Einstein–Podolsky–Rosen paradox

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https://en.wikipedia.org/wiki/Einstein%E2%80%93Podolsky%E2%80%93Rosen_paradox

Albert Einstein

The Einstein–Podolsky–Rosen (EPR) paradox is a thought experiment proposed by physicists Albert Einstein, Boris Podolsky and Nathan Rosen, which argues that the description of physical reality provided by quantum mechanics is incomplete. In a 1935 paper titled "Can Quantum-Mechanical Description of Physical Reality be Considered Complete?", they argued for the existence of "elements of reality" that were not part of quantum theory, and speculated that it should be possible to construct a theory containing these hidden variables. Resolutions of the paradox have important implications for the interpretation of quantum mechanics.

The thought experiment involves a pair of particles prepared in what would later become known as an entangled state. Einstein, Podolsky, and Rosen pointed out that, in this state, if the position of the first particle were measured, the result of measuring the position of the second particle could be predicted. If instead the momentum of the first particle were measured, then the result of measuring the momentum of the second particle could be predicted. They argued that no action taken on the first particle could instantaneously affect the other, since this would involve information being transmitted faster than light, which is impossible according to the theory of relativity. They invoked a principle, later known as the "EPR criterion of reality", which posited that: "If, without in any way disturbing a system, we can predict with certainty (i.e., with probability equal to unity) the value of a physical quantity, then there exists an element of reality corresponding to that quantity." From this, they inferred that the second particle must have a definite value of both position and of momentum prior to either quantity being measured. But quantum mechanics considers these two observables incompatible and thus does not associate simultaneous values for both to any system. Einstein, Podolsky, and Rosen therefore concluded that quantum theory does not provide a complete description of reality.

The "Paradox" paper

The term "Einstein–Podolsky–Rosen paradox" or "EPR" arose from a paper written in 1934 after Einstein joined the Institute for Advanced Study, having fled the rise of Nazi Germany. The original paper purports to describe what must happen to "two systems I and II, which we permit to interact", and after some time "we suppose that there is no longer any interaction between the two parts." The EPR description involves "two particles, A and B, [which] interact briefly and then move off in opposite directions." According to Heisenberg's uncertainty principle, it is impossible to measure both the momentum and the position of particle B exactly; however, it is possible to measure the exact position of particle A. By calculation, therefore, with the exact position of particle A known, the exact position of particle B can be known. Alternatively, the exact momentum of particle A can be measured, so the exact momentum of particle B can be worked out. As Manjit Kumar writes, "EPR argued that they had proved that ... [particle] B can have simultaneously exact values of position and momentum. ... Particle B has a position that is real and a momentum that is real. EPR appeared to have contrived a means to establish the exact values of either the momentum or the position of B due to measurements made on particle A, without the slightest possibility of particle B being physically disturbed."

EPR tried to set up a paradox to question the range of true application of quantum mechanics: quantum theory predicts that both values cannot be known for a particle, and yet the EPR thought experiment purports to show that they must both have determinate values. The EPR paper says: "We are thus forced to conclude that the quantum-mechanical description of physical reality given by wave functions is not complete." The EPR paper ends by saying: "While we have thus shown that the wave function does not provide a complete description of the physical reality, we left open the question of whether or not such a description exists. We believe, however, that such a theory is possible." The 1935 EPR paper condensed the philosophical discussion into a physical argument. The authors claim that given a specific experiment, in which the outcome of a measurement is known before the measurement takes place, there must exist something in the real world, an "element of reality", that determines the measurement outcome. They postulate that these elements of reality are, in modern terminology, local, in the sense that each belongs to a certain point in spacetime. Each element may, again in modern terminology, only be influenced by events that are located in the backward light cone of its point in spacetime (i.e. in the past). These claims are thus founded on assumptions about nature that constitute what is now known as local realism.

Article headline regarding the EPR paradox paper in the May 4, 1935, issue of The New York Times

Though the EPR paper has often been taken as an exact expression of Einstein's views, it was primarily authored by Podolsky, based on discussions at the Institute for Advanced Study with Einstein and Rosen. Einstein later expressed to Erwin Schrödinger that, "it did not come out as well as I had originally wanted; rather, the essential thing was, so to speak, smothered by the formalism." Einstein would later go on to present an individual account of his local realist ideas. Shortly before the EPR paper appeared in the Physical Review, The New York Times ran a news story about it, under the headline "Einstein Attacks Quantum Theory". The story, which quoted Podolsky, irritated Einstein, who wrote to the Times, "Any information upon which the article 'Einstein Attacks Quantum Theory' in your issue of May 4 is based was given to you without authority. It is my invariable practice to discuss scientific matters only in the appropriate forum and I deprecate advance publication of any announcement in regard to such matters in the secular press."

The Times story also sought out comment from physicist Edward Condon, who said, "Of course, a great deal of the argument hinges on just what meaning is to be attached to the word 'reality' in physics." The physicist and historian Max Jammer later noted, "[I]t remains a historical fact that the earliest criticism of the EPR paper – moreover, a criticism that correctly saw in Einstein's conception of physical reality the key problem of the whole issue – appeared in a daily newspaper prior to the publication of the criticized paper itself."

Bohr's reply

The publication of the paper prompted a response by Niels Bohr, which he published in the same journal (Physical Review), in the same year, using the same title. (This exchange was only one chapter in a prolonged debate between Bohr and Einstein about the nature of quantum reality.) He argued that EPR had reasoned fallaciously. Bohr said measurements of position and of momentum are complementary, meaning the choice to measure one excludes the possibility of measuring the other. Consequently, a fact deduced regarding one arrangement of laboratory apparatus could not be combined with a fact deduced by means of the other, and so, the inference of predetermined position and momentum values for the second particle was not valid. Bohr concluded that EPR's "arguments do not justify their conclusion that the quantum description turns out to be essentially incomplete."

