TheInfoList

Quantum mechanics is a fundamental
theory A theory is a reason, rational type of abstraction, 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 ...
in
physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical scien ...

that provides a description of the physical properties of
nature Nature, in the broadest sense, is the natural, physical, material world or universe The universe ( la, universus) is all of space and time and their contents, including planets, stars, galaxies, and all other forms of matter and ...

at the scale of
atom An atom is the smallest unit of ordinary matter In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume. All everyday objects that can be touched are ultimately composed of ato ...

s and
subatomic particle In physical sciences, subatomic particles are smaller than atom An atom is the smallest unit of ordinary matter In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume. All ...
s. It is the foundation of all quantum physics including
quantum chemistry Quantum chemistry, also called molecular quantum mechanics, is a branch of chemistry Chemistry is the scientific discipline involved with Chemical element, elements and chemical compound, compounds composed of atoms, molecules and ions: thei ...
,
quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
,
quantum technology Quantum technology is an emerging field of physics and engineering, which relies on the principles of quantum physics. Quantum computing, quantum sensors, quantum cryptography, quantum simulation, quantum metrology and quantum imaging are al ...
, and
quantum information science Quantum information science is an interdisciplinary field that seeks to understand the analysis, processing, and transmission of information using quantum mechanics Quantum mechanics is a fundamental Scientific theory, theory in physics that p ...
.
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 In physics, physical sciences, subatomic particles are smaller than atoms. They can be composite particles, such as the neutron and proton; or elementary particles, which according to the standard model are not made of other particles. Particle p ...
) 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 is the physical quantity, quantitative physical property, property that must be #Energy transfer, transferred to a physical body, body or physical system to perform Work (thermodynamics), work on the body, or to heat it. En ...

,
momentum In Newtonian mechanics, linear momentum, translational momentum, or simply momentum (plural, pl. momenta) is the product of the mass and velocity of an object. It is a Euclidean vector, vector quantity, possessing a magnitude and a direction. ...

,
angular momentum In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational equivalent of linear momentum. It is an important quantity in physics because it is a conserved quantity—the total angular momentum of a close ...
, and other quantities of a
bound Bound or bounds may refer to: Mathematics * Bound variable * Upper and lower bounds, observed limits of mathematical functions Geography *Bound Brook (Raritan River), a tributary of the Raritan River in New Jersey *Bound Brook, New Jersey, a borou ...
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 to which can be ascribed several physical property, physical or chemical , chemical properties ...

s and
wave In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physics), motion and behavior through ...

s (), 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 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 quantu ...

). 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 German German(s) may refer to: Common uses * of or related to Germany * Germans, Germanic ethnic group, citizens of Germany or people of German ancestry * For citi ...

's solution in 1900 to the
black-body radiation ) of black-body radiation scales inversely with the temperature of the black body; the locus of such colors, shown here in CIE 1931 ''x,y'' space, is known as the Planckian locus. Black-body radiation is the thermal radiation, thermal electro ...
problem, and the correspondence between energy and frequency in
Albert Einstein Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest physicists of all time. Einstein is known for developing the theory of relativity The theory ...
'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-state ...

. 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 Quantum mechanics is a fundamental Scientific theory, theory in physics that provides a description of the physical properties ...
", 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 Danish physicist A physicist is a scientist A scientist is a person who conducts Scientific method, scientific research to advance knowledge in an Branches of ...

,
Erwin Schrödinger Erwin Rudolf Josef Alexander Schrödinger (, ; ; 12 August 1887 – 4 January 1961), sometimes written as or , was a Nobel Prize-winning Austrian-Irish physicist A physicist is a scientist A scientist is a person who conducts Scientific ...
,
Werner Heisenberg Werner Karl Heisenberg (; ; 5 December 1901 – 1 February 1976) was a German theoretical physicist and one of the key pioneers of quantum mechanics Quantum mechanics is a fundamental Scientific theory, theory in physics that provides a de ...
,
Max Born Max Born (; 11 December 1882 – 5 January 1970) was a German physicist A physicist is a scientist A scientist is a person who conducts Scientific method, scientific research to advance knowledge in an Branches of science, area of in ...

and others. The modern theory is formulated in various specially developed mathematical formalisms. In one of them, a mathematical entity called the
wave function A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex number, complex-valued probability amplitude, and the probabilities for the possible results of me ...

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 quantum ...
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, 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 In a set of measurements, accuracy is closeness of the measurements to a specific value, while precision is the closeness of the measurements to each other. ''Accuracy'' has two definitions: # More commonly, it is a description of ''systematic err ...

