Economy

From Wikipedia, the free encyclopedia

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

Gross domestic product per capita of countries (2024, PPP)

•   >$60,000
•   $50,000 – $60,000
•   $40,000 – $50,000
•   $30,000 – $40,000
•   $20,000 – $30,000
•   $10,000 – $20,000
•   $5,000 – $10,000
•   $2,500 – $5,000
•   $1,000 – $2,500
•   <$1,000
•   No data

An economy is an area of the production, distribution and trade, as well as consumption of goods and services. In general, it is defined as a social domain that emphasize the practices, discourses, and material expressions associated with the production, use, and management of resources. A given economy is a set of processes that involves its culture, values, education, technological evolution, history, social organization, political structure, legal systems, and natural resources as main factors. These factors give context, content, and set the conditions and parameters in which an economy functions. In other words, the economic domain is a social domain of interrelated human practices and transactions that does not stand alone.

Economic agents can be individuals, businesses, organizations, or governments. Economic transactions occur when two groups or parties agree to the value or price of the transacted good or service, commonly expressed in a certain currency. However, monetary transactions only account for a small part of the economic domain.

Economic activity is spurred by production which uses natural resources, labor and capital. It has changed over time due to technology, innovation (new products, services, processes, expanding markets, diversification of markets, niche markets, increases revenue functions) and changes in industrial relations (most notably child labor being replaced in some parts of the world with universal access to education).

Etymology

New York City, the world's principal fintech and financial center and the epicenter of the world's principal metropolitan economy

The word economy in English is derived from the Middle French's yconomie, which itself derived from the Medieval Latin's oeconomia. The Latin word has its origin at the Ancient Greek's oikonomia or oikonomos. The word's first part oikos means "house", and the second part nemein means "to manage".

The most frequently used current sense, denoting "the economic system of a country or an area", seems not to have developed until the 1650s.

History

Main article: Economic history

Earliest roots

Ancient Roman mosaic from Bosra, depicting a merchant leading camels through the desert

As long as someone has been making, supplying and distributing goods or services, there has been some sort of economy; economies grew larger as societies grew and became more complex. Sumer developed a large-scale economy based on commodity money, while the Babylonians and their neighboring city states later developed the earliest system of economics as we think of, in terms of rules/laws on debt, legal contracts and law codes relating to business practices, and private property.

The Babylonians and their city state neighbors developed forms of economics comparable to currently used civil society (law) concepts. They developed the first known codified legal and administrative systems, complete with courts, jails, and government records.

The ancient economy was based primarily on subsistence farming. The Shekel are the first to refer to a unit of weight and currency, used by the Semitic peoples. The first usage of the term came from Mesopotamia circa 3000 BC. and referred to a specific mass of barley which related other values in a metric such as silver, bronze, copper, etc. A barley/shekel was originally both a unit of currency and a unit of weight, just as the British Pound was originally a unit denominating a one-pound mass of silver.

Most exchange of goods had occurred through social relationships. There were also traders who bartered in the marketplaces. In Ancient Greece, where the present English word 'economy' originated, many people were bond slaves of the freeholders. The economic discussion was driven by scarcity.

In Chinese economic law, the huge cycle of institutional innovation contains an idea. Serving a non-market economy promotes a firm's tenure that is legally guaranteed and protected from bureaucratic opportunities.

Middle Ages

In the Middle Ages, what is now known as an economy was not far from the subsistence level. Most exchange occurred within social groups. On top of this, the great conquerors raised what we now call venture capital (from ventura, ital.; risk) to finance their captures. The capital should be refunded by the goods they would bring up in the New World. The discoveries of Marco Polo (1254–1324), Christopher Columbus (1451–1506) and Vasco da Gama (1469–1524) led to a first global economy. The first enterprises were trading establishments. In 1513, the first stock exchange was founded in Antwerp. Economy at the time meant primarily trade.

