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Quantum mechanics is a fundamental
theory A theory is a rational type of abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinking is often associated with such processes as observational study or research. Theories may ...
in
physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical science is that depar ...

that provides a description of the physical properties of
nature Nature, in the broadest sense, is the physics, physical world or universe. "Nature" can refer to the phenomenon, phenomena of the physical world, and also to life in general. The study of nature is a large, if not the only, part of science. ...

at the scale of
atom Every atom is composed of a atomic nucleus, nucleus and one or more electrons bound to the nucleus. The nucleus is made of one or more protons and a number of neutrons. Only the most common variety of hydrogen has no neutrons. Every solid, l ...

s and
subatomic particle In physical sciences, a subatomic particle is a particle that composes an atom Every atom is composed of a atomic nucleus, nucleus and one or more electrons bound to the nucleus. The nucleus is made of one or more protons and a number of n ...
s. It is the foundation of all quantum physics including
quantum chemistry Quantum chemistry, also called molecular quantum mechanics, is a branch of physical chemistry focused on the application of quantum mechanics to chemical systems, particularly towards the quantum-mechanical calculation of electronic contributions ...
, quantum field theory, quantum technology, and quantum information science.
Classical physics Classical physics is a group of physics theories that predate modern, more complete, or more widely applicable theories. If a currently accepted theory is considered to be modern, and its introduction represented a major paradigm shift, then the ...
, the collection of theories that existed before the advent of quantum mechanics, describes many aspects of nature at an ordinary ( macroscopic) scale, but is not sufficient for describing them at small (atomic and subatomic) scales. Most theories in classical physics can be derived from quantum mechanics as an approximation valid at large (macroscopic) scale. Quantum mechanics differs from classical physics in that
energy In physics, energy (from Ancient Greek: wikt:ἐνέργεια#Ancient_Greek, ἐνέργεια, ''enérgeia'', “activity”) is the physical quantity, quantitative physical property, property that is #Energy transfer, transferred to a phy ...

,
momentum In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the Multiplication, product of the mass and velocity of an object. It is a Euclidean vector, vector quantity, possessing a magnitude and a dire ...

,
angular momentum In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of Momentum, linear momentum. It is an important physical quantity because it is a Conservation law, conserved quantity—the total angular ...

, and other quantities of a bound system are restricted to discrete values ( quantization); objects have characteristics of both
particle In the Outline of physical science, physical sciences, a particle (or corpuscule in older texts) is a small wikt:local, localized physical body, object which can be described by several physical property, physical or chemical property, chemical ...

s and
wave In physics, mathematics, and related fields, a wave is a propagating dynamic disturbance (change from List of types of equilibrium, equilibrium) of one or more quantities. Waves can be Periodic function, periodic, in which case those quantities ...

s (
wave–particle duality Wave–particle duality is the concept in quantum mechanics that every particle or quantum entity may be described as either a particle or a wave. It expresses the inability of the classical physics, classical concepts "particle" or "wave" to fu ...
); 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 In quantum mechanics, the uncertainty principle (also known as Heisenberg's uncertainty principle) is any of a variety of Inequality (mathematics), mathematical inequalities asserting a fundamental limit to the accuracy with which the values fo ...

). Quantum mechanics arose gradually from theories to explain observations which could not be reconciled with classical physics, such as
Max Planck Max Karl Ernst Ludwig Planck (, ; 23 April 1858 – 4 October 1947) was a Germans, German theoretical physicist whose discovery of quantum mechanics, energy quanta won him the Nobel Prize in Physics in 1918. Planck made many substantial con ...

's solution in 1900 to the
black-body radiation Black-body radiation is the thermal electromagnetic radiation within, or surrounding, a body in thermodynamic equilibrium with its environment, emitted by a black body (an idealized opaque, non-reflective body). It has a specific, continuous spe ...
problem, and the correspondence between energy and frequency in
Albert Einstein Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born Theoretical physics, theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for d ...

's which explained the
photoelectric effect The photoelectric effect is the emission of electrons when electromagnetic radiation, such as light, hits a material. Electrons emitted in this manner are called photoelectrons. The phenomenon is studied in condensed matter physics, and Solid-stat ...

. These early attempts to understand microscopic phenomena, now known as the "
old quantum theory The old quantum theory is a collection of results from the years 1900–1925 which predate modern quantum mechanics. The theory was never complete or self-consistent, but was rather a set of heuristic corrections to classical mechanics. The theory ...
", led to the full development of quantum mechanics in the mid-1920s by
Niels Bohr Niels Henrik David Bohr (; 7 October 1885 – 18 November 1962) was a Danes, Danish physicist who made foundational contributions to understanding atomic structure and old quantum theory, quantum theory, for which he received the Nobel ...

, Erwin Schrödinger, Werner Heisenberg,
Max Born Max Born (; 11 December 1882 – 5 January 1970) was a German physicist and mathematician who was instrumental in the development of quantum mechanics. He also made contributions to solid-state physics and optics and supervised the work of a n ...

,
Paul Dirac Paul Adrien Maurice Dirac (; 8 August 1902 – 20 October 1984) was an English theoretical physicist who is regarded as one of the most significant physicists of the 20th century. He was the Lucasian Professor of Mathematics at the Univer ...

and others. The modern theory is formulated in various specially developed mathematical formalisms. In one of them, a mathematical entity called the provides information, in the form of
probability amplitude In quantum mechanics Quantum mechanics is a fundamental Scientific theory, theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quant ...
s, about what measurements of a particle's energy, momentum, and other physical properties may yield.

# Overview and fundamental concepts

Quantum mechanics allows the calculation of properties and behaviour of physical systems. It is typically applied to microscopic systems: molecules, atoms and sub-atomic particles. It has been demonstrated to hold for complex molecules with thousands of atoms, but its application to human beings raises philosophical problems, such as
Wigner's friend Wigner's friend is a thought experiment in theoretical Quantum mechanics, quantum physics, first conceived by the physicist Eugene Wigner in 1961, Reprinted in and further developed by David Deutsch in 1985. The scenario involves an indirect obser ...
, and its application to the universe as a whole remains speculative. Predictions of quantum mechanics have been verified experimentally to an extremely high degree of
accuracy Accuracy and precision are two measures of ''observational error''. ''Accuracy'' is how close a given set of measurements (observations or readings) are to their ''true value'', while ''precision'' is how close the measurements are to each other ...