. 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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
, 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 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 an ...
, named after physicist
Max Born Max Born (; 11 December 1882 – 5 January 1970) was a German physicist A physicist is a scientist A scientist is a person who conducts Scientific method, scientific research to advance knowledge in an Branches of science, area of in ...

. For example, a quantum particle like an
electron The electron is a subatomic particle In physical sciences, subatomic particles are smaller than atom An atom is the smallest unit of ordinary matter In classical physics and general chemistry, matter is any substance that has ma ...
can be described by a
wave function A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex number, complex-valued probability amplitude, and the probabilities for the possible results of me ...

, which associates to each point in space a probability amplitude. Applying the Born rule to these amplitudes gives a
probability density function and probability density function of a normal distribution . Image:visualisation_mode_median_mean.svg, 150px, Geometric visualisation of the mode (statistics), mode, median (statistics), median and mean (statistics), mean of an arbitrary probabilit ...
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 Linearity is the property of a mathematical relationship ('' function'') that can be graphically represented as a straight line. Linearity is closely related to proportionality. Examples in physics ...
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 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 quantu ...

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, linear momentum, translational momentum, or simply momentum (plural, pl. momenta) is the product of the mass and velocity of an object. It is a Euclidean vector, vector quantity, possessing a magnitude and a direction. ...

. 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 Modern physics is a branch of physics either developed in the early 20th century and onward or branches greatly influenced by early 20th century physics. Notable branches of modern physics include quantum physics, special r ...
. 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 term "laser" originated as an acronym for "light amplification by stimulated emission of radia ...

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 The photon ( el, φῶς, phōs, light) is a type of elementary particle In particle physics, an elementary particle or fundamental particle is a subatomic particle that is not composed of other particles. Particles currently thought to be eleme ...

passes through one slit (as would a classical particle), and not through both slits (as would a wave). However, demonstrate that particles do not form the interference pattern if one detects which slit they pass through. Other atomic-scale entities, such as
electrons The electron is a subatomic particle In physical sciences, subatomic particles are smaller than atom An atom is the smallest unit of ordinary matter In classical physics and general chemistry, matter is any substance that has ma ...
, are found to exhibit the same behavior when fired towards a double slit. This behavior is known as . Another counter-intuitive phenomenon predicted by quantum mechanics is
quantum tunnelling Quantum tunnelling or tunneling (US) is the quantum mechanical phenomenon where a wavefunction can propagate through a potential barrier. The transmission through the barrier can be finite and depends exponentially on the barrier height and ba ...
: a particle that goes up against a
potential barrier 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 quantum ...
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 disintegration or nuclear disintegration) is the process by which an unstable atomic nucleus loses energy by radiation. A material containing unstable nuclei is conside ...

,
nuclear fusion 400 px, The nuclear binding energy curve. The formation of nuclei with masses up to iron-56 releases energy, as illustrated above. Nuclear fusion is a nuclear reaction, reaction in which two or more atomic nuclei are combined to form one or m ...

in stars, and applications such as
scanning tunnelling microscopy A scanning tunneling microscope (STM) is a type of microscope A microscope (from the grc, μικρός, ''mikrós'', "small" and , ''skopeîn'', "to look" or "see") is a laboratory instrument used to examine objects that are too small to ...
and the
tunnel diode A tunnel diode or Esaki diode is a type of semiconductor diode that has effectively "negative resistance" due to the quantum mechanics, quantum mechanical effect called quantum tunneling, tunneling. It was invented in August 1957 by Leo Esaki ...

. When quantum systems interact, the result can be the creation of
quantum entanglement Quantum entanglement is a physical phenomenon that occurs when a pair or group of particles is generated, interact, or share spatial proximity in a way such that the quantum state of each particle of the pair or group cannot be described indepen ...
: 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 Quantum pseudo-telepathy is the fact that in certain Bayesian games with asymmetric information, players who have access to a shared physical system in an entangled quantum state, and who are able to execute strategies that are contingent upon m ...
, 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 Quantum mechanics is a fundamental Scientific theory, theory in physics that provides a descr ...
and
superdense coding In quantum information theory Quantum information is the information of the quantum state, state of a quantum system. It is the basic entity of study in quantum information theory, and can be manipulated using quantum information processing t ...