The European captures became branches of the European states, the so-called colonies. The rising nation-states Spain, Portugal, France, Great Britain and the Netherlands tried to control the trade through custom duties and mercantilism (from mercator, lat.: merchant) was a first approach to intermediate between private wealth and public interest. The secularization in Europe allowed states to use the immense property of the church for the development of towns. The influence of the nobles decreased. The first Secretaries of State for economy started their work. Bankers like Amschel Mayer Rothschild (1773–1855) started to finance national projects such as wars and infrastructure. Economy from then on meant national economy as a topic for the economic activities of the citizens of a state.

Industrial Revolution

The first economist in the true modern meaning of the word was the Scotsman Adam Smith (1723–1790) who was inspired partly by the ideas of physiocracy, a reaction to mercantilism and also later Economics student, Adam Mari. He defined the elements of a national economy: products are offered at a natural price generated by the use of competition - supply and demand - and the division of labor. He maintained that the basic motive for free trade is human self-interest. The so-called self-interest hypothesis became the anthropological basis for economics. Thomas Malthus (1766–1834) transferred the idea of supply and demand to the problem of overpopulation.

The Industrial Revolution was a period from the 18th to the 19th century where major changes in agriculture, manufacturing, mining, and transport had a profound effect on the socioeconomic and cultural conditions starting in the United Kingdom, then subsequently spreading throughout Europe, North America, and eventually the world. The onset of the Industrial Revolution marked a major turning point in human history; almost every aspect of daily life was eventually influenced in some way. In Europe wild capitalism started to replace the system of mercantilism (today: protectionism) and led to economic growth. The period is called the Industrial Revolution because the system of production and division of labor enabled the mass production of goods.

20th century

The contemporary concept of "the economy" wasn't popularly known until the American Great Depression in the 1930s.

After the chaos of two World Wars and the devastating Great Depression, policymakers searched for new ways of controlling the course of the economy. This was explored and discussed by Friedrich August von Hayek (1899–1992) and Milton Friedman (1912–2006) who pleaded for a global free trade and are supposed to be the fathers of the so-called neoliberalism. However, the prevailing view was that held by John Maynard Keynes (1883–1946), who argued for a stronger control of the markets by the state. The theory that the state can alleviate economic problems and instigate economic growth through state manipulation of aggregate demand is called Keynesianism in his honor. In the late 1950s, the economic growth in America and Europe—often called Wirtschaftswunder (German for economic miracle) —brought up a new form of economy: mass consumption economy. In 1958, John Kenneth Galbraith (1908–2006) was the first to speak of an affluent society in his book The Affluent Society. In most of the countries the economic system is called a social market economy.

21st century

Frankfurt Stock Exchange in 2015

With the fall of the Iron Curtain and the transition of the countries of the Eastern Bloc towards democratic government and market economies, the idea of the post-industrial society is brought into importance as its role is to mark together the significance that the service sector receives instead of industrialization. Some attribute the first use of this term to Daniel Bell's 1973 book, The Coming of Post-Industrial Society, while others attribute it to social philosopher Ivan Illich's book, Tools for Conviviality. The term is also applied in philosophy to designate the fading of postmodernism in the late 90s and especially in the beginning of the 21st century.

With the spread of Internet as a mass media and communication medium especially after 2000–2001, the idea for the Internet and information economy is given place because of the growing importance of e-commerce and electronic businesses, also the term for a global information society as understanding of a new type of "all-connected" society is created. In the late 2000s, the new type of economies and economic expansions of countries like China, Brazil, and India bring attention and interest to economies different from the usually dominating Western-type economies and economic models.

Elements

Types

A market economy is one where goods and services are produced and exchanged according to demand and supply between participants (economic agents) by barter or a medium of exchange with a credit or debit value accepted within the network, such as a unit of currency. A planned economy is one where political agents directly control what is produced and how it is sold and distributed. A green economy is low-carbon and resource efficient. In a green economy, growth in income and employment is driven by public and private investments that reduce carbon emissions and pollution, enhance energy and resource efficiency, and prevent the loss of biodiversity and ecosystem services. A gig economy is one in which short-term jobs are assigned or chosen on-demand. The global economy refers to humanity's economic system or systems overall. An informal economy is neither taxed nor monitored by any form of government. A local economy is an economy centred around a particular settlement or commercial area, and may be a driver for local purchasing being promoted and practiced in its area.

