordinary component of the wave will emerge from the crystal with a different phase than the ordinary component. In mathematical language, if the incident wave is linearly polarized at an angle ${\displaystyle theta}$ with respect to the optic axis, the incident state vector can be written $${\displaystyle |\psi ⟩=\left(\begin{array}{c}\cos \theta \\ \sin \theta \end{array}\right)}$$ and the state vector for the emerging wave can be written $${\displaystyle |{\psi }^{\prime }⟩=\left(\begin{array}{c}\cos \theta \exp \left(i{\alpha }_x\right)\\ \sin \theta \exp \left(i{\alpha }_y\right)\end{array}\right)=\left(\begin{array}{cc}\exp \left(i{\alpha }_x\right)& 0\\ 0& \exp \left(i{\alpha }_y\right)\end{array}\right)\left(\begin{array}{c}\cos \theta \\ \sin \theta \end{array}\right)\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}\hat{U}|\psi ⟩.}$$

While the initial state was linearly polarized, the final state is elliptically polarized. The birefringent crystal alters the character of the polarization.

Dual of the final state

A calcite crystal laid upon a paper with some letters showing the double refraction

The initial polarization state is transformed into the final state with the operator U. The dual of the final state is given by $${\displaystyle ⟨{\psi }^{\prime }|=⟨\psi |{\hat{U}}^{†}}$$ where ${\displaystyle U^{†}}$ is the adjoint of U, the complex conjugate transpose of the matrix.

Unitary operators and energy conservation

The fraction of energy that emerges from the crystal is $${\displaystyle ⟨{\psi }^{\prime }|{\psi }^{\prime }⟩=⟨\psi |{\hat{U}}^{†}\hat{U}|\psi ⟩=⟨\psi |\psi ⟩=1.}$$

In this ideal case, all the energy impinging on the crystal emerges from the crystal. An operator U with the property that $${\displaystyle {\hat{U}}^{†}\hat{U}=I,}$$ where I is the identity operator and U is called a unitary operator. The unitary property is necessary to ensure energy conservation in state transformations.

Hermitian operators and energy conservation

Doubly refracting Calcite from Iceberg claim, Dixon, New Mexico. This 35 pound (16 kg) crystal, on display at the National Museum of Natural History, is one of the largest single crystals in the United States.

If the crystal is very thin, the final state will be only slightly different from the initial state. The unitary operator will be close to the identity operator. We can define the operator H by $${\displaystyle \hat{U}\approx I+i\hat{H}}$$ and the adjoint by $${\displaystyle {\hat{U}}^{†}\approx I-i{\hat{H}}^{†}.}$$

Energy conservation then requires $${\displaystyle I={\hat{U}}^{†}\hat{U}\approx \left(I-i{\hat{H}}^{†}\right)\left(I+i\hat{H}\right)\approx I-i{\hat{H}}^{†}+i\hat{H}.}$$

This requires that $${\displaystyle \hat{H}={\hat{H}}^{†}.}$$

Operators like this that are equal to their adjoints are called Hermitian or self-adjoint.

The infinitesimal transition of the polarization state is $${\displaystyle |{\psi }^{\prime }⟩-|\psi ⟩=i\hat{H}|\psi ⟩.}$$

Thus, energy conservation requires that infinitesimal transformations of a polarization state occur through the action of a Hermitian operator.

Photons: connection to quantum mechanics

Main article: Photon

Energy, momentum, and angular momentum of photons

Energy

The treatment to this point has been classical. It is a testament, however, to the generality of Maxwell's equations for electrodynamics that the treatment can be made quantum mechanical with only a reinterpretation of classical quantities. The reinterpretation is based on the theories of Max Planck and the interpretation by Albert Einstein of those theories and of other experiments.^{[citation needed]}

Einstein's conclusion from early experiments on the photoelectric effect is that electromagnetic radiation is composed of irreducible packets of energy, known as photons. The energy of each packet is related to the angular frequency of the wave by the relation$${\displaystyle ϵ=\hslash \omega }$$where ${\displaystyle \hslash }$ is an experimentally determined quantity known as the reduced Planck constant. If there are ${\displaystyle N}$ photons in a box of volume ${\displaystyle V}$, the energy in the electromagnetic field is$${\displaystyle N\hslash \omega }$$and the energy density is$${\displaystyle \frac{N\hslash \omega }{V}}$$