. Contrary to popular misconception, entanglement does not allow sending signals
faster than light Faster-than-light (also superluminal, FTL or supercausal) communications and travel are the conjectural propagation of information or matter faster than the speed of light. The special theory of relativity implies that only particles with zero mas ...
, as demonstrated by the
no-communication theorem In physics, the no-communication theorem or no-signaling principle is a no-go theorem from quantum information theory which states that, during measurement of an quantum entanglement, entangled quantum state, it is not possible for one observer, b ...
. 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 , 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 A Bell test, also known as Bell inequality test or Bell experiment, is a real-world physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior thr ...
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 (math ...
,
differential equation In mathematics, a differential equation is an functional equation, equation that relates one or more function (mathematics), functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives ...

s,
group theory The popular puzzle Rubik's cube invented in 1974 by Ernő Rubik has been used as an illustration of permutation group">Ernő_Rubik.html" ;"title="Rubik's cube invented in 1974 by Ernő Rubik">Rubik's cube invented in 1974 by Ernő Rubik has bee ...
, 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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
$\mathcal H$. 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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
of a Hilbert space, usually called the
complex projective space The UCL Faculty of Mathematical and Physical Sciences is one of the 11 constituent faculties of University College London , mottoeng = Let all come who by merit deserve the most reward , established = , type = Public university, Public rese ...
. 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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
functions $L^2\left(\mathbb C\right)$, while the Hilbert space for the
spin Spin or spinning may refer to: Businesses * SPIN (cable system) or South Pacific Island Network * Spin (company), an American scooter-sharing system * SPiN, a chain of table tennis lounges Computing * SPIN model checker, Gerard Holzmann's tool fo ...
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, Wikipedia list article Numerous things are named after the French mathematician Charles Hermite Charles Hermite () FRS FRSE Fellowship of the Royal Society of Edinburgh (FRSE) is an award granted to individuals that the R ...
(more precisely,
self-adjointIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
) 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 transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denote ...

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 ' Measurement is the number, numerical quantification (science), quantification of the variable an ...
, 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 an ...

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 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 sub ...
. 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 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 an ...
: 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 and probability density function of a normal distribution In probability theory, a normal (or Gaussian or Gauss or Laplace–Gauss) distribution is a type of continuous probability distribution for a real number, real-valued random variable. ...
. After the measurement, if result $\lambda$ was obtained, the quantum state is postulated to
collapse Collapse or its variants may refer to: Concepts * Collapse (structural) * Collapse (topology), a mathematical concept * Collapsing manifold * Collapse, the action of collapsing or telescoping objects * Ecosystem collapse An ecosystem ...
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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathe ...

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 A hypothesis (plural hypotheses) is a proposed explanation for a phenomenon. For a hypothesis to be a scientific hypothesis, the scientific method requires that one can t ...
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 Quantum mechanics is a fundamental Scientific theory, theory in physics that provides a description of the physical properties of ...
have been formulated that do away with the concept of "
wave function collapse 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 quantu ...
" (see, for example, the
many-worlds interpretation The many-worlds interpretation (MWI) is an interpretation of quantum mechanics An interpretation of quantum mechanics is an attempt to explain how the mathematical theory of quantum mechanics "corresponds" to reality. Although quantum mechanics ...
). 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 Quantum mechanics is a fundamental theory A theory is a reason, rational type of abstraction, abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinking is o ...
.
Chapter 8, p. 215
The time evolution of a quantum state is described by the
Schrödinger equation The Schrödinger equation is a linear Linearity is the property of a mathematical relationship ('' function'') that can be graphically represented as a straight line. Linearity is closely related to proportionality. Examples in physics ...
: :$i\hbar \psi \left(t\right) =H \psi \left(t\right).$ 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 A physical constant, sometimes fundamental physical constant or universal constant, is a physical quantity that is generally believed to be both universal in nature an ...
. 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 (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physics), motion and behavior through S ...
. 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 Unitary may refer to: * Unitary construction, in automotive design a common term for unibody (unitary body/chassis) construction * Lethal Unitary Chemical Agents and Munitions (Unitary), as chemical weapons opposite of Binary * Unitarianism, in Chr ...
. This time evolution is
deterministic Determinism is the Philosophy, philosophical view that all events are determined completely by previously existing causes. Deterministic theories throughout the history of philosophy have sprung from diverse and sometimes overlapping motives an ...
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 is a subatomic particle In physical sciences, subatomic particles are smaller than atom An atom is the smallest unit of ordinary matter In classical physics and general chemistry, matter is any substance that has ma ...
in an unexcited
atom An atom is the smallest unit of ordinary matter In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume. All everyday objects that can be touched are ultimately composed of ato ...

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 the d ...
, 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 The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary smooth potential can usually be approximated as a harmonic potential at the vicinity of a stable equilibrium point, ...
, the
particle in a box 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 quantum ...

