Quantum mechanics $\Delta x \, \Delta p \ge \frac{\hbar}{2}$ Uncertainty principle Introduction to...Mathematical formulation of... This box: view • talk • edit
Fig. In Quantum physics, the Heisenberg uncertainty principle states that locating a particle in a small region of space makes the Momentum of the particle uncertain Quantum mechanics (QM or quantum theory) is a physical science dealing with the behavior of Matter and Energy on the scale of Atoms The mathematical formulation of quantum mechanics is the body of mathematical formalisms which permits a rigorous description of Quantum mechanics. 1: The wavefunctions of an electron in a hydrogen atom possessing definite energy (increasing downward: n = 1, 2, 3, . A wave function or wavefunction is a mathematical tool used in Quantum mechanics to describe any physical system The electron is a fundamental Subatomic particle that was identified and assigned the negative charge in 1897 by J A hydrogen atom is an atom of the chemical element Hydrogen. The electrically neutral In Physics and other Sciences energy (from the Greek grc ἐνέργεια - Energeia, "activity operation" from grc ἐνεργός . . ) and angular momentum (increasing across: s, p, d,. In Physics, the angular momentum of a particle about an origin is a vector quantity equal to the mass of the particle multiplied by the Cross product of the position . . ). Brighter areas correspond to higher probability density for a position measurement. In Quantum mechanics, a probability amplitude is a complex -valued function that describes an uncertain or unknown quantity Wavefunctions like these are directly comparable to Chladni's figures of acoustic modes of vibration in classical physics and are indeed modes of oscillation as well: they possess a sharp energy and thus a keen frequency. Ernst Florens Friedrich Chladni (ˈɛʀnst ˈfloːʀɛns ˈfʀiːdʀɪç ˈkladnɪ November 30, 1756 – April 3, 1827) was a German Acoustics is the interdisciplinary science that deals with the study of Sound, Ultrasound and Infrasound (all mechanical waves in gases liquids and solids In Physics and other Sciences energy (from the Greek grc ἐνέργεια - Energeia, "activity operation" from grc ἐνεργός Frequency is a measure of the number of occurrences of a repeating event per unit Time. The angular momentum and energy are quantized, and only take on discrete values like those shown (as is the case for resonant frequencies in acoustics). In Physics, the angular momentum of a particle about an origin is a vector quantity equal to the mass of the particle multiplied by the Cross product of the position In Physics, quantization is a procedure for constructing a Quantum field theory starting from a classical field theory. In Physics, resonance is the tendency of a system to Oscillate at maximum Amplitude at certain frequencies, known as the system's

Quantum mechanics is the study of mechanical systems whose dimensions are close to or below the atomic scale, such as molecules, atoms, electrons, protons and other subatomic particles. Mechanics ( Greek) is the branch of Physics concerned with the behaviour of physical bodies when subjected to Forces or displacements History See also Atomic theory, Atomism The concept that matter is composed of discrete units and cannot be divided into arbitrarily tiny In Chemistry, a molecule is defined as a sufficiently stable electrically neutral group of at least two Atoms in a definite arrangement held together by History See also Atomic theory, Atomism The concept that matter is composed of discrete units and cannot be divided into arbitrarily tiny The electron is a fundamental Subatomic particle that was identified and assigned the negative charge in 1897 by J The proton ( Greek πρῶτον / proton "first" is a Subatomic particle with an Electric charge of one positive A subatomic particle is an elementary or composite Particle smaller than an Atom. Quantum mechanics is a fundamental branch of physics with wide applications. Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. Quantum theory generalizes classical mechanics and provides accurate descriptions for many previously unexplained phenomena such as black body radiation and stable electron orbits. Classical mechanics is used for describing the motion of Macroscopic objects from Projectiles to parts of Machinery, as well as Astronomical objects A phenomenon (from Greek φαινόμενoν, pl φαινόμενα - phenomena) is any observable occurrence The Electromagnetic radiation emitted by a Black body. You may also be looking for Incandescence, the radiation from a body The effects of quantum mechanics are typically not observable on macroscopic scales, but become evident at the atomic and subatomic level. Macroscopic is commonly used to describe physical objects that are measurable and observable by the Naked eye. History See also Atomic theory, Atomism The concept that matter is composed of discrete units and cannot be divided into arbitrarily tiny A subatomic particle is an elementary or composite Particle smaller than an Atom. There are however exceptions to this rule such as superfluidity. Superfluidity is a phase of matter or description of Heat capacity in which unusual effects are observed when Liquids, typically of Helium-4