. A fundamental feature of the theory is that it usually cannot predict with certainty what will happen, but only give probabilities. Mathematically, a probability is found by taking the square of the absolute value of a
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a ...

, known as a probability amplitude. This is known as the
Born rule The Born rule (also called Born's rule) is a key postulate of quantum mechanics which gives the probability that a measurement of a quantum system will yield a given result. In its simplest form, it states that the probability density of fin ...
, named after physicist
Max Born Max Born (; 11 December 1882 – 5 January 1970) was a German physicist and mathematician who was instrumental in the development of quantum mechanics. He also made contributions to solid-state physics and optics and supervised the work of a n ...

. For example, a quantum particle like an
electron The electron ( or ) is a subatomic particle with a negative one elementary charge, elementary electric charge. Electrons belong to the first generation (particle physics), generation of the lepton particle family, and are generally thought t ...

can be described by a , which associates to each point in space a probability amplitude. Applying the Born rule to these amplitudes gives a
probability density function In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) ...
for the position that the electron will be found to have when an experiment is performed to measure it. This is the best the theory can do; it cannot say for certain where the electron will be found. The
Schrödinger equation The Schrödinger equation is a linear differential equation, linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landm ...
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 In quantum mechanics, the uncertainty principle (also known as Heisenberg's uncertainty principle) is any of a variety of Inequality (mathematics), mathematical inequalities asserting a fundamental limit to the accuracy with which the values fo ...

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 at the same time for a measurement of its
momentum In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the Multiplication, product of the mass and velocity of an object. It is a Euclidean vector, vector quantity, possessing a magnitude and a dire ...

. 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 modern physics, the double-slit experiment is a demonstration that light and matter can display characteristics of both classically defined waves and particles; moreover, it displays the fundamentally probabilistic nature of quantum mechanics ...
. In the basic version of this experiment, a coherent light source, such as a
laser A laser is a device that emits light through a process of optical amplification based on the stimulated emission of electromagnetic radiation. The word "laser" is an acronym for "light amplification by stimulated emission of radiation". The fi ...

beam, illuminates a plate pierced by two parallel slits, and the light passing through the slits is observed on a screen behind the plate. 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. 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 A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are Massless particle, massless ...

passes through one slit (as would a classical particle), and not through both slits (as would a wave). 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. This behavior is known as
wave–particle duality Wave–particle duality is the concept in quantum mechanics that every particle or quantum entity may be described as either a particle or a wave. It expresses the inability of the classical physics, classical concepts "particle" or "wave" to fu ...
. Another counter-intuitive phenomenon predicted by quantum mechanics is
quantum tunnelling Quantum tunnelling, also known as tunneling (American English, US) is a quantum mechanics, quantum mechanical phenomenon whereby a wavefunction can propagate through a potential barrier. The transmission through the barrier can be finite and de ...

: a particle that goes up against a can cross it, even if its kinetic energy is smaller than the maximum of the potential. In classical mechanics this particle would be trapped. Quantum tunnelling has several important consequences, enabling
radioactive decay Radioactive decay (also known as nuclear decay, radioactivity, radioactive Decay chain, disintegration, or nuclear disintegration) is the process by which an unstable atomic nucleus loses energy by radiation. A material containing unstable nucl ...

,
nuclear fusion Nuclear fusion is a reaction in which two or more atomic nuclei are combined to form one or more different atomic nuclei and subatomic particles ( neutrons or protons). The difference in mass between the reactants and products is mani ...

in stars, and applications such as scanning tunnelling microscopy and the . When quantum systems interact, the result can be the creation of quantum entanglement: their properties become so intertwined that a description of the whole solely in terms of the individual parts is no longer possible. Erwin Schrödinger called entanglement "...''the'' characteristic trait of quantum mechanics, the one that enforces its entire departure from classical lines of thought". Quantum entanglement enables the counter-intuitive properties of quantum pseudo-telepathy, and can be a valuable resource in communication protocols, such as
quantum key distribution Quantum key distribution (QKD) is a secure communication method which implements a cryptographic protocol involving components of quantum mechanics. It enables two parties to produce a shared randomness, random secret key (cryptography), key known ...
and superdense coding. Contrary to popular misconception, entanglement does not allow sending signals faster than light, as demonstrated by the no-communication theorem. 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. Many Bell tests have been performed, using entangled particles, and they have shown results incompatible with the constraints imposed by local hidden variables. It is not possible to present these concepts in more than a superficial way without introducing the actual mathematics involved; understanding quantum mechanics requires not only manipulating complex numbers, but also
linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrix (mat ...
,
differential equation In mathematics, a differential equation is an functional equation, equation that relates one or more unknown function (mathematics), functions and their derivatives. In applications, the functions generally represent physical quantities, the der ...
s,
group theory In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...
, and other more advanced subjects. Accordingly, this article will present a mathematical formulation of quantum mechanics and survey its application to some useful and oft-studied examples.

