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Branch of physics describing nature on an atomic scale

Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles.[2]:1.1 It is the foundation of all quantum physics including quantum chemistry, quantum field theory, quantum technology, and quantum information science.

Classical physics, the description of physics that existed before the theory of relativity and quantum mechanics, describes many aspects of nature at an ordinary (macroscopic) scale, while quantum mechanics explains the aspects of nature at small (atomic and subatomic) scales, for which classical mechanics is insufficient. Most theories in classical physics can be derived from quantum mechanics as an approximation valid at large (macroscopic) scale.[3]

Quantum mechanics differs from classical physics in that energy, momentum, angular momentum, and other quantities of a bound system are restricted to discrete values (quantization), objects have characteristics of both particles and waves (wave-particle duality), and there are limits to how accurately the value of a physical quantity can be predicted prior to its measurement, given a complete set of initial conditions (the uncertainty principle).

Quantum mechanics arose gradually, from theories to explain observations which could not be reconciled with classical physics, such as Max Planck's solution in 1900 to the black-body radiation problem, and the correspondence between energy and frequency in Albert Einstein's 1905 paper which explained the photoelectric effect. These early attempts to understand microscopic phenomena, now known as the "old quantum theory", led to the full development of quantum mechanics in the mid-1920s by Niels Bohr, Erwin Schrdinger, Werner Heisenberg, Max Born and others. The modern theory is formulated in various specially developed mathematical formalisms. In one of them, a mathematical function, the wave function, provides information about the probability amplitude of energy, momentum, and other physical properties of a particle.

Quantum mechanics allows the calculation of probabilities for how physical systems can behave. It is typically applied to microscopic systems: molecules, atoms and sub-atomic particles. A basic mathematical feature of quantum mechanics is that a probability is found by taking the square of the absolute value of a complex number, known as a probability amplitude. This is known as the Born rule, named after physicist Max Born. For example, a quantum particle like an electron can be described by a wave function, which associates to each point in space a probability amplitude. Applying the Born rule to these amplitudes gives a probability density function for the position that the electron will be found to have when an experiment is performed to measure it. The Schrdinger equation relates the collection of probability amplitudes that pertain to one moment of time to the collection of probability amplitudes that pertain to another.

One consequence of the mathematical rules of quantum mechanics is a tradeoff in predictability between different measurable quantities. The most famous form of this uncertainty principle says that no matter how a quantum particle is prepared or how carefully experiments upon it are arranged, it is impossible to have a precise prediction for a measurement of its position and also for a measurement of its momentum.

Another consequence of the mathematical rules of quantum mechanics is the phenomenon of quantum interference, which is often illustrated with the double-slit experiment. In the basic version of this experiment, a coherent light source, such as a laser beam, illuminates a plate pierced by two parallel slits, and the light passing through the slits is observed on a screen behind the plate.[4]:102111[2]:1.11.8 The wave nature of light causes the light waves passing through the two slits to interfere, producing bright and dark bands on the screen a result that would not be expected if light consisted of classical particles.[4] However, the light is always found to be absorbed at the screen at discrete points, as individual particles rather than waves; the interference pattern appears via the varying density of these particle hits on the screen. Furthermore, versions of the experiment that include detectors at the slits find that each detected photon passes through one slit (as would a classical particle), and not through both slits (as would a wave).[4]:109[5][6] However, such experiments demonstrate that particles do not form the interference pattern if one detects which slit they pass through. Other atomic-scale entities, such as electrons, are found to exhibit the same behavior when fired towards a double slit.[2] This behavior is known as wave-particle duality.

When quantum systems interact, the result can be the creation of quantum entanglement, a type of correlation in which "the best possible knowledge of a whole" does not imply "the best possible knowledge of all its parts", as Erwin Schrdinger put it.[7] Quantum entanglement can be a valuable resource in communication protocols, as demonstrated by quantum key distribution, in which (speaking informally) the key used to encrypt a message is created in the act of observing it.[8] (Entanglement does not, however, allow sending signals faster than light.[8])

Another possibility opened by entanglement is testing for "hidden variables", hypothetical properties more fundamental than the quantities addressed in quantum theory itself, knowledge of which would allow more exact predictions than quantum theory can provide. A collection of results, most significantly Bell's theorem, have demonstrated that broad classes of such hidden-variable theories are in fact incompatible with quantum physics. According to Bell's theorem, if nature actually operates in accord with any theory of local hidden variables, then the results of a Bell test will be constrained in a particular, quantifiable way. If a Bell test is performed in a laboratory and the results are not thus constrained, then they are inconsistent with the hypothesis that local hidden variables exist. Such results would support the position that there is no way to explain the phenomena of quantum mechanics in terms of a more fundamental description of nature that is more in line with the rules of classical physics. Many types of Bell test have been performed in physics laboratories, using preparations that exhibit quantum entanglement. To date, Bell tests have found that the hypothesis of local hidden variables is inconsistent with the way that physical systems behave.[9][10]

Later sections in this article cover the practical applications of quantum mechanics, its relation to other physical theories, the history of its development, and its philosophical implications. It is not possible to address these topics in more than a superficial way without knowledge of the actual mathematics involved. As mentioned above, using quantum mechanics requires manipulating complex numbers; it also makes use of linear algebra, differential equations, group theory, and other more advanced subjects.[note 1] Accordingly, this article will present a mathematical formulation of quantum mechanics and survey its application to some useful and oft-studied examples.