Einstein's own argument

In his own publications and correspondence, Einstein indicated that he was not satisfied with the EPR paper and that Podolsky had authored most of it. He later used a different argument to insist that quantum mechanics is an incomplete theory. He explicitly de-emphasized EPR's attribution of "elements of reality" to the position and momentum of particle B, saying that "I couldn't care less" whether the resulting states of particle B allowed one to predict the position and momentum with certainty.

For Einstein, the crucial part of the argument was the demonstration of nonlocality, that the choice of measurement done in particle A, either position or momentum, would lead to two different quantum states of particle B. He argued that, because of locality, the real state of particle B could not depend on which kind of measurement was done in A and that the quantum states therefore cannot be in one-to-one correspondence with the real states. Einstein struggled unsuccessfully for the rest of his life to find a theory that could better comply with his idea of locality.

Later developments

Bohm's variant

In 1951, David Bohm proposed a variant of the EPR thought experiment in which the measurements have discrete ranges of possible outcomes, unlike the position and momentum measurements considered by EPR. The EPR–Bohm thought experiment can be explained using electron–positron pairs. Suppose we have a source that emits electron–positron pairs, with the electron sent to destination A, where there is an observer named Alice, and the positron sent to destination B, where there is an observer named Bob. According to quantum mechanics, we can arrange our source so that each emitted pair occupies a quantum state called a spin singlet. The particles are thus said to be entangled. This can be viewed as a quantum superposition of two states, which we call state I and state II. In state I, the electron has spin pointing upward along the z-axis (+z) and the positron has spin pointing downward along the z-axis (−z). In state II, the electron has spin −z and the positron has spin +z. Because it is in a superposition of states, it is impossible without measuring to know the definite state of spin of either particle in the spin singlet.

The EPR thought experiment, performed with electron–positron pairs. A source (center) sends particles toward two observers, electrons to Alice (left) and positrons to Bob (right), who can perform spin measurements.

Alice now measures the spin along the z-axis. She can obtain one of two possible outcomes: +z or −z. Suppose she gets +z. Informally speaking, the quantum state of the system collapses into state I. The quantum state determines the probable outcomes of any measurement performed on the system. In this case, if Bob subsequently measures spin along the z-axis, there is 100% probability that he will obtain −z. Similarly, if Alice gets −z, Bob will get +z. There is nothing special about choosing the z-axis: according to quantum mechanics the spin singlet state may equally well be expressed as a superposition of spin states pointing in the x direction.

Whatever axis their spins are measured along, they are always found to be opposite. In quantum mechanics, the x-spin and z-spin are "incompatible observables", meaning the Heisenberg uncertainty principle applies to alternating measurements of them: a quantum state cannot possess a definite value for both of these variables. Suppose Alice measures the z-spin and obtains +z, so that the quantum state collapses into state I. Now, instead of measuring the z-spin as well, Bob measures the x-spin. According to quantum mechanics, when the system is in state I, Bob's x-spin measurement will have a 50% probability of producing +x and a 50% probability of -x. It is impossible to predict which outcome will appear until Bob actually performs the measurement. Therefore, Bob's positron will have a definite spin when measured along the same axis as Alice's electron, but when measured in the perpendicular axis its spin will be uniformly random. It seems as if information has propagated (faster than light) from Alice's apparatus to make Bob's positron assume a definite spin in the appropriate axis.

Bell's theorem

Main article: Bell's theorem

In 1964, John Stewart Bell published a paper investigating the puzzling situation at that time: on one hand, the EPR paradox purportedly showed that quantum mechanics was nonlocal, and suggested that a hidden-variable theory could heal this nonlocality. On the other hand, David Bohm had recently developed the first successful hidden-variable theory, but it had a grossly nonlocal character. Bell set out to investigate whether it was indeed possible to solve the nonlocality problem with hidden variables, and found out that first, the correlations shown in both EPR's and Bohm's versions of the paradox could indeed be explained in a local way with hidden variables, and second, that the correlations shown in his own variant of the paradox couldn't be explained by any local hidden-variable theory. This second result became known as the Bell theorem.

To understand the first result, consider the following toy hidden-variable theory introduced later by J.J. Sakurai: in it, quantum spin-singlet states emitted by the source are actually approximate descriptions for "true" physical states possessing definite values for the z-spin and x-spin. In these "true" states, the positron going to Bob always has spin values opposite to the electron going to Alice, but the values are otherwise completely random. For example, the first pair emitted by the source might be "(+z, −x) to Alice and (−z, +x) to Bob", the next pair "(−z, −x) to Alice and (+z, +x) to Bob", and so forth. Therefore, if Bob's measurement axis is aligned with Alice's, he will necessarily get the opposite of whatever Alice gets; otherwise, he will get "+" and "−" with equal probability.

Bell showed, however, that such models can only reproduce the singlet correlations when Alice and Bob make measurements on the same axis or on perpendicular axes. As soon as other angles between their axes are allowed, local hidden-variable theories become unable to reproduce the quantum mechanical correlations. This difference, expressed using inequalities known as "Bell's inequalities", is in principle experimentally testable. After the publication of Bell's paper, a variety of experiments to test Bell's inequalities were carried out, notably by the group of Alain Aspect in the 1980s; all experiments conducted to date have found behavior in line with the predictions of quantum mechanics. The present view of the situation is that quantum mechanics flatly contradicts Einstein's philosophical postulate that any acceptable physical theory must fulfill "local realism". The fact that quantum mechanics violates Bell inequalities indicates that any hidden-variable theory underlying quantum mechanics must be non-local; whether this should be taken to imply that quantum mechanics itself is non-local is a matter of continuing debate.