## Overview

The word “quantum” came from the Latin word which means "unit of quantity". In quantum mechanics, it refers to a discrete unit that quantum theory assigns to certain physical quantities, such as the energy of an atom at rest (see Figure 1, at right). In Physics and other Sciences energy (from the Greek grc ἐνέργεια - Energeia, "activity operation" from grc ἐνεργός History See also Atomic theory, Atomism The concept that matter is composed of discrete units and cannot be divided into arbitrarily tiny The discovery that waves have discrete energy packets (called quanta) that behave in a manner similar to particles led to the branch of physics that deals with atomic and subatomic systems which we today call quantum mechanics. A wave is a disturbance that propagates through Space and Time, usually with transference of Energy. A subatomic particle is an elementary or composite Particle smaller than an Atom. It is the underlying mathematical framework of many fields of physics and chemistry, including condensed matter physics, solid-state physics, atomic physics, molecular physics, computational chemistry, quantum chemistry, particle physics, and nuclear physics. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. Chemistry (from Egyptian kēme (chem meaning "earth") is the Science concerned with the composition structure and properties Condensed matter physics is the field of Physics that deals with the macroscopic physical properties of Matter. Solid-state physics, the largest branch of Condensed matter physics, is the study of rigid Matter, or Solids The bulk of solid-state physics theory and Atomic physics (or atom physics) is the field of Physics that studies atoms as an isolated system of Electrons and an atomic nucleus. Molecular physics is the study of the physical properties of Molecules and of the Chemical bonds between Atoms that bind them Computational chemistry is a branch of Chemistry that uses computers to assist in solving chemical problems Quantum chemistry is a branch of Theoretical chemistry, which applies Quantum mechanics and Quantum field theory to address issues and problems in Particle physics is a branch of Physics that studies the elementary constituents of Matter and Radiation, and the interactions between them Nuclear physics is the field of Physics that studies the building blocks and interactions of Atomic nuclei. The foundations of quantum mechanics were established during the first half of the twentieth century by Werner Heisenberg, Max Planck, Louis de Broglie, Albert Einstein, Niels Bohr, Erwin Schrödinger, Max Born, John von Neumann, Paul Dirac, Wolfgang Pauli and others. Werner Heisenberg (5 December 1901 in Würzburg &ndash1 February 1976 in Munich) was a German theoretical physicist best known for enunciating the Louis-Victor-Pierre-Raymond 7th duc de Broglie, FRS (də bʁœj ( August 15 1892 &ndash March 19 1987) was a French Albert Einstein ( German: ˈalbɐt ˈaɪ̯nʃtaɪ̯n; English: ˈælbɝt ˈaɪnstaɪn (14 March 1879 – 18 April 1955 was a German -born theoretical Niels Henrik David Bohr (nels ˈb̥oɐ̯ˀ in Danish 7 October 1885 – 18 November 1962 was a Danish Physicist who made fundamental contributions to understanding Max Born (11 December 1882 &ndash 5 January 1970 was a German Physicist and Mathematician who was instrumental in the development of Quantum Below is a list of famous Physicists Many of these from the 20th and 21st centuries are found on the list of recipients of the Nobel Prize in physics. Some fundamental aspects of the theory are still actively studied.

Quantum mechanics is essential to understand the behavior of systems at atomic length scales and smaller. History See also Atomic theory, Atomism The concept that matter is composed of discrete units and cannot be divided into arbitrarily tiny For example, if Newtonian mechanics governed the workings of an atom, electrons would rapidly travel towards and collide with the nucleus, making stable atoms impossible. Classical mechanics is used for describing the motion of Macroscopic objects from Projectiles to parts of Machinery, as well as Astronomical objects The electron is a fundamental Subatomic particle that was identified and assigned the negative charge in 1897 by J The nucleus of an Atom is the very dense region consisting of Nucleons ( Protons and Neutrons, at the center of an atom However, in the natural world the electrons normally remain in an unknown orbital path around the nucleus, defying classical electromagnetism.