In the mathematically rigorous formulation of quantum mechanics developed by Paul Dirac,[13] David Hilbert,[14] John von Neumann,[15] and Hermann Weyl,[16] the state of a quantum mechanical system is a vector {displaystyle psi } belonging to a (separable) Hilbert space H {displaystyle {mathcal {H}}} . This vector is postulated to be normalized under the Hilbert's space inner product, that is, it obeys , = 1 {displaystyle langle psi ,psi rangle =1} , and it is well-defined up to a complex number of modulus 1 (the global phase), that is, {displaystyle psi } and e i {displaystyle e^{ialpha }psi } represent the same physical system. In other words, the possible states are points in the projective space of a Hilbert space, usually called the complex projective space. The exact nature of this Hilbert space is dependent on the system for example, for describing position and momentum the Hilbert space is the space of complex square-integrable functions L 2 ( C ) {displaystyle L^{2}(mathbb {C} )} , while the Hilbert space for the spin of a single proton is simply the space of two-dimensional complex vectors C 2 {displaystyle mathbb {C} ^{2}} with the usual inner product.

Physical quantities of interest - position, momentum, energy, spin - are represented by observables, which are Hermitian (more precisely, self-adjoint) linear operators acting on the Hilbert space. A quantum state can be an eigenvector of an observable, in which case it is called an eigenstate, and the associated eigenvalue corresponds to the value of the observable in that eigenstate. More generally, a quantum state will be a linear combination of the eigenstates, known as a quantum superposition. When an observable is measured, the result will be one of its eigenvalues with probability given by the Born rule: in the simplest case the eigenvalue {displaystyle lambda } is non-degenerate and the probability is given by | , | 2 {displaystyle |langle {vec {lambda }},psi rangle |^{2}} , where {displaystyle {vec {lambda }}} is its associated eigenvector. More generally, the eigenvalue is degenerate and the probability is given by , P {displaystyle langle psi ,P_{lambda }psi rangle } , where P {displaystyle P_{lambda }} is the projector onto its associated eigenspace.

After the measurement, if result {displaystyle lambda } was obtained, the quantum state is postulated to collapse to {displaystyle {vec {lambda }}} , in the non-degenerate case, or to P / , P {displaystyle P_{lambda }psi /{sqrt {langle psi ,P_{lambda }psi rangle }}} , in the general case. The probabilistic nature of quantum mechanics thus stems from the act of measurement. This is one of the most difficult aspects of quantum systems to understand. It was the central topic in the famous BohrEinstein debates, in which the two scientists attempted to clarify these fundamental principles by way of thought experiments. In the decades after the formulation of quantum mechanics, the question of what constitutes a "measurement" has been extensively studied. Newer interpretations of quantum mechanics have been formulated that do away with the concept of "wave function collapse" (see, for example, the many-worlds interpretation). The basic idea is that when a quantum system interacts with a measuring apparatus, their respective wave functions become entangled, so that the original quantum system ceases to exist as an independent entity. For details, see the article on measurement in quantum mechanics.[17]

The time evolution of a quantum state is described by the Schrdinger equation:

Here H {displaystyle H} denotes the Hamiltonian, the observable corresponding to the total energy of the system. The constant i {displaystyle ihbar } is introduced so that the Hamiltonian is reduced to the classical Hamiltonian in cases where the quantum system can be approximated by a classical system; the ability to make such an approximation in certain limits is called the correspondence principle.

The solution of this differential equation is given by

The operator U ( t ) = e i H t / {displaystyle U(t)=e^{-iHt/hbar }} is known as the time-evolution operator, and has the crucial property that it is unitary. This time evolution is deterministic in the sense that given an initial quantum state ( 0 ) {displaystyle psi (0)} it makes a definite prediction of what the quantum state ( t ) {displaystyle psi (t)} will be at any later time.[18]

Some wave functions produce probability distributions that are independent of time, such as eigenstates of the Hamiltonian. Many systems that are treated dynamically in classical mechanics are described by such "static" wave functions. For example, a single electron in an unexcited atom is pictured classically as a particle moving in a circular trajectory around the atomic nucleus, whereas in quantum mechanics, it is described by a static wave function surrounding the nucleus. For example, the electron wave function for an unexcited hydrogen atom is a spherically symmetric function known as an s orbital (Fig. 1).