Sectors

The economy may be considered as having developed through the following phases or degrees of precedence:

• The ancient economy was mainly based on subsistence farming.
• The Industrial Revolution phase lessened the role of subsistence farming, converting it to more extensive and mono-cultural forms of agriculture in the last three centuries. The economic growth took place mostly in mining, construction and manufacturing industries. Commerce became more significant due to the need for improved exchange and distribution of produce throughout the community.
• In the economies of modern consumer societies phase there is a growing part played by services, finance, and technology—the knowledge economy.

In modern economies, these phase precedences are somewhat differently expressed by the three-sector model:

• Primary: Involves the extraction and production of raw materials, such as corn, coal, wood and iron.
• Secondary: Involves the transformation of raw or intermediate materials into goods e.g. manufacturing steel into cars, or textiles into clothing.
• Tertiary: Involves the provision of services to consumers and businesses, such as baby-sitting, cinema and banking.

Other sectors of the developed community include:

• the public sector or state sector (which usually includes: parliament, law-courts and government centers, various emergency services, public health, shelters for impoverished and threatened people, transport facilities, air/sea ports, post-natal care, hospitals, schools, libraries, museums, preserved historical buildings, parks/gardens, nature-reserves, some universities, national sports grounds/stadiums, national arts/concert-halls or theaters and centers for various religions).
• the private sector or privately run businesses.
• the voluntary sector or social sector.

Indicators

Main article: Economic indicator

The gross domestic product (GDP) of a country is a measure of the size of its economy, or more specifically, monetary measure of the market value of all the final goods and services produced. The most conventional economic analysis of a country relies heavily on economic indicators like the GDP and GDP per capita. While often useful, GDP only includes economic activity for which money is exchanged.

Due to the growing importance of the financial sector in modern times, the term real economy is used by analysts as well as politicians to denote the part of the economy that is concerned with the actual production of goods and services, as ostensibly contrasted with the paper economy, or the financial side of the economy, which is concerned with buying and selling on the financial markets. Alternate and long-standing terminology distinguishes measures of an economy expressed in real values (adjusted for inflation), such as real GDP, or in nominal values (unadjusted for inflation).

Studies

Main article: Economics

The study of economics are roughly divided into macroeconomics and microeconomics. Today, the range of fields of study examining the economy revolves around the social science of economics, but may also include sociology, history, anthropology, and geography. Practical fields directly related to the human activities involving production, distribution, exchange, and consumption of goods and services as a whole are business, engineering, government, and health care. Macroeconomics is studied at the regional and national levels, and common analyses include income and production, money, prices, employment, international trade, and other issues.

Impact bias

From Wikipedia, the free encyclopedia

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

In the psychology of affective forecasting, the impact bias, a form of which is the durability bias, is the tendency for people to overestimate the length or the intensity of future emotional states.

Overview

People display an impact bias when they overestimate the intensity and durability of affect when making predictions about their emotional responses. It is a cognitive bias that has been found in populations ranging from college students (e.g. Dunn, Wilson, & Gilbert, 2003; Buehler & McFarland, 2001), to sports fans (Wilson et al, 2000), to registered voters (Gilbert et al, 1998).

Affective forecasting

Main article: Affective forecasting

Research shows that people often make errors about how much positive or negative effect an event will have on us. People mispredict their emotional reactions (how much pleasure or displeasure an event will bring them) when they mispredict how the event will occur. These mistaken projections can lead to mistaken assumptions about the impact of an event on their happiness. Generally, people accurately predict the valence, if an event will generate a positive or negative reaction, but people are less accurate in their predictions about the intensity and the duration of these effects.

Mistaken projections

To make decisions, people try to predict how an event will play out and what it will make them feel. However, when making these predictions, people are subject to many cognitive biases, including the impact bias. Research shows that people’s projections are often mistaken (e.g. Wilson & Gilbert, 2005; Buehler & McFarland, 2001; Loewenstein & Schake, 1999). Early studies revealed that this phenomenon is a result of people’s inability to anticipate how their feelings will be affected by external factors, and change over time (e.g. Kahneman, 1994). People are more prone to make errors about intensity and duration of affect, than about valence. These mistaken projections affect people’s ability to make decisions that maximize their happiness.