# Mathematical formulation

In the mathematically rigorous formulation of quantum mechanics, the state of a quantum mechanical system is a vector $\psi$ belonging to a ( separable) complex
Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise naturally ...
$\mathcal H$. This vector is postulated to be normalized under the Hilbert space inner product, that is, it obeys $\langle \psi,\psi \rangle = 1$, and it is well-defined up to a complex number of modulus 1 (the global phase), that is, $\psi$ and $e^\psi$ represent the same physical system. In other words, the possible states are points in the
projective space In mathematics, the concept of a projective space originated from the visual effect of perspective (graphical), perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean s ...
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 $L^2\left(\mathbb C\right)$, while the Hilbert space for the spin of a single proton is simply the space of two-dimensional complex vectors $\mathbb C^2$ with the usual inner product. Physical quantities of interestposition, momentum, energy, spinare represented by observables, which are
Hermitian {{Short description, none Numerous things are named after the French mathematician Charles Hermite (1822–1901): Hermite * Cubic Hermite spline, a type of third-degree spline * Gauss–Hermite quadrature, an extension of Gaussian quadrature meth ...
(more precisely,
self-adjoint In mathematics, and more specifically in abstract algebra, an element ''x'' of a *-algebra is self-adjoint if x^*=x. A self-adjoint element is also Hermitian matrix, Hermitian, though the reverse doesn't necessarily hold. A collection ''C'' of e ...
) linear operators acting on the Hilbert space. A quantum state can be an
eigenvector In linear algebra, an eigenvector () or characteristic vector of a Linear map, linear transformation is a nonzero Vector space, vector that changes at most by a Scalar (mathematics), scalar factor when that linear transformation is applied to i ...
of an observable, in which case it is called an
eigenstate In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement in quantum mechanics, measurement on a system. Knowledge of the quantum state together with the rul ...
, and the associated
eigenvalue In linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces ...
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 Quantum superposition is a fundamental principle of quantum mechanics. It states that, much like waves in Superposition principle#Wave superposition, classical physics, any two (or more) quantum states can be added together ("superposed") an ...
. When an observable is measured, the result will be one of its eigenvalues with probability given by the
Born rule The Born rule (also called Born's rule) is a key postulate of quantum mechanics which gives the probability that a measurement of a quantum system will yield a given result. In its simplest form, it states that the probability density of fin ...
: in the simplest case the eigenvalue $\lambda$ is non-degenerate and the probability is given by $, \langle \vec\lambda,\psi\rangle, ^2$, where $\vec\lambda$ is its associated eigenvector. More generally, the eigenvalue is degenerate and the probability is given by $\langle \psi,P_\lambda\psi\rangle$, where $P_\lambda$ is the projector onto its associated eigenspace. In the continuous case, these formulas give instead the probability density. After the measurement, if result $\lambda$ was obtained, the quantum state is postulated to collapse to $\vec\lambda$, in the non-degenerate case, or to $P_\lambda\psi/\sqrt$, in the general case. The
probabilistic Probability is the branch of mathematics concerning numerical descriptions of how likely an Event (probability theory), event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and ...
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 Bohr–Einstein debates, in which the two scientists attempted to clarify these fundamental principles by way of
thought experiment A thought experiment is a hypothetical situation in which a hypothesis, theory, or principle is laid out for the purpose of thinking through its consequences. History The ancient Greek ''deiknymi'' (), or thought experiment, "was the most anci ...
s. 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 An interpretation of quantum mechanics is an attempt to explain how the mathematical theory of quantum mechanics might correspond to experienced reality. Although quantum mechanics has held up to rigorous and extremely precise tests in an extraord ...
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 In quantum physics, a measurement is the testing or manipulation of a physical system to yield a numerical result. The predictions that quantum physics makes are in general probability, probabilistic. The mathematical tools for making predictions ...
.
Chapter 8, p. 215
The time evolution of a quantum state is described by the
Schrödinger equation The Schrödinger equation is a linear differential equation, linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landm ...
: :$i\hbar \psi \left(t\right) =H \psi \left(t\right).$ Here $H$ denotes the Hamiltonian, the observable corresponding to the total energy of the system, and $\hbar$ is the reduced
Planck constant The Planck constant, or Planck's constant, is a fundamental physical constant of foundational importance in quantum mechanics. The constant gives the relationship between the energy of a photon and its frequency, and by the mass-energy equivalenc ...
. The constant $i\hbar$ 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 In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical science ...
. The solution of this differential equation is given by :$\psi\left(t\right) = e^\psi\left(0\right).$ The operator $U\left(t\right) = e^$ is known as the time-evolution operator, and has the crucial property that it is unitary. This time evolution is
deterministic Determinism is a Philosophy, philosophical view, where all events are determined completely by previously existing causes. Deterministic theories throughout the history of philosophy have developed from diverse and sometimes overlapping motive ...
in the sense that – given an initial quantum state $\psi\left(0\right)$  – it makes a definite prediction of what the quantum state $\psi\left(t\right)$ will be at any later time. 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 The electron ( or ) is a subatomic particle with a negative one elementary charge, elementary electric charge. Electrons belong to the first generation (particle physics), generation of the lepton particle family, and are generally thought t ...

in an unexcited
atom Every atom is composed of a atomic nucleus, nucleus and one or more electrons bound to the nucleus. The nucleus is made of one or more protons and a number of neutrons. Only the most common variety of hydrogen has no neutrons. Every solid, l ...

is pictured classically as a particle moving in a circular trajectory around the
atomic nucleus The atomic nucleus is the small, dense region consisting of protons and neutrons at the center of an atom, discovered in 1911 by Ernest Rutherford based on the 1909 Geiger–Marsden experiments, Geiger–Marsden gold foil experiment. After th ...
, 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 Schrödinger equation are known for very few relatively simple model Hamiltonians including the
quantum harmonic oscillator 量子調和振動子 は、 古典調和振動子 の 量子力学 類似物です。任意の滑らかな ポテンシャル は通常、安定した 平衡点 の近くで 調和ポテンシャル として近似できるため、最 ...
, the particle in a box, the dihydrogen cation, and the
hydrogen atom A hydrogen atom is an atom of the chemical element hydrogen. The Electric charge, electrically neutral atom contains a single positively charged proton and a single negatively charged electron bound to the nucleus by the Coulomb force. Atomic ...
. Even the
helium Helium (from el, ἥλιος, helios, lit=sun) is a chemical element with the symbol (chemistry), symbol He and atomic number 2. It is a colorless, odorless, tasteless, non-toxic, inert gas, inert, monatomic gas and the first in the noble gas gr ...
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 In mathematics and applied mathematics, perturbation theory comprises methods for finding an approximation theory, approximate solution to a problem, by starting from the exact solution (equation), solution of a related, simpler problem. A crit ...
, 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 In physics, potential energy is the energy held by an object because of its position relative to other objects, stresses within itself, its electric charge, or other factors. Common types of potential energy include the gravitational potentia ...
. 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.

