Do hidden variables exist for quantum systems

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Do hidden variables exist for quantum systems?

Viewpoint: Yes, hidden variables are necessary to explain the contradictions and paradoxes inherent in quantum theory.

Viewpoint: No, experimental evidence, including the work of John Bell, Alain Aspect, and Nicolas Gisin, has continually shown that hidden variables do not exist.

Quantum physics is a daunting subject that often seems to be beyond comprehension. Nobel prize-winning quantum physicist Richard Feynman once said, "I think I can safely say that nobody understands quantum mechanics," and there would be few, if any, who would disagree with him.

Quantum theory contains many ideas that defy common sense. The popular concept of the atom is that of a tiny planetary system, with a nucleus "sun," and electron "planets" orbiting. However, quantum theory describes atoms and particles as having wavelike properties and avoids talking about specific positions and energies for particles, using instead ideas of probability. In quantum theory quantities such as energy can only exist in specific values, which contradicts the generally held notion that quantities have a continuous range and that any value in that range is possible.

Nobel prize-winning physicist Albert Einstein vehemently disliked many aspects of quantum physics, particularly the seemingly random and probabilistic nature of reality that the discipline implies, which he dismissed with his famous quote "God does not play dice." In 1932 Einstein, along with two colleagues, Boris Podolsky and Nathan Rosen, published a paper directly challenging some of the fundamental aspects of quantum theory. The EPR paper, as it came to be known, uses a thought experiment—an experiment that cannot be physically attempted, only imagined—to prove that quantum physics is an incomplete theory.

The three scientists argued that the missing bits that made quantum theory incomplete were "hidden variables" that would enable a more deterministic description of reality. Essentially, these scientists, and others, worried that quantum theory contains a number of scientific and philosophical problems and paradoxes. Examples include the infamous Schrödinger's Cat paradox, another thought experiment, in which quantum theory predicts that a cat exists in both dead and alive states until observed, or the two-slit experiment, which appears to break down the barriers of logic when single particles are used.

The Copenhagen interpretation of quantum physics, as formulated by Danish physicist Niels Bohr, German physicist Werner Heisenberg, and others, took the view that reality at the quantum level does not exist until it is measured. For example, a particle such as an electron orbiting an atomic nucleus could be in many different locations at a particular point in time. Quantum mechanics allows one to calculate the probabilities of each viable location of the electron as a wave function. However, the theory goes further, saying that until the electron is observed, it is in all possible positions, until the wave function that describes it is collapsed to a specific location by an observation. This creates some interesting philosophical problems and has been seen by some as implying that human beings create reality. Hidden variables, the EPR papers argue, would overcome these problems and allow for reality to be described with the same certainty that applies in Newtonian physics.

Hidden variables would also remove the need for "spooky" forces, as Einstein termed them—forces that act instantaneously at great distances, thereby breaking the most cherished rule of relativity theory, that nothing can travel faster than the speed of light. For example, quantum theory implies that measurement of one particle can instantaneously change another particle that may be light years away, if the particles are an entangled pair. Entangled particles are identical entities that share a common origin and have the same properties. Somehow, according to quantum theory, these particles remain in instantaneous contact with each other, no matter how far apart they separate. Hidden variables would allow two entangled particles to have specific values upon creation, thereby doing away with the need for them to be in communication with each other in some mysterious way.

The EPR paper caused many concerns for quantum theorists, but as the experiments it describes cannot be performed, the paper presented more of a philosophical problem than a scientific one. However, the later work of John Bell implied that there were specific tests that could be applied to determine whether the "spooky" forces were real or not, and therefore whether there are hidden variables after all.

In the 1980s the first such experiment was performed, and many more have been done since. The results imply that "spooky" actions-at-a-distance do indeed exist. Some scientists have challenged the validity of these experiments, and there is still some room for debate. These experiments only mean that "local" hidden variables do not exist, but would still allow "non-local" hidden variables. In this case, local effects are those that occur at or below the speed of light. You can think of the locality of an object as a sphere around it that expands at the speed of light. Outside of this sphere only non-local effects can take place, as nothing can travel faster than the speed of light. Non-local hidden variables, therefore, would have the same spookiness that the EPR paper was trying to avoid.