Analytic solutions of the Schrdinger equation are known for very few relatively simple model Hamiltonians including the quantum harmonic oscillator, the particle in a box, the dihydrogen cation, and the hydrogen atom. Even the helium atom which contains just two electrons has defied all attempts at a fully analytic treatment.

However, there are techniques for finding approximate solutions. One method, called perturbation theory, uses the analytic result for a simple quantum mechanical model to create a result for a related but more complicated model by (for example) the addition of a weak potential energy. Another method is called "semi-classical equation of motion", which applies to systems for which quantum mechanics produces only small deviations from classical behavior. These deviations can then be computed based on the classical motion. This approach is particularly important in the field of quantum chaos.

One consequence of the basic quantum formalism is the uncertainty principle. In its most familiar form, this states that no preparation of a quantum particle can imply simultaneously precise predictions both for a measurement of its position and for a measurement of its momentum.[19][20] Both position and momentum are observables, meaning that they are represented by Hermitian operators. The position operator X ^ {displaystyle {hat {X}}} and momentum operator P ^ {displaystyle {hat {P}}} do not commute, but rather satisfy the canonical commutation relation:

Given a quantum state, the Born rule lets us compute expectation values for both X {displaystyle X} and P {displaystyle P} , and moreover for powers of them. Defining the uncertainty for an observable by a standard deviation, we have

and likewise for the momentum:

The uncertainty principle states that

Either standard deviation can in principle be made arbitrarily small, but not both simultaneously.[21] This inequality generalizes to arbitrary pairs of self-adjoint operators A {displaystyle A} and B {displaystyle B} . The commutator of these two operators is

and this provides the lower bound on the product of standard deviations:

Another consequence of the canonical commutation relation is that the position and momentum operators are Fourier transforms of each other, so that a description of an object according to its momentum is the Fourier transform of its description according to its position. The fact that dependence in momentum is the Fourier transform of the dependence in position means that the momentum operator is equivalent (up to an i / {displaystyle i/hbar } factor) to taking the derivative according to the position, since in Fourier analysis differentiation corresponds to multiplication in the dual space. This is why in quantum equations in position space, the momentum p i {displaystyle p_{i}} is replaced by i x {displaystyle -ihbar {frac {partial }{partial x}}} , and in particular in the non-relativistic Schrdinger equation in position space the momentum-squared term is replaced with a Laplacian times 2 {displaystyle -hbar ^{2}} .[19]

When two different quantum systems are considered together, the Hilbert space of the combined system is the tensor product of the Hilbert spaces of the two components. For example, let A and B be two quantum systems, with Hilbert spaces H A {displaystyle {mathcal {H}}_{A}} and H B {displaystyle {mathcal {H}}_{B}} , respectively. The Hilbert space of the composite system is then

If the state for the first system is the vector A {displaystyle psi _{A}} and the state for the second system is B {displaystyle psi _{B}} , then the state of the composite system is

Not all states in the joint Hilbert space H A B {displaystyle {mathcal {H}}_{AB}} can be written in this form, however, because the superposition principle implies that linear combinations of these "separable" or "product states" are also valid. For example, if A {displaystyle psi _{A}} and A {displaystyle phi _{A}} are both possible states for system A {displaystyle A} , and likewise B {displaystyle psi _{B}} and B {displaystyle phi _{B}} are both possible states for system B {displaystyle B} , then

is a valid joint state that is not separable. States that are not separable are called entangled.[22][23]

If the state for a composite system is entangled, it is impossible to describe either component system A or system B by a state vector. One can instead define reduced density matrices that describe the statistics that can be obtained by making measurements on either component system alone. This necessarily causes a loss of information, though: knowing the reduced density matrices of the individual systems is not enough to reconstruct the state of the composite system.[22][23] Just as density matrices specify the state of a subsystem of a larger system, analogously, positive operator-valued measures (POVMs) describe the effect on a subsystem of a measurement performed on a larger system. POVMs are extensively used in quantum information theory.[22][24]

There are many mathematically equivalent formulations of quantum mechanics. One of the oldest and most common is the "transformation theory" proposed by Paul Dirac, which unifies and generalizes the two earliest formulations of quantum mechanics matrix mechanics (invented by Werner Heisenberg) and wave mechanics (invented by Erwin Schrdinger).[25] An alternative formulation of quantum mechanics is Feynman's path integral formulation, in which a quantum-mechanical amplitude is considered as a sum over all possible classical and non-classical paths between the initial and final states. This is the quantum-mechanical counterpart of the action principle in classical mechanics.