## Uncertainty principle

One consequence of the basic quantum formalism is the
uncertainty principle In quantum mechanics, the uncertainty principle (also known as Heisenberg's uncertainty principle) is any of a variety of Inequality (mathematics), mathematical inequalities asserting a fundamental limit to the accuracy with which the values fo ...

. 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. Both position and momentum are observables, meaning that they are represented by Hermitian operators. The position operator $\hat$ and momentum operator $\hat$ do not commute, but rather satisfy the canonical commutation relation: : Given a quantum state, the Born rule lets us compute expectation values for both $X$ and $P$, and moreover for powers of them. Defining the uncertainty for an observable by a
standard deviation In statistics, the standard Deviation (statistics), deviation is a measure of the amount of variation or statistical dispersion, dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean (al ...
, we have :$\sigma_X=\sqrt,$ and likewise for the momentum: :$\sigma_P=\sqrt.$ The uncertainty principle states that :$\sigma_X \sigma_P \geq \frac.$ Either standard deviation can in principle be made arbitrarily small, but not both simultaneously.Section 3.2 of . This fact is experimentally well-known for example in quantum optics; see e.g. chap. 2 and Fig. 2.1 This inequality generalizes to arbitrary pairs of self-adjoint operators $A$ and $B$. The
commutator In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory. Group theory The commutator of two elements, a ...
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 A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transform ...
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 $i/\hbar$ 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 $p_i$ is replaced by $-i \hbar \frac$, and in particular in the non-relativistic Schrödinger equation in position space the momentum-squared term is replaced with a Laplacian times $-\hbar^2$.

## Composite systems and entanglement

When two different quantum systems are considered together, the Hilbert space of the combined system is the
tensor product In mathematics, the tensor product V \otimes W of two vector spaces and (over the same Field (mathematics), field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an e ...
of the Hilbert spaces of the two components. For example, let and be two quantum systems, with Hilbert spaces $\mathcal H_A$ and $\mathcal H_B$, respectively. The Hilbert space of the composite system is then : $\mathcal H_ = \mathcal H_A \otimes \mathcal H_B.$ If the state for the first system is the vector $\psi_A$ and the state for the second system is $\psi_B$, then the state of the composite system is : $\psi_A \otimes \psi_B.$ Not all states in the joint Hilbert space $\mathcal H_$ 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 $\psi_A$ and $\phi_A$ are both possible states for system $A$, and likewise $\psi_B$ and $\phi_B$ are both possible states for system $B$, then : $\tfrac \left \left( \psi_A \otimes \psi_B + \phi_A \otimes \phi_B \right \right)$ is a valid joint state that is not separable. States that are not separable are called entangled. If the state for a composite system is entangled, it is impossible to describe either component system or system 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. 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. As described above, entanglement is a key feature of models of measurement processes in which an apparatus becomes entangled with the system being measured. Systems interacting with the environment in which they reside generally become entangled with that environment, a phenomenon known as
quantum decoherence Quantum decoherence is the loss of Coherence (physics)#Quantum coherence, quantum coherence. In quantum mechanics, particles such as electrons are described by a wave function, a mathematical representation of the quantum state of a system; a p ...
. This can explain why, in practice, quantum effects are difficult to observe in systems larger than microscopic.

## Equivalence between formulations

There are many mathematically equivalent formulations of quantum mechanics. One of the oldest and most common is the " transformation theory" proposed by
Paul Dirac Paul Adrien Maurice Dirac (; 8 August 1902 – 20 October 1984) was an English theoretical physicist who is regarded as one of the most significant physicists of the 20th century. He was the Lucasian Professor of Mathematics at the Univer ...

, which unifies and generalizes the two earliest formulations of quantum mechanics –
matrix mechanics Matrix mechanics is a formulation of quantum mechanics created by Werner Heisenberg, Max Born, and Pascual Jordan in 1925. It was the first conceptually autonomous and logically consistent formulation of quantum mechanics. Its account of Atomic el ...
(invented by Werner Heisenberg) and wave mechanics (invented by Erwin Schrödinger). An alternative formulation of quantum mechanics is Feynman's
path integral formulation The path integral formulation is a description in quantum mechanics Quantum mechanics is a fundamental Scientific theory, theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatom ...
, 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.

## Symmetries and conservation laws

The Hamiltonian $H$ is known as the ''generator'' of time evolution, since it defines a unitary time-evolution operator $U\left(t\right) = e^$ for each value of $t$. From this relation between $U\left(t\right)$ and $H$, it follows that any observable $A$ that commutes with $H$ will be ''conserved'': its expectation value will not change over time. This statement generalizes, as mathematically, any Hermitian operator $A$ can generate a family of unitary operators parameterized by a variable $t$. Under the evolution generated by $A$, any observable $B$ that commutes with $A$ will be conserved. Moreover, if $B$ is conserved by evolution under $A$, then $A$ is conserved under the evolution generated by $B$. This implies a quantum version of the result proven by
Emmy Noether Amalie Emmy NoetherEmmy (given name), Emmy is the ''Rufname'', the second of two official given names, intended for daily use. Cf. for example the résumé submitted by Noether to Erlangen University in 1907 (Erlangen University archive, ''Promot ...
in classical ( Lagrangian) mechanics: for every
differentiable In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in mo ...
symmetry Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
of a Hamiltonian, there exists a corresponding
conservation law In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical science ...
.

# Examples

## Free particle

The simplest example of a 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: :$H = \fracP^2 = - \frac \frac .$ The general solution of the Schrödinger equation is given by :$\psi \left(x,t\right)=\frac \int _^\infty\left(k,0\right)e^\mathrmk,$ which is a superposition of all possible
plane wave In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical science ...
s $e^$, which are eigenstates of the momentum operator with momentum $p = \hbar k$. The coefficients of the superposition are $\hat \left(k,0\right)$, which is the Fourier transform of the initial quantum state $\psi\left(x,0\right)$. 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. Instead, we can consider a Gaussian
wave packet In physics, a wave packet (or wave train) is a short "burst" or "Wave envelope, envelope" of localized wave action that travels as a unit. A wave packet can be analyzed into, or can be synthesized from, an infinite set of component sinusoidal ...
: :$\psi\left(x,0\right) = \frace^$ 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 wave packet 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.