, the
dihydrogen cation The dihydrogen cation or hydrogen molecular ion is a cation An ion () is a particle, atom or molecule with a net electric charge, electrical charge. The charge of the electron is considered negative by convention. The negative charge of an ...
, and the
hydrogen atom A hydrogen atom is an atom An atom is the smallest unit of ordinary matter In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume. All everyday objects that can be touch ...

. Even the
helium Helium (from el, ἥλιος, helios Helios; Homeric Greek: ), Latinized as Helius; Hyperion and Phaethon are also the names of his father and son respectively. often given the epithets Hyperion ("the one above") and Phaethon ("the shining") ...

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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
, 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 In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter ...

. 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 Quantum chaos is a branch of physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physics), ...
.

## Uncertainty principle

One consequence of the basic quantum formalism is the
uncertainty principle 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 quantu ...

. 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 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 quantum p ...
: : 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 Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin ...

, 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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

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 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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

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 quantum coherence. 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 ...
. 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. Dirac made fundamental contributions to the early developme ...

, which unifies and generalizes the two earliest formulations of quantum mechanics –
matrix mechanics#REDIRECT Matrix mechanics Matrix mechanics is a formulation of quantum mechanics created by Werner Heisenberg, Max Born Max Born (; 11 December 1882 – 5 January 1970) was a German physicist A physicist is a scientist A scientist ...
(invented by
Werner Heisenberg Werner Karl Heisenberg (; ; 5 December 1901 – 1 February 1976) was a German theoretical physicist and one of the key pioneers of quantum mechanics Quantum mechanics is a fundamental Scientific theory, theory in physics that provides a de ...
) and
wave mechanicsWave mechanics may refer to: * the mechanics of waves * the ''wave equation'' in quantum physics, see Schrödinger equation See also

* Quantum mechanics * Wave equation {{Disambiguation Wave mechanics, ...
(invented by
Erwin Schrödinger Erwin Rudolf Josef Alexander Schrödinger (, ; ; 12 August 1887 – 4 January 1961), sometimes written as or , was a Nobel Prize-winning Austrian-Irish physicist A physicist is a scientist A scientist is a person who conducts Scientific ...
). An alternative formulation of quantum mechanics is
Feynman Richard Phillips Feynman (; May 11, 1918 – February 15, 1988) was an American theoretical physicist, known for his work in the path integral formulation The path integral formulation is a description in quantum mechanics Quantum mech ...

'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 subatomic p ...
, 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 Noether 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, ''Promotionsakt Emmy Noeth ...

in classical ( Lagrangian) mechanics: for every
differentiable In calculus (a branch of mathematics), a differentiable function of one Real number, real variable is a function whose derivative exists at each point in its Domain of a function, domain. In other words, the Graph of a function, graph of a differen ...

symmetry Symmetry (from Greek συμμετρία ''symmetria'' "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 pre ...
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 (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Ph ...
.

# Examples

## Free particle

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: :$H = \fracP^2 = - \frac \frac .$ 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 (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Ph ...

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 wa ...

: :$\psi\left(x,0\right) = \frace^$ 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, : $\psi\left(x\right) = C \sin\left(kx\right) + D \cos\left(kx\right).\!$ 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 ' Measurement is the number, numerical quantification (science), quantification of the variable an ...
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, subatomic particles are smaller than atom An atom is the smallest unit of ordinary matter In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume. All ...
s that make up all forms of matter (
electron The electron is a subatomic particle In physical sciences, subatomic particles are smaller than atom An atom is the smallest unit of ordinary matter In classical physics and general chemistry, matter is any substance that has ma ...
s, protons, neutrons,
photon The photon ( el, φῶς, phōs, light) is a type of elementary particle In particle physics, an elementary particle or fundamental particle is a subatomic particle that is not composed of other particles. Particles currently thought to be eleme ...

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 chemistry Chemistry is the scientific discipline involved with Chemical element, elements and chemical compound, compounds composed of atoms, molecules and ions: thei ...
, 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 term "laser" originated as an acronym for "light amplification by stimulated emission of radia ...

, 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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
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 (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physics), motion and behavior through S ...
, 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 The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary smooth potential can usually be approximated as a harmonic potential at the vicinity of a stable equilibrium point, ...
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 Quantum chaos is a branch of physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physics), ...
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 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 sub ...
s become simply probabilistic mixtures, and
quantum entanglement Quantum entanglement is a physical phenomenon that occurs when a pair or group of particles is generated, interact, or share spatial proximity in a way such that the quantum state of each particle of the pair or group cannot be described indepen ...
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 In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
, 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 An atom is the smallest unit of ordinary matter In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume. All everyday objects that can be touch ...