The photon energy can be related to classical fields through the correspondence principle that states that for a large number of photons, the quantum and classical treatments must agree. Thus, for very large ${\displaystyle N}$, the quantum energy density must be the same as the classical energy density$${\displaystyle \frac{N\hslash \omega }{V}={\mathcal{E}}_c=\frac{|\mathbf{E}{|}^2}{8\pi }.}$$

The number of photons in the box is then$${\displaystyle N=\frac{V}{8\pi \hslash \omega }|\mathbf{E}{|}^2.}$$

Momentum

The correspondence principle also determines the momentum and angular momentum of the photon. For momentum$${\displaystyle {\mathcal{P}}_z=\frac{N\hslash \omega }{cV}=\frac{N\hslash k_z}{V}}$$where ${\displaystyle k_z}$ is the wave number. This implies that the momentum of a photon is$${\displaystyle p_z=\hslash k_z.}$$

Angular momentum and spin

Similarly for the spin angular momentum$${\displaystyle \mathcal{L}=\frac{1}{\omega }{\mathcal{E}}_c\left(|{\psi }_{\mathrm{R}}{|}^2-|{\psi }_{\mathrm{L}}{|}^2\right)=\frac{N\hslash }{V}\left(|{\psi }_{\mathrm{R}}{|}^2-|{\psi }_{\mathrm{L}}{|}^2\right)}$$where ${\displaystyle {\mathcal{E}}_c}$ is field strength. This implies that the spin angular momentum of the photon is$${\displaystyle l_z=\hslash \left(|{\psi }_{\mathrm{R}}{|}^2-|{\psi }_{\mathrm{L}}{|}^2\right).}$$the quantum interpretation of this expression is that the photon has a probability of ${\displaystyle \mid {\psi }_{\mathrm{R}}{\mid }^2}$ of having a spin angular momentum of ${\displaystyle \hslash }$ and a probability of ${\displaystyle \mid {\psi }_{\mathrm{L}}{\mid }^2}$ of having a spin angular momentum of ${\displaystyle -\hslash }$. We can therefore think of the spin angular momentum of the photon being quantized as well as the energy. The angular momentum of classical light has been verified. A photon that is linearly polarized (plane polarized) is in a superposition of equal amounts of the left-handed and right-handed states. Upon absorption by an electronic state, the angular momentum is "measured" and this superposition collapses into either right-hand or left-hand, corresponding to a raising or lowering of the angular momentum of the absorbing electronic state, respectively.

Spin operator

Main article: Spin angular momentum of light

The spin of the photon is defined as the coefficient of ${\displaystyle \hslash }$ in the spin angular momentum calculation. A photon has spin 1 if it is in the ${\displaystyle |R⟩}$ state and −1 if it is in the ${\displaystyle |L⟩}$ state. The spin operator is defined as the outer product$${\displaystyle \hat{S}\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}|\mathrm{R}⟩⟨\mathrm{R}|-|\mathrm{L}⟩⟨\mathrm{L}|=\left(\begin{array}{cc}0& -i\\ i& 0\end{array}\right).}$$

The eigenvectors of the spin operator are ${\displaystyle |\mathrm{R}⟩}$ and ${\displaystyle |\mathrm{L}⟩}$ with eigenvalues 1 and −1, respectively. These values are based on the point of view of the source as the convention to define circular polarization handedness.

The expected value of a spin measurement on a photon is then$${\displaystyle ⟨\psi |\hat{S}|\psi ⟩=|{\psi }_{\mathrm{R}}{|}^2-|{\psi }_{\mathrm{L}}{|}^2.}$$

An operator S has been associated with an observable quantity, the spin angular momentum. The eigenvalues of the operator are the allowed observable values. This has been demonstrated for spin angular momentum, but it is in general true for any observable quantity.