## Particle in a box

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. For the one-dimensional case in the $x$ direction, the time-independent Schrödinger equation may be written : $- \frac \frac = E \psi.$ With the differential operator defined by : $\hat_x = -i\hbar\frac$ the previous equation is evocative of the classic kinetic energy analogue, : $\frac \hat_x^2 = E,$ with state $\psi$ in this case having energy $E$ coincident with the kinetic energy of the particle. The general solutions of the Schrödinger equation for the particle in a box are : $\psi\left(x\right) = A e^ + B e ^ \qquad\qquad E = \frac$ or, from
Euler's formula Euler's formula, named after Leonhard Euler, is a mathematics, mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex number, complex exponential function. Euler ...
, : $\psi\left(x\right) = C \sin\left(kx\right) + D \cos\left(kx\right).\!$ The infinite potential walls of the box determine the values of $C, D,$ and $k$ at $x=0$ and $x=L$ where $\psi$ must be zero. Thus, at $x=0$, :$\psi\left(0\right) = 0 = C\sin\left(0\right) + D\cos\left(0\right) = D$ and $D=0$. At $x=L$, :$\psi\left(L\right) = 0 = C\sin\left(kL\right),$ in which $C$ cannot be zero as this would conflict with the postulate that $\psi$ has norm 1. Therefore, since $\sin\left(kL\right)=0$, $kL$ must be an integer multiple of $\pi$, :$k = \frac\qquad\qquad n=1,2,3,\ldots.$ This constraint on $k$ implies a constraint on the energy levels, yielding $E_n = \frac = \frac.$ 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 microscope, scanning tunneling microscopy.

## Harmonic oscillator

As in the classical case, the potential for the quantum harmonic oscillator is given by :$V\left(x\right)=\fracm\omega^2x^2.$ This problem can either be treated by directly solving the Schrödinger equation, which is not trivial, or by using the more elegant "ladder method" first proposed by Paul Dirac. The
eigenstate In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement in quantum mechanics, measurement on a system. Knowledge of the quantum state together with the rul ...
s are given by :$\psi_n\left(x\right) = \sqrt \cdot \left\left(\frac\right\right)^ \cdot e^ \cdot H_n\left\left(\sqrt x \right\right), \qquad$ :$n = 0,1,2,\ldots.$ where ''Hn'' are the Hermite polynomials :$H_n\left(x\right)=\left(-1\right)^n e^\frac\left\left(e^\right\right),$ and the corresponding energy levels are :$E_n = \hbar \omega \left\left(n + \right\right).$ This is another example illustrating the discretization of energy for bound states.

## Mach–Zehnder interferometer

The Mach–Zehnder interferometer (MZI) illustrates the concepts of superposition and interference with linear algebra in dimension 2, rather than differential equations. It can be seen as a simplified version of the double-slit experiment, but it is of interest in its own right, for example in the delayed choice quantum eraser, the Elitzur–Vaidman bomb tester, and in studies of quantum entanglement. We can model a photon going through the interferometer by considering that at each point it can be in a superposition of only two paths: the "lower" path which starts from the left, goes straight through both beam splitters, and ends at the top, and the "upper" path which starts from the bottom, goes straight through both beam splitters, and ends at the right. The quantum state of the photon is therefore a vector $\psi \in \mathbb^2$ that is a superposition of the "lower" path $\psi_l = \begin 1 \\ 0 \end$ and the "upper" path $\psi_u = \begin 0 \\ 1 \end$, that is, $\psi = \alpha \psi_l + \beta \psi_u$ for complex $\alpha,\beta$. In order to respect the postulate that $\langle \psi,\psi\rangle = 1$ we require that $, \alpha, ^2+, \beta, ^2 = 1$. Both beam splitters are modelled as the unitary matrix $B = \frac1\begin 1 & i \\ i & 1 \end$, which means that when a photon meets the beam splitter it will either stay on the same path with a probability amplitude of $1/\sqrt$, or be reflected to the other path with a probability amplitude of $i/\sqrt$. The phase shifter on the upper arm is modelled as the unitary matrix $P = \begin 1 & 0 \\ 0 & e^ \end$, which means that if the photon is on the "upper" path it will gain a relative phase of $\Delta\Phi$, and it will stay unchanged if it is in the lower path. A photon that enters the interferometer from the left will then be acted upon with a beam splitter $B$, a phase shifter $P$, and another beam splitter $B$, and so end up in the state :$BPB\psi_l = ie^ \begin -\sin\left(\Delta\Phi/2\right) \\ \cos\left(\Delta\Phi/2\right) \end,$ and the probabilities that it will be detected at the right or at the top are given respectively by :$p\left(u\right) = , \langle \psi_u, BPB\psi_l \rangle, ^2 = \cos^2 \frac,$ :$p\left(l\right) = , \langle \psi_l, BPB\psi_l \rangle, ^2 = \sin^2 \frac.$ One can therefore use the Mach–Zehnder interferometer to estimate the Phase (waves), phase shift by estimating these probabilities. It is interesting to consider what would happen if the photon were definitely in either the "lower" or "upper" paths between the beam splitters. This can be accomplished by blocking one of the paths, or equivalently by removing the first beam splitter (and feeding the photon from the left or the bottom, as desired). In both cases there will be no interference between the paths anymore, and the probabilities are given by $p\left(u\right)=p\left(l\right) = 1/2$, independently of the phase $\Delta\Phi$. From this we can conclude that the photon does not take one path or another after the first beam splitter, but rather that it is in a genuine quantum superposition of the two paths.