Steering

Main article: Quantum steering

Inspired by Schrödinger's treatment of the EPR paradox back in 1935, Howard M. Wiseman et al. formalised it in 2007 as the phenomenon of quantum steering. They defined steering as the situation where Alice's measurements on a part of an entangled state steer Bob's part of the state. That is, Bob's observations cannot be explained by a local hidden state model, where Bob would have a fixed quantum state in his side, which is classically correlated but otherwise independent of Alice's.

Locality

Locality has several different meanings in physics. EPR describe the principle of locality as asserting that physical processes occurring at one place should have no immediate effect on the elements of reality at another location. At first sight, this appears to be a reasonable assumption to make, as it seems to be a consequence of special relativity, which states that energy can never be transmitted faster than the speed of light without violating causality; however, it turns out that the usual rules for combining quantum mechanical and classical descriptions violate EPR's principle of locality without violating special relativity or causality. Causality is preserved because there is no way for Alice to transmit messages (i.e., information) to Bob by manipulating her measurement axis. Whichever axis she uses, she has a 50% probability of obtaining "+" and 50% probability of obtaining "−", completely at random; according to quantum mechanics, it is fundamentally impossible for her to influence what result she gets. Furthermore, Bob is able to perform his measurement only once: there is a fundamental property of quantum mechanics, the no-cloning theorem, which makes it impossible for him to make an arbitrary number of copies of the electron he receives, perform a spin measurement on each, and look at the statistical distribution of the results. Therefore, in the one measurement he is allowed to make, there is a 50% probability of getting "+" and 50% of getting "−", regardless of whether or not his axis is aligned with Alice's.

As a summary, the results of the EPR thought experiment do not contradict the predictions of special relativity. Neither the EPR paradox nor any quantum experiment demonstrates that superluminal signaling is possible; however, the principle of locality appeals powerfully to physical intuition, and Einstein, Podolsky and Rosen were unwilling to abandon it. Einstein derided the quantum mechanical predictions as "spooky action at a distance". The conclusion they drew was that quantum mechanics is not a complete theory.

Mathematical formulation

Bohm's variant of the EPR paradox can be expressed mathematically using the quantum mechanical formulation of spin. The spin degree of freedom for an electron is associated with a two-dimensional complex vector space V, with each quantum state corresponding to a vector in that space. The operators corresponding to the spin along the x, y, and z direction, denoted S_x, S_y, and S_z respectively, can be represented using the Pauli matrices:$${\displaystyle S_x=\frac{\hslash }{2}\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right],S_y=\frac{\hslash }{2}\left[\begin{array}{cc}0& -i\\ i& 0\end{array}\right],S_z=\frac{\hslash }{2}\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right],}$$ where ${\displaystyle \hslash }$ is the reduced Planck constant (or the Planck constant divided by 2π).

The eigenstates of S_z are represented as $${\displaystyle |+z⟩\leftrightarrow \left[\begin{array}{c}1\\ 0\end{array}\right],|-z⟩\leftrightarrow \left[\begin{array}{c}0\\ 1\end{array}\right]}$$ and the eigenstates of S_x are represented as $${\displaystyle |+x⟩\leftrightarrow \frac{1}{\sqrt{2}}\left[\begin{array}{c}1\\ 1\end{array}\right],|-x⟩\leftrightarrow \frac{1}{\sqrt{2}}\left[\begin{array}{c}1\\ -1\end{array}\right].}$$

The vector space of the electron-positron pair is ${\displaystyle V\otimes V}$, the tensor product of the electron's and positron's vector spaces. The spin singlet state is $${\displaystyle |\psi ⟩=\frac{1}{\sqrt{2}}(|+z⟩\otimes |-z⟩-|-z⟩\otimes |+z⟩),}$$ where the two terms on the right hand side are what we have referred to as state I and state II above.

From the above equations, it can be shown that the spin singlet can also be written as $${\displaystyle |\psi ⟩=-\frac{1}{\sqrt{2}}(|+x⟩\otimes |-x⟩-|-x⟩\otimes |+x⟩),}$$ where the terms on the right hand side are what we have referred to as state Ia and state IIa.

To illustrate the paradox, we need to show that after Alice's measurement of S_z (or S_x), Bob's value of S_z (or S_x) is uniquely determined and Bob's value of S_x (or S_z) is uniformly random. This follows from the principles of measurement in quantum mechanics. When S_z is measured, the system state ${\displaystyle |\psi ⟩}$ collapses into an eigenvector of S_z. If the measurement result is +z, this means that immediately after measurement the system state collapses to $${\displaystyle |+z⟩\otimes |-z⟩=|+z⟩\otimes \frac{|+x⟩-|-x⟩}{\sqrt{2}}.}$$

Similarly, if Alice's measurement result is −z, the state collapses to $${\displaystyle |-z⟩\otimes |+z⟩=|-z⟩\otimes \frac{|+x⟩+|-x⟩}{\sqrt{2}}.}$$ The left hand side of both equations show that the measurement of S_z on Bob's positron is now determined, it will be −z in the first case or +z in the second case. The right hand side of the equations show that the measurement of S_x on Bob's positron will return, in both cases, +x or −x with probability 1/2 each.