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 The photon ( el, φῶς, phōs, light) is a type of elementary particle In particle physics, an elementary particle or fundamental particle is a subatomic particle that is not composed of other particles. Particles currently thought to be eleme ...

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 Danish physicist A physicist is a scientist A scientist is a person who conducts Scientific method, scientific research to advance knowledge in an Branches of ...

,
Werner Heisenberg Werner Karl Heisenberg (; ; 5 December 1901 – 1 February 1976) was a German theoretical physicist and one of the key pioneers of quantum mechanics Quantum mechanics is a fundamental Scientific theory, theory in physics that provides a de ...
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 physicist, widely acknowledged to be one of the greatest physicists of all time. Einstein is known for developing the theory of relativity The theory ...
, 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 A hypothesis (plural hypotheses) is a proposed explanation for a phenomenon. For a hypothesis to be a scientific hypothesis, the scientific method requires that one can t ...
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 Linearity is the property of a mathematical relationship ('' function'') that can be graphically represented as a straight line. Linearity is closely related to proportionality. Examples in physics ...
together with the guiding equation; there is never a collapse of the wave function. This solves the measurement problem. Everett's
many-worlds interpretation The many-worlds interpretation (MWI) is an interpretation of quantum mechanics An interpretation of quantum mechanics is an attempt to explain how the mathematical theory of quantum mechanics "corresponds" to reality. Although quantum mechanics ...
, 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 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 sub ...
. 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 ) of black-body radiation scales inversely with the temperature of the black body; the locus of such colors, shown here in CIE 1931 ''x,y'' space, is known as the Planckian locus. Black-body radiation is the thermal radiation, thermal electro ...
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 German German(s) may refer to: Common uses * of or related to Germany * Germans, Germanic ethnic group, citizens of Germany or people of German ancestry * For citi ...

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 physicist, widely acknowledged to be one of the greatest physicists of all time. Einstein is known for developing the theory of relativity The theory ...
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-state ...

, 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 Danish physicist A physicist is a scientist A scientist is a person who conducts Scientific method, scientific research to advance knowledge in an Branches of ...

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 The photon ( el, φῶς, phōs, light) is a type of elementary particle In particle physics, an elementary particle or fundamental particle is a subatomic particle that is not composed of other particles. Particles currently thought to be eleme ...

), 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 term "laser" originated as an acronym for "light amplification by stimulated emission of radia ...

. 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 Quantum mechanics is a fundamental Scientific theory, theory in physics that provides a description of the physical properties ...
. 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 Werner Karl Heisenberg (; ; 5 December 1901 – 1 February 1976) was a German theoretical physicist and one of the key pioneers of quantum mechanics Quantum mechanics is a fundamental Scientific theory, theory in physics that provides a de ...
,
Max Born Max Born (; 11 December 1882 – 5 January 1970) was a German physicist A physicist is a scientist A scientist is a person who conducts Scientific method, scientific research to advance knowledge in an Branches of science, area of in ...

, 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#REDIRECT Matrix mechanics Matrix mechanics is a formulation of quantum mechanics created by Werner Heisenberg, Max Born Max Born (; 11 December 1882 – 5 January 1970) was a German physicist A physicist is a scientist A scientist ...
and the Austrian physicist
Erwin Schrödinger Erwin Rudolf Josef Alexander Schrödinger (, ; ; 12 August 1887 – 4 January 1961), sometimes written as or , was a Nobel Prize-winning Austrian-Irish physicist A physicist is a scientist A scientist is a person who conducts Scientific ...
invented
wave mechanicsWave mechanics may refer to: * the mechanics of waves * the ''wave equation'' in quantum physics, see Schrödinger equation See also

* Quantum mechanics * Wave equation {{Disambiguation 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. Dirac made fundamental contributions to the early developme ...

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 Quantum information science is an interdisciplinary field that seeks to understand the analysis, processing, and transmission of information using quantum mechanics Quantum mechanics is a fundamental Scientific theory, theory in physics that p ...
. 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 is a subatomic particle In physical sciences, subatomic particles are smaller than atom An atom is the smallest unit of ordinary matter In classical physics and general chemistry, matter is any substance that has ma ...
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 In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
, 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.

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
an
Physics
Se
8.04
an

5½ Examples in Quantum Mechanics

Imperial College Quantum Mechanics Course.
;Philosophy * * {{DEFAULTSORT:Quantum Mechanics Quantum mechanics,