# Applications

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 physics, classical methods. Quantum mechanics is often the only theory that can reveal the individual behaviors of the
subatomic particle In physical sciences, a subatomic particle is a particle that composes an atom Every atom is composed of a atomic nucleus, nucleus and one or more electrons bound to the nucleus. The nucleus is made of one or more protons and a number of n ...
s that make up all forms of matter (
electron The electron ( or ) is a subatomic particle with a negative one elementary charge, elementary electric charge. Electrons belong to the first generation (particle physics), generation of the lepton particle family, and are generally thought t ...

s, protons, neutrons,
photon A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are Massless particle, massless ...

s, and others). Solid-state physics and materials science are dependent upon quantum mechanics. In many aspects modern technology operates at a scale where quantum effects are significant. Important applications of quantum theory include
quantum chemistry Quantum chemistry, also called molecular quantum mechanics, is a branch of physical chemistry focused on the application of quantum mechanics to chemical systems, particularly towards the quantum-mechanical calculation of electronic contributions ...
, quantum optics, quantum computing, superconducting magnets, light-emitting diodes, the optical amplifier and the
laser A laser is a device that emits light through a process of optical amplification based on the stimulated emission of electromagnetic radiation. The word "laser" is an acronym for "light amplification by stimulated emission of radiation". The fi ...

, the transistor and semiconductors such as the microprocessor, medical imaging, medical and research imaging such as magnetic resonance imaging and electron microscope, electron microscopy. Explanations for many biological and physical phenomena are rooted in the nature of the chemical bond, most notably the macro-molecule DNA.

# Relation to other scientific theories

## Classical mechanics

The rules of quantum mechanics assert that the state space of a system is a
Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise naturally ...
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 In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical science ...
, a heuristic which states that the predictions of quantum mechanics reduce to those of classical mechanics in the regime of large quantum numbers. 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 Canonical quantization, quantization. When quantum mechanics was originally formulated, it was applied to models whose correspondence limit was theory of relativity, non-relativistic classical mechanics. For instance, the well-known model of the
quantum harmonic oscillator 量子調和振動子 は、 古典調和振動子 の 量子力学 類似物です。任意の滑らかな ポテンシャル は通常、安定した 平衡点 の近くで 調和ポテンシャル として近似できるため、最 ...
uses an explicitly non-relativistic expression for the kinetic energy of the oscillator, and is thus a quantum version of the harmonic oscillator, classical harmonic oscillator. Complications arise with Chaos theory, chaotic systems, which do not have good quantum numbers, and quantum chaos studies the relationship between classical and quantum descriptions in these systems. Quantum decoherence is a mechanism through which quantum systems lose quantum coherence, coherence, and thus become incapable of displaying many typically quantum effects:
quantum superposition Quantum superposition is a fundamental principle of quantum mechanics. It states that, much like waves in Superposition principle#Wave superposition, classical physics, any two (or more) quantum states can be added together ("superposed") an ...
s become simply probabilistic mixtures, and quantum entanglement becomes simply classical correlations. Quantum coherence is not typically evident at macroscopic scales, except maybe at temperatures approaching absolute zero at which quantum behavior may manifest macroscopically. Many macroscopic properties of a classical system are a direct consequence of the quantum behavior of its parts. For example, the stability of bulk matter (consisting of atoms and molecules which would quickly collapse under electric forces alone), the rigidity of solids, and the mechanical, thermal, chemical, optical and magnetic properties of matter are all results of the interaction of electric charges under the rules of quantum mechanics.

## Special relativity and electrodynamics

Early attempts to merge quantum mechanics with special relativity involved the replacement of the Schrödinger equation with a covariant equation such as the Klein–Gordon 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 electromagnetism, electromagnetic interaction. Quantum electrodynamics is, along with general relativity, one of the most accurate physical theories ever devised. 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 electric charge, charged particles as quantum mechanical objects being acted on by a classical electromagnetic field. For example, the elementary quantum model of the
hydrogen atom A hydrogen atom is an atom of the chemical element hydrogen. The Electric charge, electrically neutral atom contains a single positively charged proton and a single negatively charged electron bound to the nucleus by the Coulomb force. Atomic ...
describes the electric field of the hydrogen atom using a classical $\textstyle -e^2/\left(4 \pi\epsilon_r\right)$ Electric potential, Coulomb potential. This "semi-classical" approach fails if quantum fluctuations in the electromagnetic field play an important role, such as in the emission of
photon A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are Massless particle, massless ...

s by charged particles. Field (physics), 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.

## Relation to general relativity

Even though the predictions of both quantum theory and general relativity have been supported by rigorous and repeated empirical evidence, their abstract formalisms contradict each other and they have proven extremely difficult to incorporate into one consistent, cohesive model. 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, 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. One proposal for doing so is string theory, which posits that the Point particle, point-like particles of particle physics are replaced by Dimension (mathematics and physics), one-dimensional objects called String (physics), 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 (physics), 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 an extremely fine fabric "woven" of finite loops called spin networks. The evolution of a spin network over time is called a spin foam. The characteristic length scale of a spin foam is the Planck length, approximately 1.616×10−35 m, and so lengths shorter than the Planck length are not physically meaningful in LQG.

# Philosophical implications

Since its inception, the many counter-intuitive aspects and results of quantum mechanics have provoked strong philosophy, philosophical debates and many interpretations of quantum mechanics, interpretations. The arguments centre on the probabilistic nature of quantum mechanics, the difficulties with wavefunction collapse and the related measurement problem, and quantum nonlocality. Perhaps the only consensus that exists about these issues is that there is no consensus. Richard Feynman once said, "I think I can safely say that nobody understands quantum mechanics." According to Steven Weinberg, "There is now in my opinion no entirely satisfactory interpretation of quantum mechanics." The views of
Niels Bohr Niels Henrik David Bohr (; 7 October 1885 – 18 November 1962) was a Danes, Danish physicist who made foundational contributions to understanding atomic structure and old quantum theory, quantum theory, for which he received the Nobel ...

, Werner Heisenberg and other physicists are often grouped together as the "Copenhagen interpretation". 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 complementarity (physics), complementary nature of evidence obtained under different experimental situations. Copenhagen-type interpretations remain popular in the 21st century.
Albert Einstein Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born Theoretical physics, theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for d ...