When a measurement is performed, the introduction of a measurement device changes the Hamiltonian of the observed system. Note that such a measurement device may be any large object interacting with the observed system - including a lab measurement device, eyes, ears, cameras, microphones etc. When the measurement device is coupled to the observed system, the change in the Hamiltonian can be described by adding to the Hamiltonian a linear operator, that ties between the time evolution of the observed system with that of the measurement device. This linear operator can thus be described as the product of a measurement operator, acting on the observed system, with another operator, acting on the measurement devices.[26]

After the observed system and the measurement device interact in a manner described by this operator, they are said to be entangled, so that the quantum state of the measurement device together with the observed system is a superposition of different states, with each such state consisting of two parts: A state of the observed system with a particular measurement value, and a corresponding state of the measurement device measuring this particular value. For example, if the position of a particle is measured, the quantum state of the measurement device together with the particle will be a superposition of different states, in each of which the particle has a defined position and the measurement device shows this position; e.g. if the particle has two possible positions, x1 and x2, the overall state would be a linear combination of (particle at x1 and device showing x1) with (particle at x2 and device showing x2). The coefficients of this linear combination are called probability amplitudes; they are the inner products of the physical state with the basis vectors.[26]

Because the measurement device is a large object, the different states where it shows different measurement results can no longer interact with each other due to a process called decoherence. Any observer (e.g. the physicist) only measures one of the results, with a probability that depends on the probability amplitude of that result according to Born rule. How this happens is a matter of interpretation: Either only one of the results will continue to exist due to a hypothetical process called wavefunction collapse, or all results will co-exist in different hypothetical worlds, with the observer we know of living in one of these worlds.[26]

After a quantum state is measured, the only relevant part of it (due to decoherence and possibly also wavefunction collapse) has a well-defined value of the measurement operator. This means that it is an eigenstate of the measurement operator, with the measured value being the eigenvalue. Thus the different parts corresponding to the possible outcomes of the measurement are given by looking at the quantum state in a vector basis in which all basis vectors are eigenvectors of the measurement operator, i.e. a basis which diagonalizes this operator. Thus the measurement operator has to be diagonalizable. Further, if the possible measurement results are all real numbers, then the measurement operator must be Hermitian.[27]

As explained previously, the measurement process, e.g. measuring the position of an electron, can be described as consisting of an entanglement of the observed system with the measuring device, so that the overall physical state is a superposition of states, each of which consists of a state for the observed system (e.g. the electron) with defined measured value (e.g. position), together with a corresponding state of the measuring device showing this value. It is usually possible to analyze the possible results with the corresponding probabilities without analyzing the complete quantum description of the whole system: Only the part relevant to the observed system (the electron) should be taken into account. In order to do that, we only have to look at the probability amplitude for each possible result, and sum over all resulting probabilities. This computation can be performed through the use of the density matrix of the measured object.[19]

It can be shown that under the above definition for inner product, the time evolution operator e i H ^ t / {displaystyle e^{-i{hat {H}}t/hbar }} is unitary, a property often referred to as the unitarity of the theory.[27] This is equivalent to stating that the Hamiltonian is Hermitian:

This is desirable in order for the Hamiltonian to correspond to the classical Hamiltonian, which is why the -i factor is introduced (rather than defining the Hamiltonian with this factor included in it, which would result in an anti-Hermitian Hamiltonian). Indeed, in classical mechanics the Hamiltonian of a system is its energy, and thus in an energy measurement of an object, the measurement operator is the part of the Hamiltonian relating to this object. The energy is always a real number, and indeed the Hamiltonian is Hermitian.[19]

Let us choose a vector basis that is diagonal in a certain measurement operator; then, if this measurement is performed, the probability to get a measurement result corresponding to a particular vector basis must somehow depend on the inner product of physical state with this basis vector, i.e. the probability amplitude for this result. It turns out to be the absolute square of the probability amplitude; this is known as Born rule.

Note that the probability given by Born rule to get a particular state is simply the norm of this state. Unitarity then means that the sum of probabilities of any isolated set of state is invariant under time evolution, as long as there is no wavefunction collapse. Indeed, interpretations with no wavefunction collapse (such as the different versions of the many-worlds interpretation) always exhibit unitary time evolution, while for interpretations which include wavefunction collapse (such as the various views often grouped together as the Copenhagen interpretation) include both unitary and non-unitary time evolution, the latter happening during wavefunction collapse.[26]

The simplest example of quantum system with a position degree of freedom is a free particle in a single spatial dimension. A free particle is one which is not subject to external influences, so that its Hamiltonian consists only of its kinetic energy:

The general solution of the Schrdinger equation is given by

which is a superposition of all possible plane waves e i ( k x k 2 2 m t ) {displaystyle e^{i(kx-{frac {hbar k^{2}}{2m}}t)}} , which are eigenstates of the momentum operator with momentum p = k {displaystyle p=hbar k} . The coefficients of the superposition are ^ ( k , 0 ) {displaystyle {hat {psi }}(k,0)} , which is the Fourier transform of the initial quantum state ( x , 0 ) {displaystyle psi (x,0)} .