Example

In Gilbert et al., 1998, there was a conducted study on individuals participating in a job interview. The participants were separated into two groups; the unfair decision condition (where the decision of being hired was left up to a single MBA student with sole authority listening to the interview) and the fair decision condition (where the decision was made by a team of MBA students who had to independently and unanimously decide the fate of the interviewee). Then, certain participants were chosen to forecast how they would feel if they were chosen or not chosen for the job immediately after learning if they had been hired or fired and then they had to predict how they would feel ten minutes after hearing the news. Then following the interview, all participants were given letters notifying them they had not been selected for the job. All participants were then required to fill out a questionnaire that reported their current happiness. Then after waiting ten minutes, the experimenter presented all the participants with another questionnaire that once again asked them to report their current level of happiness. The predictions made by the participants in both the unfair and fair groups were about the same regarding how they would feel immediately after hearing the news as well as ten minutes later. Both groups accurately predicted how they would feel immediately after hearing the news. The study showed that both groups felt much better than they had originally predicted, ten minutes later, demonstrating the impact bias.

Causes

Explanations for the occurrence of the impact bias include the following:

Misconstrual problem

Misconstrual of future events: When predicting how an experience will impact us emotionally, events which have not been experienced are particularly difficult. Often what we think an event may be like does not relate to what the experience is like. People know that the future is uncertain, but fail to recognize their projections as construals, subjective perceptions or interpretations of reality. (Griffin & Ross, 1998; Wilson & Gilbert, 2003). Misconstruals can be responsible for a great range of mispredictions because there is no limit to the degree of error one can make. It leads to the impact bias when misconstruals are accurate about the valence of an event, but overpredict intensity and duration of emotional reaction.

Inaccurate theories: People have created cultural theories and had experiences that greatly influence beliefs of how an event will affect us. For example, American culture has emphasized a correlation between wealth and happiness, however despite this belief; money does not necessarily bring happiness.

Motivated Distortions: When faced with a negative event people may have forecasts that are overestimated and can evoke either comfort or fear in the present. The overestimation however can often be used to soften the effects of an event or make it easier by the reality not being as extreme as the forecasted impact.

Under correction (anchoring and adjustment): People anchor their prediction on how they currently feel and never accurately adjust their predictions. An illustrative example proposed by Wilson and Gilbert (2005), is that if you are currently in bed with a cold, and are invited to a party a month from now, it will be very difficult to separate your negative feelings from your prediction of how you will feel on a Saturday night a month later.^[4] This process is sometimes referred to as the projection bias (Loewenstein et al., 1999) whereby people’s affective forecasts are unconsciously or consciously influenced by their current state.

Focalism

Often when making a prediction of the impact of an event people focus solely on the event in question. This ignores the fact that with the passage of time, other events will occur that influence happiness. Disregarding the effect of unrelated events on future thoughts emotions, leads to erroneous predictions of our emotional reactions. Whereby, since we are focusing on the impact on one specific event, we simultaneously overestimate the intensity and duration of our emotional reaction to that event and underestimate the effect of other unrelated events. However, we fail to consider that these unrelated events can moderate our emotional responses.

Distinction bias

Cognitive bias whereby people focus too much on the differences between two future events instead of the shared features. This bias leads to the impact bias when people focus too much on a distinction that does affect their future happiness instead of focusing on features that do. As a result, people overestimate the impact of that difference on their well being. For example, a study asked college students to predict how happy they would be a year later if assigned to a desired or undesired dormitory. Results showed that the students largely overestimated their unhappiness when assigned to an undesired dormitory, as their overall happiness was nearly identical to those living in desired houses the following year.