The debate over "hidden variables" is in some sense an argument over the completeness of quantum theory. Newton's laws once seemed to describe all motion, from particles to planets. However, the laws were found to be incomplete and were replaced by relativity, with regards to planets and other large-scale objects such as humans, and by quantum physics, with regards to particles and other very small-scale objects. It seems likely that one day relativity and quantum physics will also be replaced by other theories, if only because the two of them, while explaining their respective areas extremely well, are not compatible with one another.

In another sense, the hidden variables debate is a philosophical argument over whether the universe is deterministic and concrete, or merely probabilistic and somewhat spooky. Einstein and others have argued that reality must, on some deeper level, be fixed and solid. The majority of physicists, however, see no need for this desire for physical determinism, arguing that quantum mechanics can currently explain the world of the small-scale very well without the need to add in extras such as "hidden variables."

—DAVID TULLOCH

Viewpoint: Yes, hidden variables are necessary to explain the contradictions and paradoxes inherent in quantum theory.

The modern understanding of the nature and behavior of particles is most thoroughly explained by quantum theory. The description of particles as quantum mechanical waves replaces the age-old notion of particles as "balls" or "bullets" in motion. With important limitations or uncertainties, the quantum wave interpretation of nature, and the mathematical description of the wave attributes of particles, allow accurate predictions about the state (e.g., attributes such as velocity and position) and behavior of particles. Yet, Albert Einstein and others have asserted that the quantum mechanical system is an incomplete description of nature and that there must be undiscovered internal variables, to explain what Einstein termed "spooky" forces that, in contradiction to special relativity, seemingly act instantly over great distances. Without hidden variables, quantum theory also presents a paradox of prediction because the sought-after attributes of a particle can, in fact, determine the collapse of the quantum wave itself.

The Quantum Theory

This quantum mechanical view of nature is counter-intuitive, and stretches the language used to describe theory itself. Essentially, reality, as it relates to the existence and attributes of particles, is, according to quantum theory, dependent upon whether an event is observed. Unlike other measurable waveforms, however, the quantum wave is not easily measured as a discrete entity. The very act of measurement disturbs quantum systems. The attempted observation or measurement of a quantum wave changes the wave in often mathematically indeterminate and, therefore, unpredictable ways. In fact, the act of measurement leads to the collapse of the quantum wave into traditional observations of velocity and position.

Despite this counter-intuitive nature of quantum mechanics, it is undoubtedly successful in accurately predicting the behavior of particles. Well-tested, highly verified quantum concepts serve as a cornerstone of modern physics. Although highly successful at predicting the observed properties of atomic line spectra and the results of various interference experiments, there remains, however, problems with simply adopting the irreducibility of quantum mechanisms and the philosophical acceptance of an inherently statistical interpretation of nature that must exist if there are no hidden variables in quantum systems.

The EPR Argument

Einstein, Boris Podolsky, and Nathan Rosen treated the problems of quantum mechanics with great detail in their 1935 classic paper titled "Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?" Eventually their arguments became known as the Einstein-Podolsky-Rosen (EPR) paradox. At the heart of the argument advanced by the three was an attempt to set forth a definition of reality. The EPR definition of reality is well grounded in both classical and relativistic physics (descriptions reconciled with relativity theory) and asserts that physical reality exists (as measured by physical quantities such as velocity and position) when, without disturbing a system, there is a certainty in the ability to predict the value of the physical quantity in question. Although this definition of reality is intuitive (makes sense with our common understandings based upon experience), it then required Einstein, Podolsky, and Rosen to set forth a method by which one could observe a system without disturbing that system.

The EPR paper created a thought experiment to meet that challenge. In the EPR example, two bound particles were at rest relative to the observer in a closed system. If the particles were then to suddenly separate and begin moving in opposite directions, the total momentum of the closed system must, in accordance with the law of conservation of momentum, be conserved. Given that the two particles in their bound-together state were at rest relative to the observer, the initial momentum of the system was zero. Accordingly, as the particles move in different directions, their momentum must be equal and opposite so that the sum of the particle momenta always remains zero. Because the particles move in opposite directions, it is possible that they carry the same magnitude of momentum cut with differing signs (positive or negative) related to the coordinate systems in use to describe the particle motion (i.e., one particle moves in the positive direction as the other particle moves in the negative direction). If the sum of the two particles' momentum were to exceed zero, this condition would violate the law of conservation of momentum.