Quantum mechanics was initially developed to provide a better explanation of the atom, especially the spectra of light emitted by different atomic species. A spectrum (plural spectra or spectrums) is a condition that is not limited to a specific set of values but can vary infinitely within a continuum. Light, or visible light, is Electromagnetic radiation of a Wavelength that is visible to the Human eye (about 400–700 Isotopes (Greek isos = "equal" tópos = "site place" are any of the different types of atoms ( Nuclides The quantum theory of the atom was developed as an explanation for the electron's staying in its orbital, which could not be explained by Newton's laws of motion and by Maxwell's laws of classical electromagnetism. An atomic orbital is a Mathematical function that describes the wave-like behavior of an electron in an atom In Classical electromagnetism, Maxwell's equations are a set of four Partial differential equations that describe the properties of the electric

In the formalism of quantum mechanics, the state of a system at a given time is described by a complex wave function (sometimes referred to as orbitals in the case of atomic electrons), and more generally, elements of a complex vector space. Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted A wave function or wavefunction is a mathematical tool used in Quantum mechanics to describe any physical system In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added This abstract mathematical object allows for the calculation of probabilities of outcomes of concrete experiments. Probability is the likelihood or chance that something is the case or will happen For example, it allows one to compute the probability of finding an electron in a particular region around the nucleus at a particular time. Contrary to classical mechanics, one can never make simultaneous predictions of conjugate variables, such as position and momentum, with arbitrary accuracy. In Physics, conjugate variables are pair of variables mathematically defined in such a way that they become Fourier transform duals of one-another For instance, electrons may be considered to be located somewhere within a region of space, but with their exact positions being unknown. Contours of constant probability, often referred to as “clouds” may be drawn around the nucleus of an atom to conceptualize where the electron might be located with the most probability. It should be stressed that the electron itself is not spread out over such cloud regions. It is either in a particular region of space, or it is not. Heisenberg's uncertainty principle quantifies the inability to precisely locate the particle. In Quantum physics, the Heisenberg uncertainty principle states that locating a particle in a small region of space makes the Momentum of the particle uncertain

The other exemplar that led to quantum mechanics was the study of electromagnetic waves such as light. "Exemplars" directs here For the superpowered comic book team see Exemplars (comics. Electromagnetic radiation takes the form of self-propagating Waves in a Vacuum or in Matter. When it was found in 1900 by Max Planck that the energy of waves could be described as consisting of small packets or quanta, Albert Einstein exploited this idea to show that an electromagnetic wave such as light could be described by a particle called the photon with a discrete energy dependent on its frequency. Albert Einstein ( German: ˈalbɐt ˈaɪ̯nʃtaɪ̯n; English: ˈælbɝt ˈaɪnstaɪn (14 March 1879 – 18 April 1955 was a German -born theoretical In Physics, the photon is the Elementary particle responsible for electromagnetic phenomena This led to a theory of unity between subatomic particles and electromagnetic waves called wave–particle duality in which particles and waves were neither one nor the other, but had certain properties of both. Photon polarization is the quantum mechanical description of the classical polarized sinusoidal plane electromagnetic wave In Physics and Chemistry, wave–particle duality is the concept that all Matter and Energy exhibits both Wave -like and While quantum mechanics describes the world of the very small, it also is needed to explain certain “macroscopic quantum systems” such as superconductors and superfluids. Macroscopic is commonly used to describe physical objects that are measurable and observable by the Naked eye. Superconductivity is a phenomenon occurring in certain Materials generally at very low Temperatures characterized by exactly zero electrical resistance Superfluidity is a phase of matter or description of Heat capacity in which unusual effects are observed when Liquids, typically of Helium-4

Broadly speaking, quantum mechanics incorporates four classes of phenomena that classical physics cannot account for: (i) the quantization (discretization) of certain physical quantities, (ii) wave-particle duality, (iii) the uncertainty principle, and (iv) quantum entanglement. In Physics, quantization is a procedure for constructing a Quantum field theory starting from a classical field theory. In Physics, conjugate variables are pair of variables mathematically defined in such a way that they become Fourier transform duals of one-another In Physics and Chemistry, wave–particle duality is the concept that all Matter and Energy exhibits both Wave -like and In Quantum physics, the Heisenberg uncertainty principle states that locating a particle in a small region of space makes the Momentum of the particle uncertain Quantum entanglement is a quantum mechanical Phenomenon in which the Quantum states of two or more objects are linked together so that one object Each of these phenomena is described in detail in subsequent sections.

## History

The history of quantum mechanics began essentially with the 1838 discovery of cathode rays by Michael Faraday, the 1859 statement of the black body radiation problem by Gustav Kirchhoff, the 1877 suggestion by Ludwig Boltzmann that the energy states of a physical system could be discrete, and the 1900 quantum hypothesis by Max Planck that any energy is radiated and absorbed in quantities divisible by discrete ‘energy elements’, E, such that each of these energy elements is proportional to the frequency ν with which they each individually radiate energy, as defined by the following formula:

$E = h \nu = \hbar \omega\,$

where h is Planck's Action Constant. The history of Quantum mechanics as this interlaces with history of Quantum chemistry began essentially with the 1838 discovery of Cathode rays Cathode rays (also called an electron beam or e-beam) are streams of Electrons observed in Vacuum tubes i Michael Faraday, FRS ( September 22 1791 – August 25 1867) was an English The Electromagnetic radiation emitted by a Black body. You may also be looking for Incandescence, the radiation from a body Gustav Robert Kirchhoff ( March 12, 1824 &ndash October 17, 1887) was a German Physicist who contributed to the fundamental Ludwig Eduard Boltzmann ( February 20, 1844 &ndash September 5, 1906) was an Austrian Physicist famous for his founding Frequency is a measure of the number of occurrences of a repeating event per unit Time. In Physics and other Sciences energy (from the Greek grc ἐνέργεια - Energeia, "activity operation" from grc ἐνεργός The Planck constant (denoted h\ is a Physical constant used to describe the sizes of quanta. Although Planck insisted that this was simply an aspect of the absorption and radiation of energy and had nothing to do with the physical reality of the energy itself, in 1905, to explain the photoelectric effect (1839), i. Introduction When a Metallic surface is exposed to Electromagnetic radiation above a certain threshold Frequency, the light is absorbed and Electrons e. that shining light on certain materials can function to eject electrons from the material, Albert Einstein postulated, as based on Planck’s quantum hypothesis, that light itself consists of individual quanta, which later came to be called photons (1926). Albert Einstein ( German: ˈalbɐt ˈaɪ̯nʃtaɪ̯n; English: ˈælbɝt ˈaɪnstaɪn (14 March 1879 – 18 April 1955 was a German -born theoretical Light, or visible light, is Electromagnetic radiation of a Wavelength that is visible to the Human eye (about 400–700 In Physics, the photon is the Elementary particle responsible for electromagnetic phenomena From Einstein's simple postulation was borne a flurry of debating, theorizing and testing, and thus, the entire field of quantum physics. Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons

## Relativity and quantum mechanics

The modern world of physics is notably founded on two tested and demonstrably sound theories of general relativity and quantum mechanics —theories which appear to contradict one another. The defining postulates of both Einstein's theory of relativity and quantum theory are indisputably supported by rigorous and repeated empirical evidence. However, while they do not directly contradict each other theoretically (at least with regard to primary claims), they are resistant to being incorporated within one cohesive model.

Einstein himself is well known for rejecting some of the claims of quantum mechanics. While clearly inventive in this field, he did not accept the more philosophical consequences and interpretations of quantum mechanics, such as the lack of deterministic causality and the assertion that a single subatomic particle can occupy numerous areas of space at one time. Causality (but not causation) denotes a necessary relationship between one event (called cause and another event (called effect) which is the direct consequence He also was the first to notice some of the apparently exotic consequences of entanglement and used them to formulate the Einstein-Podolsky-Rosen paradox, in the hope of showing that quantum mechanics has unacceptable implications. Quantum entanglement is a quantum mechanical Phenomenon in which the Quantum states of two or more objects are linked together so that one object In Quantum mechanics, the EPR paradox is a Thought experiment which challenged long-held ideas about the relation between the observed values of physical quantities This was 1935, but in 1964 it was shown by John Bell (see Bell inequality) that Einstein's assumption that quantum mechanics is correct, but has to be completed by hidden variables, was based on wrong philosophical assumptions: according to the paper of J. Bell's theorem is a theorem that shows that the predictions of Quantum mechanics (QM are not intuitive and touches upon fundamental philosophical issues that relate to modern Bell and the Copenhagen interpretation (the common interpretation of quantum mechanics by physicists for decades), and contrary to Einstein's ideas, quantum mechanics is

• neither a "realistic" theory (since quantum measurements do not state pre-existing properties, but rather they prepare properties)
• nor a local theory (essentially not, because the state vector $|\psi\rangle$ determines simultaneously the probability amplitudes at all sites, $|\psi\rangle\to\psi(\mathbf r), \forall \mathbf r$). The Copenhagen interpretation is an interpretation of Quantum mechanics. In Physics, the principle of locality is that distant objects cannot have direct influence on one another an object is influenced directly only by its immediate surroundings

The Einstein-Podolsky-Rosen paradox shows in any case that there exist experiments by which one can measure the state of one particle and instantaneously change the state of its entangled partner, although the two particles can be an arbitrary distance apart; however, this effect does not violate causality, since no transfer of information happens. Causality (but not causation) denotes a necessary relationship between one event (called cause and another event (called effect) which is the direct consequence These experiments are the basis of some of the most topical applications of the theory, quantum cryptography, which works well, although at small distances of typically ${\le}$ 1000 km, being on the market since 2004. Quantum cryptography, or quantum key distribution (QKD uses Quantum mechanics to guarantee secure communication