It is not possible for the solution to be a single momentum eigenstate, or a single position eigenstate, as these are not normalizable quantum states.[note 2] Instead, we can consider a Gaussian wavepacket:

which has Fourier transform, and therefore momentum distribution

We see that as we make a smaller the spread in position gets smaller, but the spread in momentum gets larger. Conversely, by making a larger we make the spread in momentum smaller, but the spread in position gets larger. This illustrates the uncertainty principle.

As we let the Gaussian wavepacket evolve in time, we see that its center moves through space at a constant velocity (like a classical particle with no forces acting on it). However, the wave packet will also spread out as time progresses, which means that the position becomes more and more uncertain. The uncertainty in momentum, however, stays constant.[28]

The particle in a one-dimensional potential energy box is the most mathematically simple example where restraints lead to the quantization of energy levels. The box is defined as having zero potential energy everywhere inside a certain region, and therefore infinite potential energy everywhere outside that region.[19]:7778 For the one-dimensional case in the x {displaystyle x} direction, the time-independent Schrdinger equation may be written

With the differential operator defined by

the previous equation is evocative of the classic kinetic energy analogue,

with state {displaystyle psi } in this case having energy E {displaystyle E} coincident with the kinetic energy of the particle.

The general solutions of the Schrdinger equation for the particle in a box are

or, from Euler's formula,

The infinite potential walls of the box determine the values of C , D , {displaystyle C,D,} and k {displaystyle k} at x = 0 {displaystyle x=0} and x = L {displaystyle x=L} where {displaystyle psi } must be zero. Thus, at x = 0 {displaystyle x=0} ,

and D = 0 {displaystyle D=0} . At x = L {displaystyle x=L} ,

in which C {displaystyle C} cannot be zero as this would conflict with the postulate that {displaystyle psi } has norm 1. Therefore, since sin ( k L ) = 0 {displaystyle sin(kL)=0} , k L {displaystyle kL} must be an integer multiple of {displaystyle pi } ,

This constraint on k {displaystyle k} implies a constraint on the energy levels, yielding

E n = 2 2 n 2 2 m L 2 = n 2 h 2 8 m L 2 . {displaystyle E_{n}={frac {hbar ^{2}pi ^{2}n^{2}}{2mL^{2}}}={frac {n^{2}h^{2}}{8mL^{2}}}.}

A finite potential well is the generalization of the infinite potential well problem to potential wells having finite depth. The finite potential well problem is mathematically more complicated than the infinite particle-in-a-box problem as the wave function is not pinned to zero at the walls of the well. Instead, the wave function must satisfy more complicated mathematical boundary conditions as it is nonzero in regions outside the well. Another related problem is that of the rectangular potential barrier, which furnishes a model for the quantum tunneling effect that plays an important role in the performance of modern technologies such as flash memory and scanning tunneling microscopy.

As in the classical case, the potential for the quantum harmonic oscillator is given by

This problem can either be treated by directly solving the Schrdinger equation, which is not trivial, or by using the more elegant "ladder method" first proposed by Paul Dirac. The eigenstates are given by

where Hn are the Hermite polynomials

and the corresponding energy levels are

This is another example illustrating the discretization of energy for bound states.

Quantum mechanics has had enormous success in explaining many of the features of our universe, with regards to small-scale and discrete quantities and interactions which cannot be explained by classical methods.[note 3] Quantum mechanics is often the only theory that can reveal the individual behaviors of the subatomic particles that make up all forms of matter (electrons, protons, neutrons, photons, and others). Quantum mechanics has strongly influenced string theories, candidates for a Theory of Everything (see reductionism).

In many aspects modern technology operates at a scale where quantum effects are significant.Important applications of quantum theory include quantum chemistry, quantum optics, quantum computing, superconducting magnets, light-emitting diodes, the optical amplifier and the laser, the transistor and semiconductors such as the microprocessor, medical and research imaging such as magnetic resonance imaging and electron microscopy.[29] Explanations for many biological and physical phenomena are rooted in the nature of the chemical bond, most notably the macro-molecule DNA.

The rules of quantum mechanics are fundamental, and predictions of quantum mechanics have been verified experimentally to an extremely high degree of accuracy.[note 4] The rules assert that the state space of a system is a Hilbert space (crucially, that the space has an inner product) and that observables of the system are Hermitian operators acting on vectors in that space although they do not tell us which Hilbert space or which operators. These can be chosen appropriately in order to obtain a quantitative description of a quantum system, a necessary step in making physical predictions. An important guide for making these choices is the correspondence principle, a heuristic which states that the predictions of quantum mechanics reduce to those of classical mechanics in the regime of large quantum numbers.[30] One can also start from an established classical model of a particular system, and then try to guess the underlying quantum model that would give rise to the classical model in the correspondence limit. This approach is known as quantization.