, himself one of the founders of Old quantum theory, quantum theory, was troubled by its apparent failure to respect some cherished metaphysical principles, such as determinism and principle of locality, locality. Einstein's long-running exchanges with Bohr about the meaning and status of quantum mechanics are now known as the Bohr–Einstein debates. Einstein believed that underlying quantum mechanics must be a theory that explicitly forbids action at a distance. He argued that quantum mechanics was incomplete, a theory that was valid but not fundamental, analogous to how thermodynamics is valid, but the fundamental theory behind it is statistical mechanics. 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 A thought experiment is a hypothetical situation in which a hypothesis, theory, or principle is laid out for the purpose of thinking through its consequences. History The ancient Greek ''deiknymi'' (), or thought experiment, "was the most anci ...
later termed the Einstein–Podolsky–Rosen paradox. In 1964, John Stewart Bell, John Bell showed that EPR's principle of locality, together with determinism, was actually incompatible with quantum mechanics: they implied constraints on the correlations produced by distance systems, now known as Bell inequalities, that can be violated by entangled particles. Since then Bell test, several experiments have been performed to obtain these correlations, with the result that they do in fact violate Bell inequalities, and thus falsify the conjunction of locality with determinism. Bohmian mechanics shows that it is possible to reformulate quantum mechanics to make it deterministic, at the price of making it explicitly nonlocal. It attributes not only a wave function to a physical system, but in addition a real position, that evolves deterministically under a nonlocal guiding equation. The evolution of a physical system is given at all times by the
Schrödinger equation The Schrödinger equation is a linear differential equation, linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landm ...
together with the guiding equation; there is never a collapse of the wave function. This solves the measurement problem. Everett's 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. This is a consequence of removing the axiom of the collapse of the wave packet. All possible states of the measured system and the measuring apparatus, together with the observer, are present in a real physical
quantum superposition Quantum superposition is a fundamental principle of quantum mechanics. It states that, much like waves in Superposition principle#Wave superposition, classical physics, any two (or more) quantum states can be added together ("superposed") an ...
. While the multiverse is deterministic, we perceive non-deterministic behavior governed by probabilities, because we don't observe the multiverse as a whole, but only one parallel universe at a time. Exactly how this is supposed to work has been the subject of much debate. Several attempts have been made to make sense of this and derive the Born rule, with no consensus on whether they have been successful. Relational quantum mechanics appeared in the late 1990s as a modern derivative of Copenhagen-type ideas, and QBism was developed some years later.

# History

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. In 1803 English polymath Thomas Young (scientist), Thomas Young described the famous Young's interference experiment, double-slit experiment. This experiment played a major role in the general acceptance of the wave theory of light. During the early 19th century, chemistry, chemical research by John Dalton and Amedeo Avogadro lent weight to the atomic theory of matter, an idea that James Clerk Maxwell, Ludwig Boltzmann and others built upon to establish the kinetic theory of gases. The successes of kinetic theory gave further credence to the idea that matter is composed of atoms, yet the theory also had shortcomings that would only be resolved by the development of quantum mechanics. While the early conception of atoms from Greek philosophy had been that they were indivisible units the word "atom" deriving from the Greek for "uncuttable" the 19th century saw the formulation of hypotheses about subatomic structure. One important discovery in that regard was Michael Faraday's 1838 observation of a glow caused by an electrical discharge inside a glass tube containing gas at low pressure. Julius Plücker, Johann Wilhelm Hittorf and Eugen Goldstein carried on and improved upon Faraday's work, leading to the identification of cathode rays, which J. J. Thomson found to consist of subatomic particles that would be called electrons. The
black-body radiation Black-body radiation is the thermal electromagnetic radiation within, or surrounding, a body in thermodynamic equilibrium with its environment, emitted by a black body (an idealized opaque, non-reflective body). It has a specific, continuous spe ...
problem was discovered by Gustav Kirchhoff in 1859. In 1900,
Max Planck Max Karl Ernst Ludwig Planck (, ; 23 April 1858 – 4 October 1947) was a Germans, German theoretical physicist whose discovery of quantum mechanics, energy quanta won him the Nobel Prize in Physics in 1918. Planck made many substantial con ...

proposed the hypothesis that energy is radiated and absorbed in discrete "quanta" (or energy packets), yielding a calculation that precisely matched the observed patterns of black-body radiation. The word ''quantum'' derives from the Latin language, Latin, meaning "how great" or "how much". According to Planck, quantities of energy could be thought of as divided into "elements" whose size (''E'') would be proportional to their frequency (''ν''): :$E = h \nu\$, where ''h'' is Planck constant, 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. In fact, he considered his quantum hypothesis a mathematical trick to get the right answer rather than a sizable discovery. However, in 1905
Albert Einstein Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born Theoretical physics, theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for d ...

interpreted Planck's quantum hypothesis local realism, realistically and used it to explain the
photoelectric effect The photoelectric effect is the emission of electrons when electromagnetic radiation, such as light, hits a material. Electrons emitted in this manner are called photoelectrons. The phenomenon is studied in condensed matter physics, and Solid-stat ...

, in which shining light on certain materials can eject electrons from the material.
Niels Bohr Niels Henrik David Bohr (; 7 October 1885 – 18 November 1962) was a Danes, Danish physicist who made foundational contributions to understanding atomic structure and old quantum theory, quantum theory, for which he received the Nobel ...

then developed Planck's ideas about radiation into a Bohr model, model of the hydrogen atom that successfully predicted the spectral lines of hydrogen. 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 A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are Massless particle, massless ...

), with a discrete amount of energy that depends on its frequency. 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, which became the basis of the
laser A laser is a device that emits light through a process of optical amplification based on the stimulated emission of electromagnetic radiation. The word "laser" is an acronym for "light amplification by stimulated emission of radiation". The fi ...