There do exist quantum theories which incorporate special relativity—for example, quantum electrodynamics (QED), which is currently the most accurately tested physical theory [1]—and these lie at the very heart of modern particle physics. Quantum electrodynamics ( QED) is a relativistic Quantum field theory of Electrodynamics. Particle physics is a branch of Physics that studies the elementary constituents of Matter and Radiation, and the interactions between them 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 applications. However, the lack of a correct theory of quantum gravity is an important issue in cosmology. Quantum gravity is the field of Theoretical physics attempting to unify Quantum mechanics, which describes three of the fundamental forces of nature Cosmology (from Greek grc κοσμολογία - grc κόσμος kosmos, "universe" and grc -λογία -logia) is study

## Attempts at a unified theory

Main article: Quantum gravity

Inconsistencies arise when one tries to join the quantum laws with general relativity, a more elaborate description of spacetime which incorporates gravitation. Quantum gravity is the field of Theoretical physics attempting to unify Quantum mechanics, which describes three of the fundamental forces of nature Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons General relativity or the general theory of relativity is the geometric theory of Gravitation published by Albert Einstein in 1916 SpaceTime is a patent-pending three dimensional graphical user interface that allows end users to search their content such as Google Google Images Yahoo! YouTube eBay Amazon and RSS Gravitation is a natural Phenomenon by which objects with Mass attract one another Resolving these inconsistencies has been a major goal of twentieth- and twenty-first-century physics. The twentieth century of the Common Era began on The 21st century is the current century of the Christian Era or Common Era in accordance with the Gregorian calendar. Many prominent physicists, including Stephen Hawking, have labored in the attempt to discover a "Grand Unification Theory" that combines not only different models of subatomic physics, but also derives the universe's four forces—the strong force, electromagnetism, weak force, and gravity— from a single force or phenomenon. Stephen William Hawking CH, CBE, FRS, FRSA (born 8 January 1942 is a British theoretical physicist. Grand Unification, grand unified theory, or GUT refers to any of several very similar unified field theories or models in Physics that In particle physics the strong interaction, or strong force, or color force, holds Quarks and Gluons together to form Protons and Electromagnetism is the Physics of the Electromagnetic field: a field which exerts a Force on particles that possess the property of The weak interaction (often called the weak force or sometimes the weak nuclear force) is one of the four Fundamental interactions of nature Gravitation is a natural Phenomenon by which objects with Mass attract one another

## Quantum mechanics and classical physics

Predictions of quantum mechanics have been verified experimentally to a very high degree of accuracy. Thus, the current logic of correspondence principle between classical and quantum mechanics is that all objects obey laws of quantum mechanics, and classical mechanics is just a quantum mechanics of large systems (or a statistical quantum mechanics of a large collection of particles). This article discusses quantum theory For other uses see Correspondence principle (disambiguation. Laws of classical mechanics thus follow from laws of quantum mechanics at the limit of large systems or large quantum numbers.

Main differences between classical and quantum theories have already been mentioned above in the remarks on the Einstein-Podolsky-Rosen paradox. In Quantum mechanics, the EPR paradox is a Thought experiment which challenged long-held ideas about the relation between the observed values of physical quantities Essentially the difference boils down to the statement that quantum mechanics is coherent (addition of amplitudes), whereas classical theories are incoherent (addition of intensities). In Physics, coherence is a property of waves that enables stationary (i Thus, such quantities as coherence lengths and coherence times come into play. For microscopic bodies the extension of the system is certainly much smaller than the coherence length; for macroscopic bodies one expects that it should be the other way round.

This is in accordance with the following observations:

Many “macroscopic” properties of “classic” systems are direct consequences of quantum behavior of its parts. For example, stability of bulk matter (which consists of atoms and molecules which would quickly collapse under electric forces alone), rigidity of this matter, mechanical, thermal, chemical, optical and magnetic properties of this matter—they are all results of interaction of electric charges under the rules of quantum mechanics. In Chemistry, a molecule is defined as a sufficiently stable electrically neutral group of at least two Atoms in a definite arrangement held together by Electric charge is a fundamental conserved property of some Subatomic particles which determines their Electromagnetic interaction.

Because seemingly exotic behavior of matter posited by quantum mechanics and relativity theory become more apparent when dealing with extremely fast-moving or extremely tiny particles, the laws of classical “Newtonian” physics still remain accurate in predicting the behavior of surrounding (“large”) objects—of the order of the size of large molecules and bigger.