Sense-making

People fail to recognize how quickly they will make sense of an event, thereby, failing to anticipate the deceleration of emotional reactions. Research suggests that there are four processes by which our psychological immune system deals with unpredicted and poorly understood events: First, they pay a lot of attention to the event, then, they react emotionally, they attempt to make sense of the event, and finally, they adapt emotionally. Failure to recognize that these processes will occur or how fast they will occur causes people to overestimate the impact (and the duration of impact) of such events.

Immune Neglect: We have unconscious psychological processes such as ego defense, dissonance reductions, self-serving biases, etc. that will cushion the effects of a negative event. When making predictions, people are generally unaware of these unconscious processes, and fail to take them into account when making affective forecasts.

Consequences

In the context of decision-making, the impact bias has important consequences. When making decisions (ranging from deciding whether to move to California or not to deciding whether to bike or drive to work), people attempt to predict the outcome of their decisions by projecting their emotional reactions to future events (e.g.: “How will this make me feel?”). Thereby, they base their decisions on affective forecasts (Wilson & Gilbert, 2005). Erroneous projections about future emotional reactions, such as overestimating the intensity and durability of affect (i.e.: impact bias) can lead to mistaken projections. These mistakes impact decisions, and misguide people into making decisions that are not compatible with their future states and may be harmful to their wellbeing.

The impact bias can also result in error in recollecting memories. People display retrospective impact bias when they overestimate the intensity and duration of an emotional reaction to a past event. This can lead to errors in decision making because it can lead people overestimating how an event positively or negatively impacts their wellbeing. Furthermore, people are influenced by their current emotion when recalling their past emotions. This can explain why people do not learn from their mistakes, and make more accurate forecasts. This phenomenon can cause people to make irrational or unbalanced decisions because they recollect that an example was positive (or negative), but fail to recollect the degree of positive (or negative) effect, leading to an inaccurate cost-benefit analysis.

Development in children

Recent evidence suggests that 3, 4 and 5 year-old children show an impact bias for the intensity of their negative future emotions, but not their positive future emotions.

Pauli exclusion principle

From Wikipedia, the free encyclopedia

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

Wolfgang Pauli during a lecture in Copenhagen (1929). Wolfgang Pauli formulated the Pauli exclusion principle.

In quantum mechanics, the Pauli exclusion principle (German: Pauli-Ausschlussprinzip) states that two or more identical particles with half-integer spins (i.e. fermions) cannot simultaneously occupy the same quantum state within a system that obeys the laws of quantum mechanics. This principle was formulated by Austrian physicist Wolfgang Pauli in 1925 for electrons, and later extended to all fermions with his spin–statistics theorem of 1940.

In the case of electrons in atoms, the exclusion principle can be stated as follows: in a poly-electron atom it is impossible for any two electrons to have the same two values of all four of their quantum numbers, which are: n, the principal quantum number; ℓ, the azimuthal quantum number; m_ℓ, the magnetic quantum number; and m_s, the spin quantum number. For example, if two electrons reside in the same orbital, then their values of n, ℓ, and m_ℓ are equal. In that case, the two values of m_s (spin) pair must be different. Since the only two possible values for the spin projection m_s are +1/2 and −1/2, it follows that one electron must have m_s = +1/2 and one m_s = −1/2.

Particles with an integer spin (bosons) are not subject to the Pauli exclusion principle. Any number of identical bosons can occupy the same quantum state, such as photons produced by a laser, or atoms found in a Bose–Einstein condensate.

A rigorous statement which justifies the exclusion principle is: under the exchange of two identical particles, the total (many-particle) wave function is antisymmetric for fermions and symmetric for bosons. This means that if the space and spin coordinates of two identical particles are interchanged, then the total wave function changes sign (from positive to negative or vice versa) for fermions, but does not change sign for bosons. So, if hypothetically two fermions were in the same state—for example, in the same atom in the same orbital with the same spin—then interchanging them would change nothing and the total wave function would be unchanged. However, the only way a total wave function can both change sign (which is required for fermions), and also remain unchanged, is that such the function must be zero everywhere, which means such a state cannot exist. This reasoning does not apply to bosons because the sign does not change.