Spin states

See also: Circular polarization § Quantum mechanics

We can write the circularly polarized states as$${\displaystyle |s⟩}$$where s = 1 for ${\displaystyle |\mathrm{R}⟩}$ and s = −1 for ${\displaystyle |\mathrm{L}⟩}$. An arbitrary state can be written$${\displaystyle |\psi ⟩=\sum _{s=-1,1}{a}_s\exp \left(i{\alpha }_x-is\theta \right)|s⟩}$$where ${\displaystyle {\alpha }_1}$ and ${\displaystyle {\alpha }_{-1}}$ are phase angles, θ is the angle by which the frame of reference is rotated, and$${\displaystyle \sum _{s=-1,1}|a_s{|}^2=1.}$$

Spin and angular momentum operators in differential form

When the state is written in spin notation, the spin operator can be written$${\displaystyle {\hat{S}}_d\to i\frac{\mathrm{\partial }}{\mathrm{\partial }\theta }}$$$${\displaystyle {\hat{S}}_d^{†}\to -i\frac{\mathrm{\partial }}{\mathrm{\partial }\theta }.}$$

The eigenvectors of the differential spin operator are$${\displaystyle \exp \left(i{\alpha }_x-is\theta \right)|s⟩.}$$

To see this, note$${\displaystyle {\hat{S}}_d\exp \left(i{\alpha }_x-is\theta \right)|s⟩\to i\frac{\mathrm{\partial }}{\mathrm{\partial }\theta }\exp \left(i{\alpha }_x-is\theta \right)|s⟩=s\left[\exp \left(i{\alpha }_x-is\theta \right)|s⟩\right].}$$

The spin angular momentum operator is$${\displaystyle {\hat{l}}_z=\hslash {\hat{S}}_d.}$$

Nature of probability in quantum mechanics

Probability for a single photon

There are two ways in which probability can be applied to the behavior of photons; probability can be used to calculate the probable number of photons in a particular state, or probability can be used to calculate the likelihood of a single photon to be in a particular state. The former interpretation violates energy conservation. The latter interpretation is the viable, if nonintuitive, option. Dirac explains this in the context of the double-slit experiment:

Some time before the discovery of quantum mechanics people realized that the connection between light waves and photons must be of a statistical character. What they did not clearly realize, however, was that the wave function gives information about the probability of one photon being in a particular place and not the probable number of photons in that place. The importance of the distinction can be made clear in the following way. Suppose we have a beam of light consisting of a large number of photons split up into two components of equal intensity. On the assumption that the beam is connected with the probable number of photons in it, we should have half the total number going into each component. If the two components are now made to interfere, we should require a photon in one component to be able to interfere with one in the other. Sometimes these two photons would have to annihilate one another and other times they would have to produce four photons. This would contradict the conservation of energy. The new theory, which connects the wave function with probabilities for one photon gets over the difficulty by making each photon go partly into each of the two components. Each photon then interferes only with itself. Interference between two different photons never occurs.
—Paul Dirac, The Principles of Quantum Mechanics, 1930, Chapter 1

Probability amplitudes

The probability for a photon to be in a particular polarization state depends on the fields as calculated by the classical Maxwell's equations. The polarization state of the photon is proportional to the field. The probability itself is quadratic in the fields and consequently is also quadratic in the quantum state of polarization. In quantum mechanics, therefore, the state or probability amplitude contains the basic probability information. In general, the rules for combining probability amplitudes look very much like the classical rules for composition of probabilities: [The following quote is from Baym, Chapter 1]

1. The probability amplitude for two successive probabilities is the product of amplitudes for the individual possibilities. For example, the amplitude for the x polarized photon to be right circularly polarized and for the right circularly polarized photon to pass through the y-polaroid is ${\displaystyle ⟨R|x⟩⟨y|R⟩,}$ the product of the individual amplitudes.
2. The amplitude for a process that can take place in one of several indistinguishable ways is the sum of amplitudes for each of the individual ways. For example, the total amplitude for the x polarized photon to pass through the y-polaroid is the sum of the amplitudes for it to pass as a right circularly polarized photon, ${\displaystyle ⟨y|R⟩⟨R|x⟩,}$ plus the amplitude for it to pass as a left circularly polarized photon, ${\displaystyle ⟨y|L⟩⟨L|x⟩\dots }$
3. The total probability for the process to occur is the absolute value squared of the total amplitude calculated by 1 and 2.