Despite the proposal of many novel ideas, the unification of quantum mechanics—which reigns in the domain of the very small—and general relativity—a superb description of the very large—remains, tantalizingly, a future possibility. (See quantum gravity, string theory. Quantum gravity is the field of Theoretical physics attempting to unify Quantum mechanics, which describes three of the fundamental forces of nature String theory is a still-developing scientific approach to Theoretical physics, whose original building blocks are one-dimensional extended objects called strings )

## Theory

There are numerous mathematically equivalent formulations of quantum mechanics. One of the oldest and most commonly used formulations is the transformation theory proposed by Cambridge theoretical physicist Paul Dirac, which unifies and generalizes the two earliest formulations of quantum mechanics, matrix mechanics (invented by Werner Heisenberg)[2] and wave mechanics (invented by Erwin Schrödinger). The term transformation theory refers to a procedure used by P Matrix mechanics is a formulation of Quantum mechanics created by Werner Heisenberg, Max Born, and Pascual Jordan in 1925 Werner Heisenberg (5 December 1901 in Würzburg &ndash1 February 1976 in Munich) was a German theoretical physicist best known for enunciating the In Physics, especially Quantum mechanics, the Schrödinger equation is an equation that describes how the Quantum state of a Physical system

In this formulation, the instantaneous state of a quantum system encodes the probabilities of its measurable properties, or "observables". In Quantum physics, a quantum state is a mathematical object that fully describes a quantum system. In Physics, particularly in Quantum physics, a system observable is a property of the system state that can be determined by some sequence of physical Examples of observables include energy, position, momentum, and angular momentum. In Physics and other Sciences energy (from the Greek grc ἐνέργεια - Energeia, "activity operation" from grc ἐνεργός In Classical mechanics, momentum ( pl momenta SI unit kg · m/s, or equivalently N · s) is the product In Physics, the angular momentum of a particle about an origin is a vector quantity equal to the mass of the particle multiplied by the Cross product of the position Observables can be either continuous (e. In Mathematics, a continuous function is a function for which intuitively small changes in the input result in small changes in the output g. , the position of a particle) or discrete (e. Discrete mathematics, also called finite mathematics, is the study of mathematical structures that are fundamentally discrete in the sense of not supporting or requiring the g. , the energy of an electron bound to a hydrogen atom).

Generally, quantum mechanics does not assign definite values to observables. Instead, it makes predictions about probability distributions; that is, the probability of obtaining each of the possible outcomes from measuring an observable. In Probability theory and Statistics, a probability distribution identifies either the probability of each value of an unidentified Random variable Naturally, these probabilities will depend on the quantum state at the instant of the measurement. There are, however, certain states that are associated with a definite value of a particular observable. These are known as "eigenstates" of the observable ("eigen" can be roughly translated from German as inherent or as a characteristic). The German language (de ''Deutsch'') is a West Germanic language and one of the world's major languages. In the everyday world, it is natural and intuitive to think of everything being in an eigenstate of every observable. Everything appears to have a definite position, a definite momentum, and a definite time of occurrence. However, quantum mechanics does not pinpoint the exact values for the position or momentum of a certain particle in a given space in a finite time; rather, it only provides a range of probabilities of where that particle might be. Therefore, it became necessary to use different words for (a) the state of something having an uncertainty relation and (b) a state that has a definite value. The latter is called the "eigenstate" of the property being measured.

For example, consider a free particle. In Physics, a free particle is a particle that in some sense is not bound In quantum mechanics, there is wave-particle duality so the properties of the particle can be described as a wave. In Physics and Chemistry, wave–particle duality is the concept that all Matter and Energy exhibits both Wave -like and Therefore, its quantum state can be represented as a wave, of arbitrary shape and extending over all of space, called a wave function. In Quantum physics, a quantum state is a mathematical object that fully describes a quantum system. A wave is a disturbance that propagates through Space and Time, usually with transference of Energy. A wave function or wavefunction is a mathematical tool used in Quantum mechanics to describe any physical system The position and momentum of the particle are observables. The Uncertainty Principle of quantum mechanics states that both the position and the momentum cannot simultaneously be known with infinite precision at the same time. In Quantum physics, the Heisenberg uncertainty principle states that locating a particle in a small region of space makes the Momentum of the particle uncertain However, one can measure just the position alone of a moving free particle creating an eigenstate of position with a wavefunction that is very large at a particular position x, and almost zero everywhere else. The Dirac delta or Dirac's delta is a mathematical construct introduced by the British theoretical physicist Paul Dirac. If one performs a position measurement on such a wavefunction, the result x will be obtained with almost 100% probability. In other words, the position of the free particle will almost be known. This is called an eigenstate of position (mathematically more precise: a generalized eigenstate (eigendistribution) ). In Mathematical analysis, distributions (also known as generalized functions) are objects which generalize functions and Probability distributions If the particle is in an eigenstate of position then its momentum is completely unknown. An eigenstate of momentum, on the other hand, has the form of a plane wave. In the Physics of Wave propagation (especially Electromagnetic waves, a plane wave (also spelled planewave) is a constant-frequency wave whose It can be shown that the wavelength is equal to h/p, where h is Planck's constant and p is the momentum of the eigenstate. In Physics wavelength is the distance between repeating units of a propagating Wave of a given Frequency. The Planck constant (denoted h\ is a Physical constant used to describe the sizes of quanta. In Mathematics, given a Linear transformation, an of that linear transformation is a nonzero vector which when that transformation is applied to it changes If the particle is in an eigenstate of momentum then its position is completely blurred out.