Overview

The Pauli exclusion principle describes the behavior of all fermions (particles with half-integer spin), while bosons (particles with integer spin) are subject to other principles. Fermions include elementary particles such as quarks, electrons and neutrinos. Additionally, baryons such as protons and neutrons (subatomic particles composed from three quarks) and some atoms (such as helium-3) are fermions, and are therefore described by the Pauli exclusion principle as well. Atoms can have different overall spin, which determines whether they are fermions or bosons: for example, helium-3 has spin 1/2 and is therefore a fermion, whereas helium-4 has spin 0 and is a boson. The Pauli exclusion principle underpins many properties of everyday matter, from its large-scale stability to the chemical behavior of atoms.

Half-integer spin means that the intrinsic angular momentum value of fermions is ${\displaystyle \hslash =h/2\pi }$ (reduced Planck constant) times a half-integer (1/2, 3/2, 5/2, etc.). In the theory of quantum mechanics, fermions are described by antisymmetric states. In contrast, particles with integer spin (bosons) have symmetric wave functions and may share the same quantum states. Bosons include the photon, the Cooper pairs which are responsible for superconductivity, and the W and Z bosons. Fermions take their name from the Fermi–Dirac statistical distribution, which they obey, and bosons take theirs from the Bose–Einstein distribution.

History

In the early 20th century it became evident that atoms and molecules with even numbers of electrons are more chemically stable than those with odd numbers of electrons. In the 1916 article "The Atom and the Molecule" by Gilbert N. Lewis, for example, the third of his six postulates of chemical behavior states that the atom tends to hold an even number of electrons in any given shell, and especially to hold eight electrons, which he assumed to be typically arranged symmetrically at the eight corners of a cube.^[3] In 1919 chemist Irving Langmuir suggested that the periodic table could be explained if the electrons in an atom were connected or clustered in some manner. Groups of electrons were thought to occupy a set of electron shells around the nucleus. In 1922, Niels Bohr updated his model of the atom by assuming that certain numbers of electrons (for example 2, 8 and 18) corresponded to stable "closed shells".

Pauli looked for an explanation for these numbers, which were at first only empirical. At the same time he was trying to explain experimental results of the Zeeman effect in atomic spectroscopy and in ferromagnetism. He found an essential clue in a 1924 paper by Edmund C. Stoner, which pointed out that, for a given value of the principal quantum number (n), the number of energy levels of a single electron in the alkali metal spectra in an external magnetic field, where all degenerate energy levels are separated, is equal to the number of electrons in the closed shell of the noble gases for the same value of n. This led Pauli to realize that the complicated numbers of electrons in closed shells can be reduced to the simple rule of one electron per state if the electron states are defined using four quantum numbers. For this purpose he introduced a new two-valued quantum number, identified by Samuel Goudsmit and George Uhlenbeck as electron spin.

Connection to quantum state symmetry

In his Nobel lecture, Pauli clarified the importance of quantum state symmetry to the exclusion principle:

Among the different classes of symmetry, the most important ones (which moreover for two particles are the only ones) are the symmetrical class, in which the wave function does not change its value when the space and spin coordinates of two particles are permuted, and the antisymmetrical class, in which for such a permutation the wave function changes its sign...[The antisymmetrical class is] the correct and general wave mechanical formulation of the exclusion principle.

The Pauli exclusion principle with a single-valued many-particle wavefunction is equivalent to requiring the wavefunction to be antisymmetric with respect to exchange. If ${\displaystyle |x⟩}$ and ${\displaystyle |y⟩}$ range over the basis vectors of the Hilbert space describing a one-particle system, then the tensor product produces the basis vectors ${\displaystyle |x,y⟩=|x⟩\otimes |y⟩}$ of the Hilbert space describing a system of two such particles. Any two-particle state can be represented as a superposition (i.e. sum) of these basis vectors:

${\displaystyle |\psi ⟩=\sum _{x,y}A(x,y)|x,y⟩,}$

where each A(x, y) is a (complex) scalar coefficient. Antisymmetry under exchange means that A(x, y) = −A(y, x). This implies A(x, y) = 0 when x = y, which is Pauli exclusion. It is true in any basis since local changes of basis keep antisymmetric matrices antisymmetric.