Stern–Gerlach experiment

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Stern%E2%80%93Gerlach_experiment 

Stern–Gerlach experiment: Silver atoms travelling through an inhomogeneous magnetic field, and being deflected up or down depending on their spin; (1) furnace, (2) beam of silver atoms, (3) inhomogeneous magnetic field, (4) classically expected result, (5) observed result

In quantum physics, the Stern–Gerlach experiment demonstrated that the spatial orientation of angular momentum is quantized. Thus an atomic-scale system was shown to have intrinsically quantum properties. In the original experiment, silver atoms were sent through a spatially-varying magnetic field, which deflected them before they struck a detector screen, such as a glass slide. Particles with non-zero magnetic moment were deflected, owing to the magnetic field gradient, from a straight path. The screen revealed discrete points of accumulation, rather than a continuous distribution, owing to their quantized spin. Historically, this experiment was decisive in convincing physicists of the reality of angular-momentum quantization in all atomic-scale systems.

After its conception by Otto Stern in 1921, the experiment was first successfully conducted with Walther Gerlach in early 1922.

Description

See also: Spin quantum number

The Stern–Gerlach experiment involves sending silver atoms through an inhomogeneous magnetic field and observing their deflection. Silver atoms were evaporated using an electric furnace in a vacuum. Using thin slits, the atoms were guided into a flat beam and the beam sent through an inhomogeneous magnetic field before colliding with a metallic plate. The laws of classical physics predict that the collection of condensed silver atoms on the plate should form a thin solid line in the same shape as the original beam. However, the inhomogeneous magnetic field caused the beam to split in two separate directions, creating two lines on the metallic plate.

The results show that particles possess an intrinsic angular momentum that is closely analogous to the angular momentum of a classically spinning object, but that takes only certain quantized values. Another important result is that only one component of a particle's spin can be measured at one time, meaning that the measurement of the spin along the z-axis destroys information about a particle's spin along the x and y axis.

The experiment is normally conducted using electrically neutral particles such as silver atoms. This avoids the large deflection in the path of a charged particle moving through a magnetic field and allows spin-dependent effects to dominate.

If the particle is treated as a classical spinning magnetic dipole, it will precess in a magnetic field because of the torque that the magnetic field exerts on the dipole (see torque-induced precession). If it moves through a homogeneous magnetic field, the forces exerted on opposite ends of the dipole cancel each other out and the trajectory of the particle is unaffected. However, if the magnetic field is inhomogeneous then the force on one end of the dipole will be slightly greater than the opposing force on the other end, so that there is a net force which deflects the particle's trajectory. If the particles were classical spinning objects, one would expect the distribution of their spin angular momentum vectors to be random and continuous. Each particle would be deflected by an amount proportional to the dot product of its magnetic moment with the external field gradient, producing some density distribution on the detector screen. Instead, the particles passing through the Stern–Gerlach apparatus are deflected either up or down by a specific amount. This was a measurement of the quantum observable now known as spin angular momentum, which demonstrated possible outcomes of a measurement where the observable has a discrete set of values or point spectrum.

Although some discrete quantum phenomena, such as atomic spectra, were observed much earlier, the Stern–Gerlach experiment allowed scientists to directly observe separation between discrete quantum states for the first time.

Theoretically, quantum angular momentum of any kind has a discrete spectrum, which is sometimes briefly expressed as "angular momentum is quantized".

Experiment using particles with +1/2 or −1/2 spin

If the experiment is conducted using charged particles like electrons, there will be a Lorentz force that tends to bend the trajectory in a circle. This force can be cancelled by an electric field of appropriate magnitude oriented transverse to the charged particle's path.

Spin values for fermions

Electrons are spin-1/2 particles. These have only two possible spin angular momentum values measured along any axis, ${\displaystyle +\frac{\hslash }{2}}$ or ${\displaystyle -\frac{\hslash }{2}}$, a purely quantum mechanical phenomenon. Because its value is always the same, it is regarded as an intrinsic property of electrons, and is sometimes known as "intrinsic angular momentum" (to distinguish it from orbital angular momentum, which can vary and depends on the presence of other particles). If one measures the spin along a vertical axis, electrons are described as "spin up" or "spin down", based on the magnetic moment pointing up or down, respectively.

To mathematically describe the experiment with spin-1/2 particles, it is easiest to use Dirac's bra–ket notation. As the particles pass through the Stern–Gerlach device, they are deflected either up or down, and observed by the detector which resolves to either spin up or spin down. These are described by the angular momentum quantum number ${\displaystyle j}$, which can take on one of the two possible allowed values, either +1/2 or -1/2. The act of observing (measuring) the momentum along the ${\displaystyle z}$ axis corresponds to the ${\displaystyle z}$-axis angular momentum operator, often denoted ${\displaystyle J_z}$. In mathematical terms, the initial state of the particles is

${\displaystyle |\psi ⟩=c_1|{\psi }_{j=+\frac{1}{2}}⟩+c_2|{\psi }_{j=-\frac{1}{2}}⟩}$

where constants ${\displaystyle c_1}$ and ${\displaystyle c_2}$ are complex numbers. This initial state spin can point in any direction. The squares of the absolute values ${\displaystyle |c_1{|}^2}$ and ${\displaystyle |c_2{|}^2}$ are respectively the probabilities for a system in the state ${\displaystyle |\psi ⟩}$ to be found in ${\displaystyle |{\psi }_{j=+\frac{1}{2}}⟩}$ and ${\displaystyle |{\psi }_{j=-\frac{1}{2}}⟩}$ after the measurement along ${\displaystyle z}$ axis is made. The constants ${\displaystyle c_1}$ and ${\displaystyle c_2}$ must also be normalized in order that the probability of finding either one of the values be unity, that is we must ensure that ${\displaystyle |c_1{|}^2+|c_2{|}^2=1}$. However, this information is not sufficient to determine the values of ${\displaystyle c_1}$ and ${\displaystyle c_2}$, because they are complex numbers. Therefore, the measurement yields only the squared magnitudes of the constants, which are interpreted as probabilities.