Usually, a system will not be in an eigenstate of whatever observable we are interested in. However, if one measures the observable, the wavefunction will instantaneously be an eigenstate (or generalized eigenstate) of that observable. This process is known as wavefunction collapse. In certain interpretations of quantum mechanics, wave function collapse is one of two processes by which Quantum systems apparently evolve according to the laws of It involves expanding the system under study to include the measurement device, so that a detailed quantum calculation would no longer be feasible and a classical description must be used. If one knows the corresponding wave function at the instant before the measurement, one will be able to compute the probability of collapsing into each of the possible eigenstates. For example, the free particle in the previous example will usually have a wavefunction that is a wave packet centered around some mean position x0, neither an eigenstate of position nor of momentum. In physics a wave packet is an envelope or packet containing an arbitrary number of wave forms When one measures the position of the particle, it is impossible to predict with certainty the result that we will obtain. It is probable, but not certain, that it will be near x0, where the amplitude of the wave function is large. After the measurement is performed, having obtained some result x, the wave function collapses into a position eigenstate centered at x.

Wave functions can change as time progresses. An equation known as the Schrödinger equation describes how wave functions change in time, a role similar to Newton's second law in classical mechanics. In Physics, especially Quantum mechanics, the Schrödinger equation is an equation that describes how the Quantum state of a Physical system Newton's laws of motion are three Physical laws which provide relationships between the Forces acting on a body and the motion of the The Schrödinger equation, applied to the aforementioned example of the free particle, predicts that the center of a wave packet will move 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 uncertain. This also has the effect of turning position eigenstates (which can be thought of as infinitely sharp wave packets) into broadened wave packets that are no longer position eigenstates.

Some wave functions produce probability distributions that are constant in time. Many systems that are treated dynamically in classical mechanics are described by such "static" wave functions. For example, a single electron in an unexcited atom is pictured classically as a particle moving in a circular trajectory around the atomic nucleus, whereas in quantum mechanics it is described by a static, spherically symmetric wavefunction surrounding the nucleus (Fig. 1). The electron is a fundamental Subatomic particle that was identified and assigned the negative charge in 1897 by J History See also Atomic theory, Atomism The concept that matter is composed of discrete units and cannot be divided into arbitrarily tiny The nucleus of an Atom is the very dense region consisting of Nucleons ( Protons and Neutrons, at the center of an atom In Mathematics, the spherical coordinate system is a Coordinate system for representing geometric figures in three dimensions using three coordinates the radial (Note that only the lowest angular momentum states, labeled s, are spherically symmetric).

The time evolution of wave functions is deterministic in the sense that, given a wavefunction at an initial time, it makes a definite prediction of what the wavefunction will be at any later time. Determinism is the philosophical Proposition that every event including human cognition and behaviour decision and action is causally determined During a measurement, the change of the wavefunction into another one is not deterministic, but rather unpredictable, i. The framework of Quantum mechanics requires a careful definition of measurement, and a thorough discussion of its practical and philosophical implications e. , random. Randomness is a lack of order Purpose, cause, or predictability

The probabilistic nature of quantum mechanics thus stems from the act of measurement. Probability is the likelihood or chance that something is the case or will happen 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 experiments. In the decades after the formulation of quantum mechanics, the question of what constitutes a "measurement" has been extensively studied. Interpretations of quantum mechanics have been formulated to do away with the concept of "wavefunction collapse"; see, for example, the relative state interpretation. An interpretation of quantum mechanics is a statement which attempts to explain how Quantum mechanics informs our Understanding of Nature. The many-worlds interpretation or MWI (also known as relative state formulation, theory of the universal wavefunction, parallel universes, The basic idea is that when a quantum system interacts with a measuring apparatus, their respective wavefunctions become entangled, so that the original quantum system ceases to exist as an independent entity. Quantum entanglement is a quantum mechanical Phenomenon in which the Quantum states of two or more objects are linked together so that one object For details, see the article on measurement in quantum mechanics. The framework of Quantum mechanics requires a careful definition of measurement, and a thorough discussion of its practical and philosophical implications