Conversely, if the diagonal quantities A(x, x) are zero in every basis, then the wavefunction component

${\displaystyle A(x,y)=⟨\psi |x,y⟩=⟨\psi |(|x⟩\otimes |y⟩)}$

is necessarily antisymmetric. To prove it, consider the matrix element

${\displaystyle ⟨\psi |((|x⟩+|y⟩)\otimes (|x⟩+|y⟩)).}$

This is zero, because the two particles have zero probability to both be in the superposition state ${\displaystyle |x⟩+|y⟩}$. But this is equal to

${\displaystyle ⟨\psi |x,x⟩+⟨\psi |x,y⟩+⟨\psi |y,x⟩+⟨\psi |y,y⟩.}$

The first and last terms are diagonal elements and are zero, and the whole sum is equal to zero. So the wavefunction matrix elements obey:

${\displaystyle ⟨\psi |x,y⟩+⟨\psi |y,x⟩=0,}$

or

${\displaystyle A(x,y)=-A(y,x).}$

For a system with n > 2 particles, the multi-particle basis states become n-fold tensor products of one-particle basis states, and the coefficients of the wavefunction ${\displaystyle A(x_1,x_2,\dots ,x_n)}$ are identified by n one-particle states. The condition of antisymmetry states that the coefficients must flip sign whenever any two states are exchanged: ${\displaystyle A(\dots ,x_i,\dots ,x_j,\dots )=-A(\dots ,x_j,\dots ,x_i,\dots )}$ for any ${\displaystyle i\ne j}$. The exclusion principle is the consequence that, if ${\displaystyle x_i=x_j}$ for any ${\displaystyle i\ne j,}$ then ${\displaystyle A(\dots ,x_i,\dots ,x_j,\dots )=0.}$ This shows that none of the n particles may be in the same state.

Advanced quantum theory

According to the spin–statistics theorem, particles with integer spin occupy symmetric quantum states, and particles with half-integer spin occupy antisymmetric states; furthermore, only integer or half-integer values of spin are allowed by the principles of quantum mechanics. In relativistic quantum field theory, the Pauli principle follows from applying a rotation operator in imaginary time to particles of half-integer spin.

In one dimension, bosons, as well as fermions, can obey the exclusion principle. A one-dimensional Bose gas with delta-function repulsive interactions of infinite strength is equivalent to a gas of free fermions. The reason for this is that, in one dimension, the exchange of particles requires that they pass through each other; for infinitely strong repulsion this cannot happen. This model is described by a quantum nonlinear Schrödinger equation. In momentum space, the exclusion principle is valid also for finite repulsion in a Bose gas with delta-function interactions, as well as for interacting spins and Hubbard model in one dimension, and for other models solvable by Bethe ansatz. The ground state in models solvable by Bethe ansatz is a Fermi sphere.

Applications

Atoms

The Pauli exclusion principle helps explain a wide variety of physical phenomena. One particularly important consequence of the principle is the elaborate electron shell structure of atoms and the way atoms share electrons, explaining the variety of chemical elements and their chemical combinations. An electrically neutral atom contains bound electrons equal in number to the protons in the nucleus. Electrons, being fermions, cannot occupy the same quantum state as other electrons, so electrons have to "stack" within an atom, i.e. have different spins while at the same electron orbital as described below.

An example is the neutral helium atom (He), which has two bound electrons, both of which can occupy the lowest-energy (1s) states by acquiring opposite spin; as spin is part of the quantum state of the electron, the two electrons are in different quantum states and do not violate the Pauli principle. However, the spin can take only two different values (eigenvalues). In a lithium atom (Li), with three bound electrons, the third electron cannot reside in a 1s state and must occupy a higher-energy state instead. The lowest available state is 2s, so that the ground state of Li is 1s²2s. Similarly, successively larger elements must have shells of successively higher energy. The chemical properties of an element largely depend on the number of electrons in the outermost shell; atoms with different numbers of occupied electron shells but the same number of electrons in the outermost shell have similar properties, which gives rise to the periodic table of the elements.