Sequential experiments

If we link multiple Stern–Gerlach apparatuses (the rectangles containing S-G), we can clearly see that they do not act as simple selectors, i.e. filtering out particles with one of the states (pre-existing to the measurement) and blocking the others. Instead they alter the state by observing it (as in light polarization). In the figure below, x and z name the directions of the (inhomogenous) magnetic field, with the x-z-plane being orthogonal to the particle beam. In the three S-G systems shown below, the cross-hatched squares denote the blocking of a given output, i.e. each of the S-G systems with a blocker allows only particles with one of two states to enter the next S-G apparatus in the sequence.

Exp. 1 - Notice that no z- neutrons are detected at the second S-G analyzer

Experiment 1

The top illustration shows that when a second, identical, S-G apparatus is placed at the exit of the first apparatus, only z+ is seen in the output of the second apparatus. This result is expected since all particles at this point are expected to have z+ spin, as only the z+ beam from the first apparatus entered the second apparatus.

Exp. 2 - The z-spin is known, now measuring the x-spin.

Experiment 2

The middle system shows what happens when a different S-G apparatus is placed at the exit of the z+ beam resulting of the first apparatus, the second apparatus measuring the deflection of the beams on the x axis instead of the z axis. The second apparatus produces x+ and x- outputs. Now classically we would expect to have one beam with the x characteristic oriented + and the z characteristic oriented +, and another with the x characteristic oriented - and the z characteristic oriented +.

Exp. 3 - Neutrons thought to have only z+ spin are measured again, finding that the z-spin has been 'reset'.

Experiment 3

The bottom system contradicts that expectation. The output of the third apparatus which measures the deflection on the z axis again shows an output of z- as well as z+. Given that the input to the second S-G apparatus consisted only of z+, it can be inferred that a S-G apparatus must be altering the states of the particles that pass through it. This experiment can be interpreted to exhibit the uncertainty principle: since the angular momentum cannot be measured on two perpendicular directions at the same time, the measurement of the angular momentum on the x direction destroys the previous determination of the angular momentum in the z direction. That's why the third apparatus measures renewed z+ and z- beams like the x measurement really made a clean slate of the z+ output.

History

A plaque at the Frankfurt institute commemorating the experiment

The Stern–Gerlach experiment was conceived by Otto Stern in 1921 and performed by him and Walther Gerlach in Frankfurt in 1922. At the time of the experiment, the most prevalent model for describing the atom was the Bohr-Sommerfeld model, which described electrons as going around the positively charged nucleus only in certain discrete atomic orbitals or energy levels. Since the electron was quantized to be only in certain positions in space, the separation into distinct orbits was referred to as space quantization. The Stern–Gerlach experiment was meant to test the Bohr–Sommerfeld hypothesis that the direction of the angular momentum of a silver atom is quantized.

The experiment was first performed with an electromagnet that allowed the non-uniform magnetic field to be turned on gradually from a null value. When the field was null, the silver atoms were deposited as a single band on the detecting glass slide. When the field was made stronger, the middle of the band began to widen and eventually to split into two, so that the glass-slide image looked like a lip-print, with an opening in the middle, and closure at either end. In the middle, where the magnetic field was strong enough to split the beam into two, statistically half of the silver atoms had been deflected by the non-uniformity of the field.

Note that the experiment was performed several years before George Uhlenbeck and Samuel Goudsmit formulated their hypothesis about the existence of electron spin in 1925. Even though the result of the Stern−Gerlach experiment has later turned out to be in agreement with the predictions of quantum mechanics for a spin-1/2 particle, the experimental result was also consistent with the Bohr–Sommerfeld theory.

In 1927, T.E. Phipps and J.B. Taylor reproduced the effect using hydrogen atoms in their ground state, thereby eliminating any doubts that may have been caused by the use of silver atoms. However, in 1926 the non-relativistic scalar Schrödinger equation had incorrectly predicted the magnetic moment of hydrogen to be zero in its ground state. To correct this problem Wolfgang Pauli considered a spin-1/2 version of the Schrödinger equation using the 3 Pauli matrices which now bear his name, which was later shown by Paul Dirac in 1928 to be a consequence of his relativistic Dirac equation.

In the early 1930s Stern, together with Otto Robert Frisch and Immanuel Estermann improved the molecular beam apparatus sufficiently to measure the magnetic moment of the proton, a value nearly 2000 times smaller than the electron moment. In 1931, theoretical analysis by Gregory Breit and Isidor Isaac Rabi showed that this apparatus could be used to measure nuclear spin whenever the electronic configuration of the atom was known. The concept was applied by Rabi and Victor W. Cohen in 1934 to determine the ${\displaystyle 3/2}$ spin of sodium atoms.

In 1938 Rabi and coworkers inserted an oscillating magnetic field element into their apparatus, inventing nuclear magnetic resonance spectroscopy. By tuning the frequency of the oscillator to the frequency of the nuclear precessions they could selectively tune into each quantum level of the material under study. Rabi was awarded the Nobel Prize in 1944 for this work.