Because the sum of the momenta of the particles must equal zero, Einstein, Podolsky, and Rosen argued that by measuring the momentum of one particle, the momentum of the other particle can be determined with absolute certainty. If the velocity of one particle is known, the velocity of the other particle can be exactly determined without uncertainty. Correspondingly, if the position of one particle is known, the other can also be exactly determined. Given observation of one particle, no interaction on the part of the observer with the second particle is required to know with certainty the state or relevant physical quantity of the particle. In essence, in opposition to fundamental quantum theory assertions, no observation is necessary to determine the state of the particle (e.g., either the particle's velocity or position).

In accord with the uncertainty principle, it remains impossible to simultaneously determine both the velocity and the position of the second particle because the measurement of the first particle's velocity would alter that particle's velocity, and then subject it to different conditions than the second particle—essentially rupturing the special bound relationship of the two particles in which the sum of their respective momenta must remain zero.

In the EPR experiment, the fact that the velocity and position of the second particle can be specified imparts a physical reality to the second particle. More importantly, that reality (the physical states of the second particle) is determined apart from any influence of the observer. These factors directly challenge and stand in contrast to the inability of quantum theory to provide descriptions of the state of the second particle. Quantum theory can only describe these attributes in terms of the quantum wave. The EPR paradox then directly challenges this inability of quantum theory by asserting that some unknown variables must exist, unaccounted for by quantum theory, that allow for the determination of the second particle's state.

Some quantum theorists respond with the tautology (circular reasoning) that the observation of the first particle somehow determines the state of the second particle—without accounting for a lack of observer interaction or other mechanism of determination. Hidden variable proponents, however, counter that that argument only strengthens the assertion that hidden variables, unaccounted for by quantum theory, must operate to determine the state of the second particle. An attack on the EPR premise and definition that physical reality exists when physical states are independent of observation is an inadequate response to the EPR paradox because it simply leaves open the definition of reality without providing a testable alternative.

More importantly, if, as quantum theory dictates, the observation of the first particle serves to determine the state of the second particle, there is no accounting for the distance between the particles and the fact that the determination of state in the second particle must be instantaneous with any change in the state of the first particle. Given the speed of light limitations of relativity theory, any transmission over any distance that is simultaneous (i.e., requires zero time of transmission) violates relativity theory.

Hidden variable interpretations of quantum theory accept the validity and utility of quantum predictions while maintaining that the theory remains an incomplete description of nature. In accord with deterministic physical theory, these hidden variables lie inside the so-called black box of quantum theory and are determinant to the states currently described only in terms of statistical probability or the quantum wave. Moreover, the sum influence of these quantum hidden variables becomes the quantum wave.

The Limitations of Quantum Theory

Although quantum theory is mathematically complete, the assertion that no hidden variables exist leaves an inherently non-deterministic, probability-based explanation of the physical world. Hidden variable proponents, while granting that quantum mechanics represents the best and most useful model for predicting the behavior of particles, assert that only the discovery and explanation of the hidden variables in quantum theory will allow a complete and deterministic (where known causes lead to known effects) account of particle behavior that will remove the statistical uncertainties that lie at the heart of modern quantum theory.

The reliance on an indeterminate probability-based foundation for quantum theory rests heavily on the work of physicist and mathematician John von Neumann's 1932 elegant mathematical proof that deterministic mechanisms are not compatible with quantum theory. Other physicists and mathematicians, however, were able to construct and assert models based upon deterministic hidden variables that also completely explained the empirical results. David Bohm in the 1950s argued that von Neumann's assumptions, upon which his proof rested, may not have been entirely correct and that hidden variables could exist—but only under certain conditions not empirically demonstrated. Although subsequently John Bell's theorem and experiments testing Bell's inequalities are often touted as definitive proof that hidden variables do not exist, Bell's inequalities test only for local hidden variables and are, therefore, more properly only a test of locality.

Bell went on to revisit the EPR paradox and compare it to several popular hidden variable models. Bell's work demonstrated that for certain experiments, classical (i.e., deterministic) hidden variable theories predicted different results than those predicted by standard quantum theory. Although Bell's results were heralded as decisive in favor of quantum theory, without the need for hidden variables, they did not completely explain quantum entanglements, nor did they rule out non-local hidden variables. As a result, Bell's findings properly assert only that if hidden variables exist, they must be non-local (i.e., an effect in one reference frame that has the ability to simultaneously influence an event in another reference frame, even of the two reference frames are light years apart).

