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.

Biological naturalism

From Wikipedia, the free encyclopedia

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

Biological naturalism is a theory about, among other things, the relationship between consciousness and body (i.e., brain), and hence an approach to the mind–body problem. It was first proposed by the philosopher John Searle in 1980 and is defined by two main theses: 1) all mental phenomena, ranging from pains, tickles, and itches to the most abstruse thoughts, are caused by lower-level neurobiological processes in the brain; and 2) mental phenomena are higher-level features of the brain.

This entails that the brain has the right causal powers to produce intentionality. However, Searle's biological naturalism does not entail that brains and only brains can cause consciousness. Searle is careful to point out that while it appears to be the case that certain brain functions are sufficient for producing conscious states, our current state of neurobiological knowledge prevents us from concluding that they are necessary for producing consciousness. In his own words:

"The fact that brain processes cause consciousness does not imply that only brains can be conscious. The brain is a biological machine, and we might build an artificial machine that was conscious; just as the heart is a machine, and we have built artificial hearts. Because we do not know exactly how the brain does it we are not yet in a position to know how to do it artificially." ("Biological Naturalism", 2004)

Overview

John Searle

Searle denies Cartesian dualism, the idea that the mind is a separate kind of substance to the body, as this contradicts our entire understanding of physics, and unlike Descartes, he does not bring God into the problem. Indeed, Searle denies any kind of dualism, the traditional alternative to monism, claiming the distinction is a mistake. He rejects the idea that because the mind is not objectively viewable, it does not fall under the rubric of physics.

Searle believes that consciousness "is a real part of the real world and it cannot be eliminated in favor of, or reduced to, something else" whether that something else is a neurological state of the brain or a computer program. He contends, for example, that the software known as Deep Blue knows nothing about chess. He also believes that consciousness is both a cause of events in the body and a response to events in the body.

On the other hand, Searle doesn't treat consciousness as a ghost in the machine. He treats it, rather, as a state of the brain. The causal interaction of mind and brain can be described thus in naturalistic terms: Events at the micro-level (perhaps at that of individual neurons) cause consciousness. Changes at the macro-level (the whole brain) constitute consciousness. Micro-changes cause and then are impacted by holistic changes, in much the same way that individual football players cause a team (as a whole) to win games, causing the individuals to gain confidence from the knowledge that they are part of a winning team.

He articulates this distinction by pointing out that the common philosophical term 'reducible' is ambiguous. Searle contends that consciousness is "causally reducible" to brain processes without being "ontologically reducible". He hopes that making this distinction will allow him to escape the traditional dilemma between reductive materialism and substance dualism; he affirms the essentially physical nature of the universe by asserting that consciousness is completely caused by and realized in the brain, but also doesn't deny what he takes to be the obvious facts that humans really are conscious, and that conscious states have an essentially first-person nature.

It can be tempting to see the theory as a kind of property dualism, since, in Searle's view, a person's mental properties are categorically different from his or her micro-physical properties. The latter have "third-person ontology" whereas the former have "first-person ontology." Micro-structure is accessible objectively by any number of people, as when several brain surgeons inspect a patient's cerebral hemispheres. But pain or desire or belief are accessible subjectively by the person who has the pain or desire or belief, and no one else has that mode of access. However, Searle holds mental properties to be a species of physical property—ones with first-person ontology. So this sets his view apart from a dualism of physical and non-physical properties. His mental properties are putatively physical. (see "Property dualism" under the "Criticism" section below.)

Criticism

There have been several criticisms of Searle's idea of biological naturalism.

Jerry Fodor suggests that Searle gives us no account at all of exactly why he believes that a biochemistry like, or similar to, that of the human brain is indispensable for intentionality. Fodor thinks that it seems much more plausible to suppose that it is the way in which an organism (or any other system for that matter) is connected to its environment that is indispensable in the explanation of intentionality. It is easier to see "how the fact that my thought is causally connected to a tree might bear on its being a thought about a tree. But it's hard to imagine how the fact that (to put it crudely) my thought is made out of hydrocarbons could matter, except on the unlikely hypothesis that only hydrocarbons can be causally connected to trees in the way that brains are."