Importance

The Stern–Gerlach experiment was the first direct evidence of angular-momentum quantization in quantum mechanics, and it strongly influenced later developments in modern physics:

• In the decade that followed, scientists showed using similar techniques, that the nuclei of some atoms also have quantized angular momentum. It is the interaction of this nuclear angular momentum with the spin of the electron that is responsible for the hyperfine structure of the spectroscopic lines.
• Norman F. Ramsey later modified the Rabi apparatus to improve its sensitivity (using the separated oscillatory field method). In the early sixties, Ramsey, H. Mark Goldenberg, and Daniel Kleppner used a Stern–Gerlach system to produce a beam of polarized hydrogen as the source of energy for the hydrogen maser. This led to developing an extremely stable clock based on a hydrogen maser. From 1967 until 2019, the second was defined based on 9,192,631,770 Hz hyperfine transition of a cesium-133 atom; the atomic clock which is used to set this standard is an application of Ramsey's work.
• The Stern–Gerlach experiment has become a prototype for quantum measurement, demonstrating the observation of a discrete value (eigenvalue) of a physical property, previously assumed to be continuous. Entering the Stern–Gerlach magnet, the direction of the silver atom's magnetic moment is indefinite, but when the atom is registered at the screen, it is observed to be at either one spot or the other, and this outcome cannot be predicted in advance. Because the experiment illustrates the character of quantum measurements, The Feynman Lectures on Physics use idealized Stern–Gerlach apparatuses to explain the basic mathematics of quantum theory.

Empathy gap

From Wikipedia, the free encyclopedia

For cognitive bias, see Hot-cold empathy gap.

An empathy gap, sometimes referred to as an empathy bias, is a breakdown or reduction in empathy (the ability to recognize, understand, and share another's thoughts and feelings) where it might otherwise be expected to occur. Empathy gaps may occur due to a failure in the process of empathizing or as a consequence of stable personality characteristics, and may reflect either a lack of ability or motivation to empathize.

Empathy gaps can be interpersonal (toward others) or intrapersonal (toward the self, e.g. when predicting one's own future preferences). A great deal of social psychological research has focused on intergroup empathy gaps, their underlying psychological and neural mechanisms, and their implications for downstream behavior (e.g. prejudice toward outgroup members).

Classification

Cognitive empathy gaps

Failures in cognitive empathy (also referred to as perspective-taking) may sometimes result from a lack of ability. For example, young children often engage in failures of perspective-taking (e.g., on false belief tasks) due to underdeveloped social cognitive abilities. Neurodivergent individuals often face difficulties inferring others' emotional and cognitive states, though the double empathy problem proposes that the problem is mutual, occurring as well in non-neurodivergent individuals' struggle to understand and relate to neurodivergent people. Failures in cognitive empathy may also result from cognitive biases that impair one's ability to understand another's perspective (for example, see the related concept of naive realism.)

One's ability to perspective-take may be limited by one's current emotional state. For example, behavioral economics research has described a number of failures in empathy that occur due to emotional influences on perspective-taking when people make social predictions. People may either fail to accurately predict one's own preferences and decisions (intrapersonal empathy gaps), or to consider how others' preferences might differ from one's own (interpersonal empathy gaps). For example, people not owning a certain good underestimate their attachment to that good were they to own it.

In other circumstances, failures in cognitive empathy may occur due to a lack of motivation. For example, people are less likely to take the perspective of outgroup members with whom they disagree.

Affective empathy gaps

Further information: Hot-cold empathy gap

Affective (i.e. emotional) empathy gaps may describe instances in which an observer and target do not experience similar emotions, or when an observer does not experience anticipated emotional responses toward a target, such as sympathy and compassion.

Certain affective empathy gaps may be driven by a limited ability to share another's emotions. For example, psychopathy is characterized by impairments in emotional empathy.

Individuals may be motivated to avoid empathizing with others' emotions due to the emotional costs of doing so. For example, according to C. D. Batson's model of empathy, empathizing with others may either result in empathic concern (i.e. feelings of warmth and concern for another) or personal distress (i.e. when another's distress causes distress for the self). A trait-level tendency to experience personal distress (vs. empathic concern) may motivate individuals to avoid situations which would require them to empathize with others, and indeed predicts reduced helping behavior.

Notable examples

Intergroup empathy gaps

Humans are less likely to help outgroup members in need, as compared to ingroup members. People are also less likely to value outgroup members' lives as highly as those of ingroup members. These effects are indicative of an ingroup empathy bias, in which people empathize more with ingroup (vs. outgroup) members.

Intergroup empathy gaps are often affective or cognitive in nature, but also extend to other domains such as pain. For example, a great deal of research has demonstrated that people show reduced responses (e.g. neural activity) when observing outgroup (vs. ingroup) members in pain. These effects may occur for real-world social groups such as members of different races. In one study utilizing a minimal groups paradigm (in which groups are randomly assigned, ostensibly based on an arbitrary distinction), individuals also judged the perceived pain of ingroup members to be more painful than that of outgroup members.

Intergroup schadenfreude

Perhaps the most well-known "counter-empathic" emotion—i.e., an emotion that reflects an empathy gap for the target—is schadenfreude, or the experience of pleasure when observing or learning about another's suffering or misfortune. Schadenfreude frequently occurs in intergroup contexts. In fact, the two factors that most strongly predict schadenfreude are identification with one's group and the presence of competition between groups in conflict. Competition may be explicit; for example, one study found that soccer fans were less likely to help an injured stranger wearing a rival team shirt than someone wearing an ingroup team shirt. However, schadenfreude may also be directed toward members of groups associated with high-status, competitive stereotypes. These findings correspond with the stereotype content model, which proposes that such groups elicit envy, thereby precipitating schadenfreude.