The acceptance of the argument that there are no hidden variables also entails the acceptance of quantum entanglement and superposition wherein particles may exist in a number of different states at the same time. These "Schrödinger's cat" arguments (e.g., that under a given set of circumstances a cat could be both dead and alive) when applied to particle behavior mean that particles can, for example with regard to radioactive decay, exist simultaneously in a decayed and non-decayed state. Moreover, the particle can also exist in innumerable superpositioned states where it exists in all possible states or decay. Although investigation of quantum entanglement holds great promise for communication systems and advances in thermodynamics, the exact extent to which these entangled states can be used or manipulated remains unknown. Although the EPR paradox powerfully argues that quantum entanglement means that quantum theory is incomplete and that hidden variables must exist, the fact that these variables must violate special relativity assertions is an admittedly powerful reason for modern physicists to assert that hidden variables do not exist.

Despite the weight of empirical evidence against the existence of hidden variables, it is philosophically important to remember that both relativity theory and quantum theory must be fully correct to assert that there are no undiscovered hidden variables. Without hidden variables, quantum theory remains a statistical, probability-based description of particle theory without the completeness of classical deterministic physics.

—BRENDA WILMOTH LERNER

Viewpoint: No, experimental evidence, including the work of John Bell, Alain Aspect, and Nicolas Gisin, has continually shown that hidden variables do not exist.

Although the standard model of quantum physics offers a theoretically and mathematically sound model of particle behavior consistent with experiment, the possible existence of hidden variables in quantum theory remained a subject of serious scientific debate during the twentieth century.

Based upon our everyday experience, well explained by the deterministic concepts of classical physics, it is intuitive that there be hidden variables to determine quantum states. Nature is not, however, obliged to act in accord with what is convenient or easy to understand. Although the existence and understanding of heretofore hidden variables might seemingly explain Albert Einstein's "spooky" forces, the existence of such variables would simply provide the need to determine whether they, too, included their own hidden variables. Quantum theory breaks this never-ending chain of causality by asserting (with substantial empirical evidence) that there are no hidden variables. Moreover, quantum theory replaces the need for a deterministic evaluation of natural phenomena with an understanding of particles and particle behavior based upon statistical probabilities. Although some philosophers and philosophically minded physicists would like to keep the hidden variable argument alive, the experimental evidence is persuasive, compelling, and conclusive that such hidden variables do not exist.

The EPR Paradox

The classic 1935 paper written by Einstein, Boris Podolsky, and Nathan Rosen (EPR) and titled "Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?" presented a Gedanken-experiment (German for "thought experiment") that seemingly mandates hidden variables. What eventually became known as the EPR paradox struck at the ability of particles to remain correlated in entangled states even though those particles might be separated by a great distance. Quantum entanglement is a concept of quantum theory that relies on the superposition of possible states for particles. In a two-particle entangled system, the act of measuring one of the entangled particles causes that particle's quantum wave to collapse to a definite state (e.g., a defined velocity or position). Simultaneous with the collapse of the first particle's wave state, the quantum wave of the second particle also collapses to a definite state. Such correlations must be instantaneous, and EPR argued that if there were any distance between the particles, any force acting between the particles would have to exceed the speed of light. Einstein called these forces "spooky actions at a distance."

EPR specifically identified three main problems with the standard interpretations of quantum mechanics that did not allow for the existence of hidden variables. Because of the limitations of special relativity, EPR argued that there could be no transacting force that instantaneously determines the state of the second particle in a two-particle system where the particles were separated and moving in opposite directions. EPR also challenged the uncertainty limitations found in quantum systems wherein the measurement of one state (e.g., velocity) makes impossible the exact determination of a second state (e.g., position). Most importantly, the EPR paper challenged the quantum view of nature as, at the quantum level, a universe explained only by probability rather than classical deterministic predictability where known causes produce known results. Einstein in particular objected to the inherent fundamental randomness of quantum theory (explaining his often quoted "God does not play dice!" challenge to Niels Bohr and other quantum theorists) and argued that for all its empirical usefulness in predicting line spectra and other physical phenomena, quantum theory was incomplete and that the discovery of hidden variables would eventually force modifications to the theory that would bring it into accord with relativity theory (especially concerning the absolute limitation of the speed of light).