When quantum mechanics was originally formulated, it was applied to models whose correspondence limit was non-relativistic classical mechanics. For instance, the well-known model of the quantum harmonic oscillator uses an explicitly non-relativistic expression for the kinetic energy of the oscillator, and is thus a quantum version of the classical harmonic oscillator.

Complications arise with chaotic systems, which do not have good quantum numbers, and quantum chaos studies the relationship between classical and quantum descriptions in these systems.

Quantum coherence is an essential difference between classical and quantum theories as illustrated by the EinsteinPodolskyRosen (EPR) paradox an attack on a certain philosophical interpretation of quantum mechanics by an appeal to local realism.[31] Quantum interference involves adding together probability amplitudes, whereas classical "waves" infer that there is an adding together of intensities. For microscopic bodies, the extension of the system is much smaller than the coherence length, which gives rise to long-range entanglement and other nonlocal phenomena characteristic of quantum systems.[32] Quantum coherence is not typically evident at macroscopic scales, except maybe at temperatures approaching absolute zero at which quantum behavior may manifest macroscopically.[note 5] This is in accordance with the following observations:

Early attempts to merge quantum mechanics with special relativity involved the replacement of the Schrdinger equation with a covariant equation such as the KleinGordon equation or the Dirac equation. While these theories were successful in explaining many experimental results, they had certain unsatisfactory qualities stemming from their neglect of the relativistic creation and annihilation of particles. A fully relativistic quantum theory required the development of quantum field theory, which applies quantization to a field (rather than a fixed set of particles). The first complete quantum field theory, quantum electrodynamics, provides a fully quantum description of the electromagnetic interaction. Quantum electrodynamics is, along with general relativity, one of the most accurate physical theories ever devised.[35][36]

The full apparatus of quantum field theory is often unnecessary for describing electrodynamic systems. A simpler approach, one that has been used since the inception of quantum mechanics, is to treat charged particles as quantum mechanical objects being acted on by a classical electromagnetic field. For example, the elementary quantum model of the hydrogen atom describes the electric field of the hydrogen atom using a classical e 2 / ( 4 0 r ) {displaystyle textstyle -e^{2}/(4pi epsilon _{_{0}}r)} Coulomb potential. This "semi-classical" approach fails if quantum fluctuations in the electromagnetic field play an important role, such as in the emission of photons by charged particles.

Quantum field theories for the strong nuclear force and the weak nuclear force have also been developed. The quantum field theory of the strong nuclear force is called quantum chromodynamics, and describes the interactions of subnuclear particles such as quarks and gluons. The weak nuclear force and the electromagnetic force were unified, in their quantized forms, into a single quantum field theory (known as electroweak theory), by the physicists Abdus Salam, Sheldon Glashow and Steven Weinberg.[37]

A Grand Unified Theory (GUT) is a model in particle physics in which, at high energies, the three gauge interactions of the Standard Model comprising the electromagnetic, weak, and strong forces are merged into a single force. Although this unified force has not been directly observed, the many GUT models theorize its existence. If unification of these three interactions is possible, it raises the possibility that there was a grand unification epoch in the very early universe in which these three fundamental interactions were not yet distinct.

Experiments have confirmed that at high energy the electromagnetic interaction and weak interaction unify into a single electroweak interaction. GUT models predict that at even higher energy, the strong interaction and the electroweak interaction will unify into a single electronuclear interaction. This interaction is characterized by one larger gauge symmetry and thus several force carriers, but one unified coupling constant. The novel particles predicted by GUT models are expected to have extremely high massesaround the GUT scale of 10 16 {displaystyle 10^{16}} GeV (just a few orders of magnitude below the Planck scale of 10 19 {displaystyle 10^{19}} GeV)and so are well beyond the reach of any foreseen particle collider experiments. Therefore, the particles predicted by GUT models will be unable to be observed directly, and instead the effects of grand unification might be detected through indirect observations such as proton decay, electric dipole moments of elementary particles, or the properties of neutrinos.[38]

Even with the defining postulates of both Einstein's theory of general relativity and quantum theory being indisputably supported by rigorous and repeated empirical evidence, and while they do not directly contradict each other theoretically (at least with regard to their primary claims), they have proven extremely difficult to incorporate into one consistent, cohesive model.[note 6]

Gravity is negligible in many areas of particle physics, so that unification between general relativity and quantum mechanics is not an urgent issue in those particular applications. However, the lack of a correct theory of quantum gravity is an important issue in physical cosmology and the search by physicists for an elegant "Theory of Everything" (TOE). Consequently, resolving the inconsistencies between both theories has been a major goal of 20th- and 21st-century physics. This TOE would combine not only the models of subatomic physics, but also derive the four fundamental forces of nature from a single force or phenomenon.