Occupational burnout

Stress related to the experience of empathy may cause empathic distress fatigue and occupational burnout, particularly among those in the medical profession. Expressing empathy is an important component of patient-centered care, and can be expressed through behaviors such as concern, attentiveness, sharing emotions, vulnerability, understanding, dialogue, reflection, and authenticity. However, expressing empathy can be cognitively and emotionally demanding for providers. Physicians who lack proper support may experience depression and burnout, particularly in the face of the extended or frequent experiences of personal distress.

Forecasting failures

Within the domain of social psychology, "empathy gaps" typically describe breakdowns in empathy toward others (interpersonal empathy gaps). However, research in behavioral economics has also identified a number of intrapersonal empathy gaps (i.e. toward one's self). For example, "hot-cold empathy gaps" describe a breakdown in empathy for one's future self—specifically, a failure to anticipate how one's future affective states will affect one's preferences. Such failures can negatively impact decision-making, particularly in regards to health outcomes. Hot-cold empathy gaps are related to the psychological concepts of affective forecasting and temporal discounting.

Psychological factors

Mentalizing processes

Both affective and cognitive empathy gaps can occur due to a breakdown in the process of mentalizing others' states. For example, breakdowns in mentalizing may include but are not limited to:

• Mind attribution: People may fail to take another's perspective due to a failure to attribute a mind or agency to that person. Behavioral research has found that individuals are less likely to assign mental states to outgroup compared to ingroup members.
• Episodic simulation: People may find it difficult to empathize with others if they struggle (due to a lack of ability or motivation) to episodically simulate others' mental states—i.e. to imagine events from others' lives which occur at a specific time and place. The ability to engage in episodic simulation is predictive of greater affective empathy and prosocial behavior towards others.

Neural evidence also supports the key role of mentalizing in supporting empathic responses, particularly in an intergroup context. For example, a meta-analysis of neuroimaging studies of intergroup social cognition found that thinking about ingroup members (in comparison to outgroup members) was more frequently related to brain regions known to underlie mentalizing.

Gender norms

Further information: Gender empathy gap

Gender differences in the experience of empathy have been a subject of debate. In particular, scientists have sought to determine whether observed gender differences in empathy are due to variance in ability, motivation, or both between men and women. Research to date raises the possibility that gender norms regarding the experience and expression of empathy may decrease men's willingness to empathize with others, and therefore their tendency to engage in empathy.

A number of studies, primarily utilizing self-report, have found gender differences in men's and women's empathy. A 1977 review of nine studies found women to be more empathic than men on average. A 1983 review found a similar result, although differences in scores were stronger for self-report, as compared to observational, measures. In recent decades, a number of studies utilizing self-reported empathy have shown gender differences in empathy. According to the results of a nationally representative survey, men reported less willingness to give money or volunteer time to a poverty relief organization as compared to women, a finding mediated by men's lower self-reported feelings of empathic concern toward others.

However, more recent work has found little evidence that gender differences in self-reported empathy are related to neurophysiological measures (hemodynamic responses and pupil dilation). This finding raises the possibility that self-reported empathy may not be driven by biological differences in responses, but rather gender differences in willingness to report empathy. Specifically, women may be more likely to report experiencing empathy because it is more gender-normative for women than men. In support of this idea, a study found that manipulating the perceived gender normativity of empathy eliminated gender differences in men and women's self-reported empathy. Specifically, assigning male and female participants to read a narrative describing fictitious neurological research evidence which claimed that males score higher on measures of empathy eliminated the gender gap in self-reported empathy.

Trait differences

Psychological research has identified a number of trait differences associated with reduced empathic responses, including but not limited to:

• Social dominance orientation: Individuals high in social dominance orientation (SDO; i.e., those who endorse inequality and hierarchy between groups), are more likely to be high in prejudice and have less empathic concern for outgroup members. In addition to predicting greater intergroup empathy bias, high SDO scores correlated with greater counter-empathy (i.e. schadenfreude) toward outgroup targets, including Asian and Black targets (compared to ingroup White targets) when group boundaries were previously made salient, as well as toward competitive outgroup members (compared to ingroup members) in a novel group setting.
• Reduced importance of social ideals and relationships: Reduced familial and religious importance also appear to be predictive of reduced empathic responses. In a sample of adults aged 18 to 35 (N = 722), family importance was positively associated with affective empathy and perspective taking, particularly among non-Hispanic whites. Religious importance was significantly related to affective empathy, especially among Black, Indigenous, and/or People of Color.
• Conservative political orientation: In an analysis of data from the 2004 American General Social Survey, researchers found conservatives to have lower levels of empathy as compared to liberals, but only among individuals with low (vs. high) levels of religiosity.
• Higher social class: Some studies have found that people from upper-class backgrounds are less likely to experience feelings of compassion or to engage in empathetic behaviors, such as helping others. Education may play a role in this, wealthy and low-income students often attend different schools and do not get a chance to interact with one another. There is growing evidence to suggest that greater economic inequality is linked with lower empathy among the wealthy.

Neural mechanisms

Neural simulation

According to the perception–action-model of empathy, perception–action-coupling (i.e., the vicarious activation of the neural system for action during the perception of action) allows humans to understand others' actions, intentions, and emotions. According to this theory, when a "subject" individual observes an "object" individual, the object's physical movements and facial expressions activate corresponding neural mechanisms in the subject. That is, by neurally simulating the object's observed states, the subject also experiences these states, the basis of empathy.

The mirror neuron system has been proposed as a neural mechanism supporting perception-action coupling and empathy, although such claims remain a subject of scientific debate. Although the exact (if any) role of mirror neurons in supporting empathy is unclear, evidence suggests that neural simulation (i.e., recreating neural states associated with a process observed in another) may generally support a variety of psychological processes in humans, including disgust, pain, touch, and facial expressions.