Uncertainty principle

Main article: Uncertainty principle

Cauchy–Schwarz inequality in Euclidean space. ${\displaystyle \mathbf{V}\cdot \mathbf{W}=\Vert \mathbf{V}\Vert \Vert \mathbf{W}\Vert \cos a.}$ This implies ${\displaystyle \mathbf{V}\cdot \mathbf{W}\le \Vert \mathbf{V}\Vert \Vert \mathbf{W}\Vert .}$

Mathematical preparation

For any legal^{[clarification needed]} operators the following inequality, a consequence of the Cauchy–Schwarz inequality, is true. $${\displaystyle \frac{1}{4}{\left|⟨(\hat{A}\hat{B}-\hat{B}\hat{A})x|x⟩\right|}^2\le {\Vert \hat{A}x\Vert }^2{\Vert \hat{B}x\Vert }^2.}$$

If B A ψ and A B ψ are defined, then by subtracting the means and re-inserting in the above formula, we deduce $${\displaystyle {\mathrm{\Delta }}_{\psi }\hat{A}{\mathrm{\Delta }}_{\psi }\hat{B}\ge \frac{1}{2}\left|{⟨\left[\hat{A},\hat{B}\right]⟩}_{\psi }\right|}$$ where $${\displaystyle {⟨\hat{X}⟩}_{\psi }=⟨\psi |\hat{X}|\psi ⟩}$$ is the operator mean of observable X in the system state ψ and $${\displaystyle {\mathrm{\Delta }}_{\psi }\hat{X}=\sqrt{⟨{\hat{X}}^2{⟩}_{\psi }-⟨\hat{X}{⟩}_{\psi }^2}.}$$

Here $${\displaystyle \left[\hat{A},\hat{B}\right]\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}\hat{A}\hat{B}-\hat{B}\hat{A}}$$ is called the commutator of A and B.

This is a purely mathematical result. No reference has been made to any physical quantity or principle. It simply states that the uncertainty of one operator times the uncertainty of another operator has a lower bound.

Application to angular momentum

The connection to physics can be made if we identify the operators with physical operators such as the angular momentum and the polarization angle. We have then $${\displaystyle {\mathrm{\Delta }}_{\psi }{\hat{L}}_z{\mathrm{\Delta }}_{\psi }\theta \ge \frac{\hslash }{2},}$$ which means that angular momentum and the polarization angle cannot be measured simultaneously with infinite accuracy. (The polarization angle can be measured by checking whether the photon can pass through a polarizing filter oriented at a particular angle, or a polarizing beam splitter. This results in a yes/no answer that, if the photon was plane-polarized at some other angle, depends on the difference between the two angles.)

States, probability amplitudes, unitary and Hermitian operators, and eigenvectors

Much of the mathematical apparatus of quantum mechanics appears in the classical description of a polarized sinusoidal electromagnetic wave. The Jones vector for a classical wave, for instance, is identical with the quantum polarization state vector for a photon. The right and left circular components of the Jones vector can be interpreted as probability amplitudes of spin states of the photon. Energy conservation requires that the states be transformed with a unitary operation. This implies that infinitesimal transformations are transformed with a Hermitian operator. These conclusions are a natural consequence of the structure of Maxwell's equations for classical waves.

Quantum mechanics enters the picture when observed quantities are measured and found to be discrete rather than continuous. The allowed observable values are determined by the eigenvalues of the operators associated with the observable. In the case angular momentum, for instance, the allowed observable values are the eigenvalues of the spin operator.

These concepts have emerged naturally from Maxwell's equations and Planck's and Einstein's theories. They have been found to be true for many other physical systems. In fact, the typical program is to assume the concepts of this section and then to infer the unknown dynamics of a physical system. This was done, for instance, with the dynamics of electrons. In that case, working back from the principles in this section, the quantum dynamics of particles were inferred, leading to the Schrödinger equation, a departure from Newtonian mechanics. The solution of this equation for atoms led to the explanation of the Balmer series for atomic spectra and consequently formed a basis for all of atomic physics and chemistry.

This is not the only occasion in which Maxwell's equations have forced a restructuring of Newtonian mechanics. Maxwell's equations are relativistically consistent. Special relativity resulted from attempts to make classical mechanics consistent with Maxwell's equations (see, for example, Moving magnet and conductor problem).