John Haugeland takes on the central notion of some set of special "right causal powers" that Searle attributes to the biochemistry of the human brain. He asks us to imagine a concrete situation in which the "right" causal powers are those that our neurons have to reciprocally stimulate one another. In this case, silicon-based alien life forms can be intelligent just in case they have these "right" causal powers; i.e. they possess neurons with synaptics connections that have the power to reciprocally stimulate each other. Then we can take any speaker of the Chinese language and cover his neurons in some sort of wrapper which prevents them from being influenced by neurotransmitters and, hence, from having the right causal powers. At this point, "Searle's demon" (an English speaking nanobot, perhaps) sees what is happening and intervenes: he sees through the covering and determines which neurons would have been stimulated and which not and proceeds to stimulate the appropriate neurons and shut down the others himself. The experimental subject's behavior is unaffected. He continues to speak perfect Chinese as before the operation but now the causal powers of his neurotransmitters have been replaced by someone who does not understand the Chinese language. The point is generalizable: for any causal powers, it will always be possible to hypothetically replace them with some sort of Searlian demon which will carry out the operations mechanically. His conclusion is that Searle's is necessarily a dualistic view of the nature of causal powers, "not intrinsically connected with the actual powers of physical objects."

Searle himself does not rule out the possibility for alternate arrangements of matter bringing forth consciousness other than biological brains.

Property dualism

Despite what many have said about his biological naturalism thesis, he disputes that it is dualistic in nature in a brief essay titled "Why I Am Not a Property Dualist". Firstly, he rejects the idea that the mental and physical are primary ontological categories, instead claiming that the act of categorisation is simply a way of speaking about our one world, so whether something is mental or physical is a matter of the vocabulary that one employs. He believes that a more useful distinction can be made between the biological and non-biological, in which case consciousness is a biological process. Secondly, he accepts that the mental is ontologically irreducible to the physical for the simple reason that the former has a first-person ontology and the latter a third-person ontology, but he rejects the property dualist notion of "over and above"; in other words, he believes that, causally speaking, consciousness is entirely reducible to and is nothing more than the neurobiology of the brain (again, because both are biological processes).

Thus, for Searle, the dilemma between epiphenomenalism and causal overdetermination that plagues the property dualist simply does not arise because, causally speaking, there is nothing there except the neurobiology of the brain, but because of the different ontologies of the mental and physical, the former is irreducible to the latter:

I say consciousness is a feature of the brain. The property dualist says consciousness is a feature of the brain. This creates the illusion that we are saying the same thing. But we are not, […]. The property dualist means that in addition to all the neurobiological features of the brain, there is an extra, distinct, non-physical feature of the brain, whereas I mean that consciousness is a state the brain can be in, in the way that liquidity and solidity are states that water can be in.

Buddhism and abortion

From Wikipedia, the free encyclopedia

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

There is no single Buddhist view concerning abortion, although it is generally regarded negatively.

Scriptural views and the monastic code

Inducing or otherwise causing an abortion is regarded as a serious matter in the monastic rules followed by both Theravada and Mahayana monks; monks can be expelled for assisting a woman in procuring an abortion. Traditional sources do not recognize a distinction between early- and late-term abortion, but in Sri Lanka and Thailand the "moral stigma" associated with an abortion grows with the development of the fetus. While traditional sources do not seem to be aware of the possibility of abortion as relevant to the health of the mother, modern Buddhist teachers from many traditions – and abortion laws in many Buddhist countries – recognize a threat to the life or physical health of the mother as an acceptable justification for abortion as a practical matter, though it may still be seen as a deed with negative moral or karmic consequences.

Regional views

Views on abortion vary a great deal between different regions, reflecting the influence of the various Buddhist traditions, as well as the influence of other religious and philosophical traditions and contact with Western thought.