Reduced neural simulation of responses to suffering may account in part for observed empathy gaps, particularly in an intergroup context. This possibility is supported by research demonstrating that people show reduced neural activity when they witness ethnic outgroup (vs. ingroup) members in physical or emotional pain. In one study, Chinese and Causian participants viewed videos of Chinese and Causasian targets, who displayed neutral facial expressions as they received either painful or non-painful stimulation to their cheeks. Witnessing racial ingroup faces receive painful stimulation increased activity in the dorsal anterior cingulate cortex and anterior insula (two regions which generally activate during the experience of pain.) However, these responses were diminished toward outgroup members in pain. These results replicated among White-Italian and Black-African participants. Additionally, EEG work has shown reduced neural simulation of movement (in primary motor cortex) for outgroup members, compared to in-group members. This effect was magnified by prejudice and toward disliked groups (i.e. South-Asians, Blacks, and East Asians).

Oxytocin

A great deal of social neuroscience research has been conducted to investigate the social functions of the hormone oxytocin, including its role in empathy. Generally speaking, oxytocin is associated with cooperation between individuals (in both humans and non-human animals). However, these effects interact with group membership in intergroup settings: oxytocin is associated with increased bonding with ingroup, but not outgroup, members, and may thereby contribute to ingroup favoritism and intergroup empathy bias. However, in one study of Israelis and Palestinians, intranasal oxytocin administration improved opposing partisans' empathy for outgroup members by increasing the salience of their pain.

In addition to temporary changes in oxytocin levels, the influence of oxytocin on empathic responses may also be influenced by an oxytocin receptor gene polymorphism, such that certain individuals may differ in the extent to which oxytocin promotes ingroup favoritism.

Specific neural correlates

A number of studies have been conducted to identify the neural regions implicated in intergroup empathy biases. This work has highlighted candidate regions supporting psychological processes such as mentalizing for ingroup members, deindividuation of outgroup members, and the pleasure associated with the experience of schadenfreude.

Role of dmPFC

A meta-analysis of 50 fMRI studies of intergroup social cognition found more consistent activation in dorsomedial prefrontal cortex (dmPFC) during ingroup (vs. outgroup) social cognition. dmPFC has previously been linked to the ability to infer others' mental states, which suggests that individuals may be more likely to engage in mentalizing for ingroup (as compared to outgroup) members. dmPFC activity has also been linked to prosocial behavior; thus, dmPFC's association with cognition about ingroup members suggests a potential neurocognitive mechanism underlying ingroup favoritism.

Role of anterior insula

Activation patterns in the anterior insula (AI) have been observed when thinking about both ingroup and outgroup members. For example, greater activity in the anterior insula has been observed when participants view ingroup members on a sports team receiving pain, compared to outgroup members receiving pain. In contrast, the meta-analysis referenced previously found that anterior insula activation was more reliably related to social cognition about outgroup members.

These seemingly divergent results may be due in part to functional differences between anatomic subregions of the anterior insula. Meta-analyses have identified two distinct subregions of the anterior insula: ventral AI, which is linked to emotional and visceral experiences (e.g. subjective arousal); and dorsal AI, which has been associated with exogenous attention processes such as attention orientation, salience detection, and task performance monitoring. Therefore, anterior insula activation may occur more often when thinking about outgroup members because doing is more attentionally demanding than thinking about ingroup members.

Lateralization of function within the anterior insula may also help account for divergent results, due to differences in connectivity between left and right AI. The right anterior insula has greater connectivity with regions supporting attentional orientation and arousal (e.g. postcentral gyrus and supramarginal gyrus), compared to the left anterior insula, which has greater connectivity with regions involved in perspective-taking and cognitive motor control (e.g. dmPFC and superior frontal gyrus). The previously referenced meta-analysis found right lateralization of anterior insula for outgroup compared to ingroup processing. These findings raise the possibility that when thinking about outgroup members, individuals may use their attention to focus on targets' salient outgroup status, as opposed to thinking about the outgroup member as an individual. In contrast, the meta-analysis found left lateralization of anterior insula activity for thinking about ingroup compared to outgroup members. This finding suggests that left anterior insula may help support perspective-taking and mentalizing about ingroup members, and thinking about them in an individuated way. However, these possibilities are speculative and lateralization may vary due to characteristics such as age, gender, and other individual differences, which should be accounted for in future research.

Role of ventral striatum

A number of fMRI studies have attempted to identify the neural activation patterns underlying the experience of intergroup schadenfreude, particularly toward outgroup members in pain. These studies have found increased activation in the ventral striatum, a region related to reward processing and pleasure.

Consequences

Helping behavior

Breakdowns in empathy may reduce helping behavior, a phenomenon illustrated by the identifiable victim effect. Specifically, humans are less likely to assist others who are not identifiable on an individual level. A related concept is psychological distance—that is, we are less likely to help those who feel more psychologically distant from us.

Reduced empathy for outgroup members is associated with a reduction in willingness to entertain another's points of view, the likelihood of ignoring a customer's complaints, the likelihood of helping others during a natural disaster, and the chance that one opposes social programs designed to benefit disadvantaged individuals.

Prejudice

Empathy gaps may contribute to prejudicial attitudes and behavior. However, training people in perspective-taking, for example by providing instructions about how to take an outgroup member's perspective, has been shown to increase intergroup helping and the recognition of group disparities. Perspective-taking interventions are more likely to be effective when a multicultural approach is used (i.e., an approach that appreciates intergroup differences), as opposed to a "colorblind" approach (e.g. an approach that attempts to emphasize a shared group identity).