Quantum theory, highly dependent on mathematical descriptions, depicts the wave nature of matter with a wave function (quantum waves). The wave function is used to calculate probabilities associated with finding a particle in a given state (e.g., position or velocity). When an observer interacts with a particle by attempting to measure a particular state, the wave function collapses and the particle takes on a determinable state that can be measured with a high degree of accuracy. If a fundamental particle such as an electron is depicted as a quantum wave, then it has a certain probability of being at any two points at the same time. If, however, an observer attempts to determine the location of the particle and determines it to be at a certain point, then the wave function has collapsed in that the probability of finding the electron at any other location is, in this measured state, zero.

The EPR paradox seemingly demands that for the wave function to collapse at the second point, some signal must be, in violation of special relativity, instantaneously transmitted from the point of measurement (i.e., the point of interaction between the observer and the particle) to any other point, no matter how far away that point may be, so that at that point the wave function collapses to zero.

David Bohm's subsequent support of EPR through a reconciliation of quantum theory with relativity theory was based upon the existence of local hidden variables. Bohm's hypothesis, however, suffering from a lack of empirical validation, smoldered on the back burners of theoretical physics until John Bell's inequalities provided a mechanism to empirically test the hidden variable hypothesis versus the standard interpretation of quantum mechanics.

John Bell's Inequalities

Bell's theorem (a set of inequalities) and work dispelled the idea that there are undiscovered hidden variables in quantum theory that determine particle states. Bell's inequalities, verified by subsequent studies of photon behavior, predicted testable differences between entangled photon pairs that were in superposition and entangled photons whose subsequent states were determined by local hidden variables.

Most importantly, Bell provided a very specific mechanism, based upon the polarization of photons, to test Bohm's local hidden variable hypothesis. Polarized photons are created by passing photons through optical filters or prisms that allow the transmission of light polarized in one direction (a particular orientation of the planes of the perpendicular electromagnetic wave) while blocking differently oriented photons. Most useful to tests of the EPR assertions are polarized photons produced by atomic cascades. Such photons are produced as electron decay from higher energy orbitals toward their ground state via a series of quantum jumps from one allowable orbital level to another. The law of the conservation of energy dictates that as electrons instantaneously transition from one orbital level to another they must give off a photon of light with exactly the same amount of energy as the difference between the two orbitals. An electron moving toward the ground state that makes that transition through two discreet orbital jumps (e.g., from the fourth orbital to the third and then from the third to the first) must produce two photons with energy (frequency and wavelength differences) directly related to the differences in total energy of the various oribitals. Of particular interest to EPR studies, however, is the fact that in cascades where there is no net rotational motion, the photons produced are quantum-entangled photons with regard to the fact that they must have specifically correlated polarizations. If the polarization of one photon can be determined, the other can be exactly known without any need for measurement.

Although the details of the measurement process, based upon the angles of various filters and measurement of arrival times of polarized photon pairs taking different paths, are beyond the scope of this article, the most critical aspect is that the outcomes predicted by standard quantum theory are different than the outcomes predicted if hidden variables exist. This difference in predicted outcomes makes it possible to test Bell's inequalities, and, in fact, a number of experiments have been performed to exactly test for these differences. In every experiment to date, the results are consistent with the predictions made by the standard interpretation of quantum mechanics and inconsistent for the existence of any local hidden variables as proposed by Bohm.

In 1982, the French physicist Alain Aspect, along with others, performed a series of experiments that demonstrated that between photons separated by short distances there was "action at a distance."

In 1997, Nicolas Gisin and colleagues at the University of Geneva extended the distances between entangled photons to a few kilometers. Measurements of particle states at the two laboratory sites showed that the photons adopted the correct state faster than light could have possibly traveled between the two laboratories.

Any Empirical Evidence?

In modern physics, Einstein's "spooky" actions underpin the concept of non-locality. Local, in this context, means forces that operate within the photons. Although Bell's inequality does not rule out the existence of non-local hidden variables that could act instantaneously over even great distances, such non-local hidden variables or forces would have a seemingly impossible theoretical and empirical barrier to surmount. If such non-local hidden variables exist, they must act or move faster than the speed of light and this, of course, would violate one of the fundamental assertions of special relativity. Just as quantum theory is well supported by empirical evidence, so too is relativity theory. Accordingly, for hidden variables to exist, both quantum and relativity theories would need to be rewritten. Granting that quantum and relativity theories are incompatible and that both may become components of a unified theory at some future date, this is certainly not tantamount to evidence for hidden variables.