. This phase is known as the
old quantum theory The old quantum theory is a collection of results from the years 1900–1925 which predate modern quantum mechanics. The theory was never complete or self-consistent, but was rather a set of heuristic corrections to classical mechanics. The theory ...
. Never complete or self-consistent, the old quantum theory was rather a set of heuristic corrections to classical mechanics. The theory is now understood as a WKB approximation#Application to the Schr.C3.B6dinger equation, semi-classical approximation to modern quantum mechanics. Notable results from this period include, in addition to the work of Planck, Einstein and Bohr mentioned above, Einstein and Peter Debye's work on the specific heat of solids, Bohr and Hendrika Johanna van Leeuwen's Bohr–Van Leeuwen theorem, proof that classical physics cannot account for diamagnetism, and Arnold Sommerfeld's extension of the Bohr model to include special-relativistic effects. In the mid-1920s quantum mechanics was developed to become the standard formulation for atomic physics. In 1923, the French physicist Louis-Victor de Broglie, Louis de Broglie put forward his theory of matter waves by stating that particles can exhibit wave characteristics and vice versa. Building on de Broglie's approach, modern quantum mechanics was born in 1925, when the German physicists Werner Heisenberg,
Max Born Max Born (; 11 December 1882 – 5 January 1970) was a German physicist and mathematician who was instrumental in the development of quantum mechanics. He also made contributions to solid-state physics and optics and supervised the work of a n ...

, and Pascual JordanDavid Edwards,"The Mathematical Foundations of Quantum Mechanics", ''Synthese'', Volume 42, Number 1/September, 1979, pp. 1–70.D. Edwards, "The Mathematical Foundations of Quantum Field Theory: Fermions, Gauge Fields, and Super-symmetry, Part I: Lattice Field Theories", ''International J. of Theor. Phys.'', Vol. 20, No. 7 (1981). developed
matrix mechanics Matrix mechanics is a formulation of quantum mechanics created by Werner Heisenberg, Max Born, and Pascual Jordan in 1925. It was the first conceptually autonomous and logically consistent formulation of quantum mechanics. Its account of Atomic el ...
and the Austrian physicist Erwin Schrödinger invented wave mechanics. Born introduced the probabilistic interpretation of Schrödinger's wave function in July 1926. Thus, the entire field of quantum physics emerged, leading to its wider acceptance at the Fifth Solvay Conference in 1927. By 1930 quantum mechanics had been further unified and formalized by David Hilbert,
Paul Dirac Paul Adrien Maurice Dirac (; 8 August 1902 – 20 October 1984) was an English theoretical physicist who is regarded as one of the most significant physicists of the 20th century. He was the Lucasian Professor of Mathematics at the Univer ...

and John von Neumann with greater emphasis on measurement in quantum mechanics, measurement, the statistical nature of our knowledge of reality, and Interpretations of quantum mechanics, 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, periodic table of elements, and describes the behaviors of atoms during chemical bonding and the flow of
electron The electron ( or ) is a subatomic particle with a negative one elementary charge, elementary electric charge. Electrons belong to the first generation (particle physics), generation of the lepton particle family, and are generally thought t ...

s 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 superconductivity, superconductors and superfluids.

* Bra–ket notation * Einstein's thought experiments * List of textbooks on classical and quantum mechanics * Macroscopic quantum phenomena * Phase-space formulation * Regularization (physics) * Two-state quantum system

# References

The following titles, all by working physicists, attempt to communicate quantum theory to lay people, using a minimum of technical apparatus. * Marvin Chester, Chester, Marvin (1987). ''Primer of Quantum Mechanics''. John Wiley. * * Richard Feynman, 1985. ''QED: The Strange Theory of Light and Matter'', Princeton University Press. . Four elementary lectures on quantum electrodynamics and quantum field theory, yet containing many insights for the expert. * Giancarlo Ghirardi, Ghirardi, GianCarlo, 2004. ''Sneaking a Look at God's Cards'', Gerald Malsbary, trans. Princeton Univ. Press. The most technical of the works cited here. Passages using algebra, trigonometry, and bra–ket notation can be passed over on a first reading. * N. David Mermin, 1990, "Spooky actions at a distance: mysteries of the QT" in his ''Boojums All the Way Through''. Cambridge University Press: 110–76. * Victor Stenger, 2000. ''Timeless Reality: Symmetry, Simplicity, and Multiple Universes''. Buffalo, NY: Prometheus Books. Chpts. 5–8. Includes cosmological and philosophical considerations. More technical: * * * * * Bryce DeWitt, R. Neill Graham, eds., 1973. ''The Many-Worlds Interpretation of Quantum Mechanics'', Princeton Series in Physics, Princeton University Press. * * * Daniel Greenberger, D. Greenberger, Klaus Hentschel, K. Hentschel, F. Weinert, eds., 2009. ''Compendium of quantum physics, Concepts, experiments, history and philosophy'', Springer-Verlag, Berlin, Heidelberg. * A standard undergraduate text. * Max Jammer, 1966. ''The Conceptual Development of Quantum Mechanics''. McGraw Hill. * Hagen Kleinert, 2004. ''Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets'', 3rd ed. Singapore: World Scientific
Draft of 4th edition.
*
Online copy
* * Gunther Ludwig, 1968. ''Wave Mechanics''. London: Pergamon Press. * George Mackey (2004). ''The mathematical foundations of quantum mechanics''. Dover Publications. . * * Albert Messiah, 1966. ''Quantum Mechanics'' (Vol. I), English translation from French by G.M. Temmer. North Holland, John Wiley & Sons. Cf. chpt. IV, section III
online
* * Eric R. Scerri, Scerri, Eric R., 2006. ''The Periodic Table: Its Story and Its Significance''. Oxford University Press. Considers the extent to which chemistry and the periodic system have been reduced to quantum mechanics. * * * * Martinus J. G. Veltman, Veltman, Martinus J.G. (2003), ''Facts and Mysteries in Elementary Particle Physics''. On Wikibooks
This Quantum World

* J. O'Connor and E. F. Robertson

Introduction to Quantum Theory at Quantiki.

Quantum Physics Made Relatively Simple
three video lectures by Hans Bethe ; Course material
Quantum Cook Book
an
PHYS 201: Fundamentals of Physics II
by Ramamurti Shankar, Yale OpenCourseware
The Modern Revolution in Physics
– an online textbook. * MIT OpenCourseWare
Chemistry
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5½ Examples in Quantum Mechanics

Imperial College Quantum Mechanics Course.
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