To test the Pauli exclusion principle for the helium atom, Gordon Drake of the University of Windsor carried out very precise calculations for hypothetical states of the He atom that violate it, which are called paronic states. Later, K. Deilamian et al. used an atomic beam spectrometer to search for the paronic state 1s2s ¹S₀ calculated by Drake. The search was unsuccessful and showed that the statistical weight of this paronic state has an upper limit of 5×10⁻⁶. (The exclusion principle implies a weight of zero.)

Solid state properties

In conductors and semiconductors, there are very large numbers of molecular orbitals which effectively form a continuous band structure of energy levels. In strong conductors (metals) electrons are so degenerate that they cannot even contribute much to the thermal capacity of a metal. Many mechanical, electrical, magnetic, optical and chemical properties of solids are the direct consequence of Pauli exclusion.

Stability of matter

Further information: Stability of matter

The stability of each electron state in an atom is described by the quantum theory of the atom, which shows that close approach of an electron to the nucleus necessarily increases the electron's kinetic energy, an application of the uncertainty principle of Heisenberg. However, stability of large systems with many electrons and many nucleons is a different question, and requires the Pauli exclusion principle.

It has been shown that the Pauli exclusion principle is responsible for the fact that ordinary bulk matter is stable and occupies volume. This suggestion was first made in 1931 by Paul Ehrenfest, who pointed out that the electrons of each atom cannot all fall into the lowest-energy orbital and must occupy successively larger shells. Atoms, therefore, occupy a volume and cannot be squeezed too closely together.

The first rigorous proof was provided in 1967 by Freeman Dyson and Andrew Lenard (de), who considered the balance of attractive (electron–nuclear) and repulsive (electron–electron and nuclear–nuclear) forces and showed that ordinary matter would collapse and occupy a much smaller volume without the Pauli principle. A much simpler proof was found later by Elliott H. Lieb and Walter Thirring in 1975. They provided a lower bound on the quantum energy in terms of the Thomas-Fermi model, which is stable due to a theorem of Teller. The proof used a lower bound on the kinetic energy which is now called the Lieb–Thirring inequality.

The consequence of the Pauli principle here is that electrons of the same spin are kept apart by a repulsive exchange interaction, which is a short-range effect, acting simultaneously with the long-range electrostatic or Coulombic force. This effect is partly responsible for the everyday observation in the macroscopic world that two solid objects cannot be in the same place at the same time.

Astrophysics

Dyson and Lenard did not consider the extreme magnetic or gravitational forces that occur in some astronomical objects. In 1995 Elliott Lieb and coworkers showed that the Pauli principle still leads to stability in intense magnetic fields such as in neutron stars, although at a much higher density than in ordinary matter. It is a consequence of general relativity that, in sufficiently intense gravitational fields, matter collapses to form a black hole.

Astronomy provides a spectacular demonstration of the effect of the Pauli principle, in the form of white dwarf and neutron stars. In both bodies, the atomic structure is disrupted by extreme pressure, but the stars are held in hydrostatic equilibrium by degeneracy pressure, also known as Fermi pressure. This exotic form of matter is known as degenerate matter. The immense gravitational force of a star's mass is normally held in equilibrium by thermal pressure caused by heat produced in thermonuclear fusion in the star's core. In white dwarfs, which do not undergo nuclear fusion, an opposing force to gravity is provided by electron degeneracy pressure. In neutron stars, subject to even stronger gravitational forces, electrons have merged with protons to form neutrons. Neutrons are capable of producing an even higher degeneracy pressure, neutron degeneracy pressure, albeit over a shorter range. This can stabilize neutron stars from further collapse, but at a smaller size and higher density than a white dwarf. Neutron stars are the most "rigid" objects known; their Young modulus (or more accurately, bulk modulus) is 20 orders of magnitude larger than that of diamond. However, even this enormous rigidity can be overcome by the gravitational field of a neutron star mass exceeding the Tolman–Oppenheimer–Volkoff limit, leading to the formation of a black hole.