The only hope for hidden variable proponents is if the hidden variables can act non-locally, or if particles have a way to predict their future state and make the needed transformations as appropriate. Such transactional interpretations of quantum theory use a reverse-causality argument to allow the existence of hidden variables that does not violate Bell's inequality. Other "many worlds" interpretations transform the act of measurement into the selection of a physical reality among a myriad of possibilities. Not only is there no empirical evidence to support this hypothesis, but also it severely strains Ockham's razor (the idea that given equal alternative explanations, the simpler is usually correct).

In common, hidden variable proponents essentially argue that particles are of unknown rather than undefined state when in apparent superposition. Although the hidden variable, transactional, or "many worlds" interpretations of quantum theory would make the quantum world more understandable in terms of conventional experience and philosophical understanding, there is simply no experimental evidence that such interpretations of quantum theory have any basis or validity. The mere possibility that any argument may be true does not in any way provide evidence that a particular argument is true.

In contrast to the apparent EPR paradox, it is a mistake to assume that quantum theory demands or postulates faster-than-light forces or signals (superluminal signals). Both quantum theory and relativity theory preclude the possibility of superluminal transmission, and, to this extent, quantum theory is normalized with relativity theory. For example, the instantaneous transformation of electrons from one allowed orbital (energy state) to another are most properly understood in terms of wave collapse rather than through some faster-than-light travel. The proper mathematical interpretation of the wave collapse completely explains quantum leaps, without any need for faster-than-light forces or signal transmission. Instead of a physical form or independent reality, the waveform is best understood as the state of an observer's knowledge about the state of a particle or system.

Most importantly, although current quantum theory does not completely rule out the existence of hidden variables under every set of conceivable circumstances, the mere possibility that hidden variables might exist under such special circumstances is in no way proof that hidden variables do exist. There is simply no empirical evidence that such hidden variables exist.

More importantly, quantum theory makes no claim to impart any form of knowing or consciousness on the behavior of particles. Although it is trendy to borrow selected concepts from quantum theory to prop up many New Age interpretations of nature, quantum theory does not provide for any mystical mechanisms. The fact that quantum theory makes accurate depictions and predictions of particle behavior does not mean that the mathematical constructs of quantum theory depict the actual physical reality of the quantum wave. Simply put, there is no demand that the universe present us with easy-to-understand mechanisms of action.

—K. LEE LERNER

KEY TERMS

DETERMINISM (DETERMINISTIC):

Causes precede effects—and there is a clear chain of causality that can be used to explain events. In essence, if a set of conditions is completely known, then an accurate prediction can be made of future events. Correspondingly, behavior of particles could be explained if all of the variables or mechanisms (causes) influencing the behavior of a particle were completely known.

NON-LOCALITY:

The phenomena wherein a force, act (observation), or event in one place simultaneously (instantly) influences an event or particle in another place, even if there is a vast (e.g., light years) distance between them.

PROBABILITY FUNCTION:

In quantum theory, not all possible states of matter—attributes such as position, velocity, spin, etc.—have equal probabilities. Although states are undetermined until measured, some are more likely than others. Quantum theory allows predictions of states based upon the probabilities represented in the quantum wave function.

QUANTUM ENTANGLEMENT:

The ability to link the states of two particles. Entanglement can be produced by random or probability-based processes (e.g., under special conditions two photons with correlated states can sometimes be produced by passing one photon through a crystal). Quantum entanglement is essentially a test for non-locality. Recently NIST researchers were able to entangle an ion's internal spin to its external motion and then entangle (correlate) those states to the motion and spin of another atom. The concept of quantum entanglement holds great promise for quantum computing.

QUANTUM WAVE:

The properties of matter can be described in terms of waves and particles. De Broglie waves describe the wave properties of matter related to momentum. Waves can also be described as a function of probability density. The quantum wave represents all states and all potentialities. The differential equation for quantum waves is the Schrödinger equation (also termed the quantum wave function) that treats time, energy, and position.

SUPERLUMINAL:

Faster-than-light transmission or motion.

SUPERPOSITION:

A concept related to quantum entanglement. For example, if one of two particles with opposite spins, that in a combined state would have zero spin, is measured and determined to be spinning in a particular direction, the spin of the other particle must be equal in magnitude but in the opposite direction. Superposition allows a particle to exist in all possible states of spin simultaneously, and the spin of a particle is not determined until measured.

Science in Dispute