Beyond the "grand unification" of the electromagnetic and nuclear forces, it is speculated that it may be possible to merge gravity with the other three gauge symmetries, expected to occur at roughly 1019 GeV. However and while special relativity is parsimoniously incorporated into quantum electrodynamics general relativity, currently the best theory describing the gravitational force, has not been fully incorporated into quantum theory. One proposal for doing so is string theory, which posits that the point-like particles of particle physics are replaced by one-dimensional objects called strings. String theory describes how these strings propagate through space and interact with each other. On distance scales larger than the string scale, a string looks just like an ordinary particle, with its mass, charge, and other properties determined by the vibrational state of the string. In string theory, one of the many vibrational states of the string corresponds to the graviton, a quantum mechanical particle that carries gravitational force.

Another popular theory is loop quantum gravity (LQG), which describes quantum properties of gravity and is thus a theory of quantum spacetime. LQG is an attempt to merge and adapt standard quantum mechanics and standard general relativity. This theory describes space as granular analogous to the granularity of photons in the quantum theory of electromagnetism and the discrete energy levels of atoms. More precisely, space is an extremely fine fabric or networks "woven" of finite loops called spin networks. The evolution of a spin network over time is called a spin foam. The predicted size of this structure is the Planck length, which is approximately 1.6161035 m. According to this theory, there is no meaning to length shorter than this (cf. Planck scale energy).

Since its inception, the many counter-intuitive aspects and results of quantum mechanics have provoked strong philosophical debates and many interpretations. Even fundamental issues, such as Max Born's basic rules about probability amplitudes and probability distributions, took decades to be appreciated by society and many leading scientists. Richard Feynman once said, "I think I can safely say that nobody understands quantum mechanics."[40] According to Steven Weinberg, "There is now in my opinion no entirely satisfactory interpretation of quantum mechanics."[41]

The views of Niels Bohr, Werner Heisenberg and other physicists are often grouped together as the "Copenhagen interpretation".[42][43] According to these views, the probabilistic nature of quantum mechanics is not a temporary feature which will eventually be replaced by a deterministic theory, but is instead a final renunciation of the classical idea of "causality". Bohr in particular emphasized that any well-defined application of the quantum mechanical formalism must always make reference to the experimental arrangement, due to the conjugate nature of evidence obtained under different experimental situations. Copenhagen-type interpretations remain popular in the 21st century.[44]

Albert Einstein, himself one of the founders of quantum theory, did not accept some of the more philosophical or metaphysical interpretations of quantum mechanics, such as rejection of determinism and of causality. Einstein believed that underlying quantum mechanics must be a theory that thoroughly and directly expresses the rule against action at a distance; in other words, he insisted on the principle of locality. He argued that quantum mechanics was incomplete, a currently valid but not a permanently definitive theory about nature. Einstein's long-running exchanges with Bohr about the meaning and status of quantum mechanics are now known as the BohrEinstein debates. In 1935, Einstein and his collaborators Boris Podolsky and Nathan Rosen published an argument that the principle of locality implies the incompleteness of quantum mechanics, a thought experiment later termed the EinsteinPodolskyRosen paradox.[note 7]

John Bell showed that the EPR paradox led to experimentally testable differences between quantum mechanics and theories that rely on local hidden variables. Experiments confirmed the accuracy of quantum mechanics, thereby showing that quantum mechanics cannot be improved upon by addition of local hidden variables.[49] Alain Aspect's experiments in 1982 and many later experiments definitively verified quantum entanglement. Entanglement, as demonstrated in Bell-type experiments, does not violate causality, since it does not involve transfer of information. By the early 1980s, experiments had shown that such inequalities were indeed violated in practice so that there were in fact correlations of the kind suggested by quantum mechanics. At first these just seemed like isolated esoteric effects, but by the mid-1990s, they were being codified in the field of quantum information theory, and led to constructions with names like quantum cryptography and quantum teleportation.[22][23] Quantum cryptography is proposed for use in high-security applications in banking and government.

The Everett many-worlds interpretation, formulated in 1956, holds that all the possibilities described by quantum theory simultaneously occur in a multiverse composed of mostly independent parallel universes.[50] This is not accomplished by introducing a "new axiom" to quantum mechanics, but by removing the axiom of the collapse of the wave packet. All possible consistent states of the measured system and the measuring apparatus (including the observer) are present in a real physical not just formally mathematical, as in other interpretations quantum superposition. Such a superposition of consistent state combinations of different systems is called an entangled state. While the multiverse is deterministic, we perceive non-deterministic behavior governed by probabilities, because we can only observe the universe (i.e., the consistent state contribution to the aforementioned superposition) that we, as observers, inhabit. The role of probability in many-worlds interpretations has been the subject of much debate. Why we should assign probabilities at all to outcomes that are certain to occur in some worlds, and why should the probabilities be given by the Born rule?[51] Everett tried to answer both questions in the paper that introduced many-worlds; his derivation of the Born rule has been criticized as relying on unmotivated assumptions.[52] Since then several other derivations of the Born rule in the many-worlds framework have been proposed. There is no consensus on whether this has been successful.[53][54]