Northern Buddhism

Abortion is generally regarded extremely negatively among ethnic Tibetan Buddhists. Prior to the emergence of the Tibetan diaspora in the 1950s, Tibetans do not seem to have been familiar with abortion for reasons of medical necessity, and, facing little population pressure, saw little reason to engage in what they saw as the destruction of innocent life. Though no systematic information is available, abortion appears to be very rare among exiled Tibetans living in areas where abortion is legal. Tibetan Buddhists believe that a person who has had an abortion should be treated compassionately, and guided to atone for the negative act through appropriate good deeds and religious practices; these acts are aimed at improving the karmic outcome for both the mother and the aborted fetus, but authorities warn that they will not be effective if one has undertaken an abortion while planning to 'negate' it by atoning for it later. The Dalai Lama has said that abortion is "negative," but there are exceptions. He said, "I think abortion should be approved or disapproved according to each circumstance."

Southern Buddhism

Laws and views on abortion vary greatly in Theravada Buddhist nations. Attitudes and laws in Thailand are generally more favourable towards abortion than in Sri Lanka. While abortion is still viewed as negative in Burma (Myanmar), it is allegedly also employed with some frequency to prevent out-of-wedlock births. Regarding attitudes towards abortion in Thailand, Peter Harvey notes:

...abortion is discussed not in the language of rights – to life or choice – but of 'benefit and harm, with the intent of relieving as much human suffering in all its states, stages and situations as circumstances allow', with an emphasis on reducing the circumstances leading women to feel that they need to have an abortion.

In November 2010, the issue of abortion and Buddhism in Thailand was thrust onto the front pages after 2000 fetuses were discovered stored at a temple in Bangkok. At this time, abortion was illegal in the country except in cases of rape or risk to the woman's health. Following the scandal, leading politicians and monks spoke out to reaffirm their opposition to abortion laws. Phramaha Vudhijaya Vajiramedhi was unequivocal: "In [the] Buddhist view, both having an abortion and performing an abortion amount to murder. Those involved in abortions will face distress in both this life and the next because their sins will follow them." Prime Minister Abhisit announced a crackdown on illegal abortion clinics and refused calls to change the law, saying that current laws were "good enough." However, in October 2022, Thailand's Public Health Ministry legalized abortions up to the 20th week of pregnancy – an extension of a previous law which allowed termination of pregnancy within the first 12 weeks. Pro-choice advocates in Thailand and around the world celebrated the new rules as a positive development but noted that more needed to be done to ensure doctors were trained and the public was made aware of their rights to an abortion. Experts note that Thailand's move to expand abortion access comes amid a wave of global expansion of abortion rights in recent years.

Peter Harvey relates attitudes towards abortion in Burma to Melford Spiro's observation that Buddhists in Myanmar recognize a clear distinction between what may be regarded as 'ultimate good' in a religious sense and what is a 'worldly good' or utilitarian act. Despite the prevalence of illegal abortions in Myanmar due to economic difficulty, many Buddhists consider it against their religious beliefs. A 1995 survey on women in Myanmar showed that 99% thought abortion was against their religious beliefs.

East Asia

Buddhists in Japan are said to be more tolerant of abortion than those who live elsewhere. In Japan, women sometimes participate in the Shinto-Buddhist ritual of mizuko kuyō (水子供養, lit. 'fetus memorial service') after an induced abortion or an abortion as the result of a miscarriage.

Similarly, in Taiwan, women sometimes pray to appease ghosts of aborted fetuses and assuage feelings of guilt due to having an abortion; this type of ritual is called yingling gongyang. The modern practice emerged in the mid-1970s and grew significantly in popularity in the 1980s, particularly following the full legalization of abortion in 1985. It draws both from traditional antecedents dating back to the Han dynasty, and the Japanese practice. These modern practices emerged in the context of demographic change associated with modernization – rising population, urbanization, and decreasing family size – together with changing attitudes towards sexuality, which occurred first in Japan, and then in Taiwan, hence the similar response and Taiwan's taking inspiration from Japan.