Relational quantum mechanics appeared in the late 1990s as a modern derivative of Copenhagen-type ideas,[55] and QBism was developed some years later.[56]

Quantum mechanics was developed in the early decades of the 20th century, driven by the need to explain phenomena that, in some cases, had been observed in earlier times. Scientific inquiry into the wave nature of light began in the 17th and 18th centuries, when scientists such as Robert Hooke, Christiaan Huygens and Leonhard Euler proposed a wave theory of light based on experimental observations.[57] In 1803 English polymath Thomas Young described the famous double-slit experiment.[58] This experiment played a major role in the general acceptance of the wave theory of light.

In 1838 Michael Faraday discovered cathode rays. These studies were followed by the 1859 statement of the black-body radiation problem by Gustav Kirchhoff, the 1877 suggestion by Ludwig Boltzmann that the energy states of a physical system can be discrete, and the 1900 quantum hypothesis of Max Planck.[59] Planck's hypothesis that energy is radiated and absorbed in discrete "quanta" (or energy packets) precisely matched the observed patterns of black-body radiation. The word quantum derives from the Latin, meaning "how great" or "how much".[60] According to Planck, quantities of energy could be thought of as divided into "elements" whose size (E) would be proportional to their frequency ():

where h is Planck's constant. Planck cautiously insisted that this was only an aspect of the processes of absorption and emission of radiation and was not the physical reality of the radiation.[61] In fact, he considered his quantum hypothesis a mathematical trick to get the right answer rather than a sizable discovery.[62] However, in 1905 Albert Einstein interpreted Planck's quantum hypothesis realistically and used it to explain the photoelectric effect, in which shining light on certain materials can eject electrons from the material. Niels Bohr then developed Planck's ideas about radiation into a model of the hydrogen atom that successfully predicted the spectral lines of hydrogen.[63] Einstein further developed this idea to show that an electromagnetic wave such as light could also be described as a particle (later called the photon), with a discrete amount of energy that depends on its frequency.[64] In his paper "On the Quantum Theory of Radiation," Einstein expanded on the interaction between energy and matter to explain the absorption and emission of energy by atoms. Although overshadowed at the time by his general theory of relativity, this paper articulated the mechanism underlying the stimulated emission of radiation,[65] which became the basis of the laser.

This phase is known as the old quantum theory. Never complete or self-consistent, the old quantum theory was rather a set of heuristic corrections to classical mechanics.[66] The theory is now understood as a semi-classical approximation[67] to modern quantum mechanics.[68] Notable results from this period include, in addition to the work of Planck, Einstein and Bohr mentioned above, Einstein and Debye's work on the specific heat of solids, Bohr and van Leeuwen's proof that classical physics cannot account for diamagnetism, and Arnold Sommerfeld's extension of the Bohr model to include relativistic effects.

In the mid-1920s quantum mechanics was developed to become the standard formulation for atomic physics. In the summer of 1925, Bohr and Heisenberg published results that closed the old quantum theory. Heisenberg, Max Born, and Pascual Jordan pioneered matrix mechanics. The following year, Erwin Schrdinger suggested a partial differential equation for the wave functions of particles like electrons. And when effectively restricted to a finite region, this equation allowed only certain modes, corresponding to discrete quantum states whose properties turned out to be exactly the same as implied by matrix mechanics. Born introduced the probabilistic interpretation of Schrdinger's wave function in July 1926.[69] Thus, the entire field of quantum physics emerged, leading to its wider acceptance at the Fifth Solvay Conference in 1927.[70]

By 1930 quantum mechanics had been further unified and formalized by David Hilbert, Paul Dirac and John von Neumann[71] with greater emphasis on measurement, the statistical nature of our knowledge of reality, and philosophical speculation about the 'observer'. It has since permeated many disciplines, including quantum chemistry, quantum electronics, quantum optics, and quantum information science. It also provides a useful framework for many features of the modern periodic table of elements, and describes the behaviors of atoms during chemical bonding and the flow of electrons in computer semiconductors, and therefore plays a crucial role in many modern technologies. While quantum mechanics was constructed to describe the world of the very small, it is also needed to explain some macroscopic phenomena such as superconductors[72] and superfluids.[73]

Its speculative modern developments include string theory and other attempts to build a quantum theory of gravity.

The following titles, all by working physicists, attempt to communicate quantum theory to lay people, using a minimum of technical apparatus.

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Quantum mechanics - Wikipedia