### Mathematical formulation

In the mathematically rigorous formulation of quantum mechanics, developed by Paul Dirac and John von Neumann, the possible states of a quantum mechanical system are represented by unit vectors (called "state vectors") residing in a complex separable Hilbert space (variously called the "state space" or the "associated Hilbert space" of the system) well defined up to a complex number of norm 1 (the phase factor). The mathematical formulation of quantum mechanics is the body of mathematical formalisms which permits a rigorous description of Quantum mechanics. In Mathematical physics and Quantum mechanics, quantum logic is a set of rules for Reasoning about propositions which takes the principles of Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted In Mathematics a Topological space is called separable if it contains a countable dense subset that is there exists a sequence \{ x_n This article assumes some familiarity with Analytic geometry and the concept of a limit. In other words, the possible states are points in the projectivization of a Hilbert space. In Mathematics a projective space is a set of elements constructed from a vector space such that a distinct element of the projective space consists of all non-zero vectors which The exact nature of this Hilbert space is dependent on the system; for example, the state space for position and momentum states is the space of square-integrable functions, while the state space for the spin of a single proton is just the product of two complex planes. In Mathematics, an integrable function is a function whose Integral exists Each observable is represented by a maximally-Hermitian (precisely: by a self-adjoint) linear operator acting on the state space. A number of Mathematical entities are named Hermitian, after the Mathematician Charles Hermite: Hermitian adjoint In Mathematics, on a finite-dimensional Inner product space, a self-adjoint operator is one that is its own adjoint, or equivalently one whose matrix In Mathematics, an operator is a function which operates on (or modifies another function Each eigenstate of an observable corresponds to an eigenvector of the operator, and the associated eigenvalue corresponds to the value of the observable in that eigenstate. In Mathematics, given a Linear transformation, an of that linear transformation is a nonzero vector which when that transformation is applied to it changes In Mathematics, given a Linear transformation, an of that linear transformation is a nonzero vector which when that transformation is applied to it changes If the operator's spectrum is discrete, the observable can only attain those discrete eigenvalues.

The time evolution of a quantum state is described by the Schrödinger equation, in which the Hamiltonian, the operator corresponding to the total energy of the system, generates time evolution. In Physics, especially Quantum mechanics, the Schrödinger equation is an equation that describes how the Quantum state of a Physical system In Quantum mechanics, the Hamiltonian H is the Observable corresponding to the Total energy of the system In physics an operator is a function acting on the space of Physical states As a resultof its application on a physical state another physical state is obtained

The inner product between two state vectors is a complex number known as a probability amplitude. In Mathematics, an inner product space is a Vector space with the additional Structure of inner product. In Quantum mechanics, a probability amplitude is a complex -valued function that describes an uncertain or unknown quantity During a measurement, the probability that a system collapses from a given initial state to a particular eigenstate is given by the square of the absolute value of the probability amplitudes between the initial and final states. In Mathematics, the absolute value (or modulus) of a Real number is its numerical value without regard to its sign. The possible results of a measurement are the eigenvalues of the operator - which explains the choice of Hermitian operators, for which all the eigenvalues are real. We can find the probability distribution of an observable in a given state by computing the spectral decomposition of the corresponding operator. In Mathematics, particularly Linear algebra and Functional analysis, the spectral theorem is any of a number of results about Linear operators Heisenberg's uncertainty principle is represented by the statement that the operators corresponding to certain observables do not commute. In Quantum physics, the Heisenberg uncertainty principle states that locating a particle in a small region of space makes the Momentum of the particle uncertain In Mathematics, the commutator gives an indication of the extent to which a certain Binary operation fails to be Commutative.

The Schrödinger equation acts on the entire probability amplitude, not merely its absolute value. Whereas the absolute value of the probability amplitude encodes information about probabilities, its phase encodes information about the interference between quantum states. The phase of an oscillation or wave is the fraction of a complete cycle corresponding to an offset in the displacement from a specified reference point at time t = 0 In physics interference is the addition ( superposition) of two or more Waves that result in a new wave pattern This gives rise to the wave-like behavior of quantum states.

It turns out that analytic solutions of Schrödinger's equation are only available for a small number of model Hamiltonians, of which the quantum harmonic oscillator, the particle in a box, the hydrogen-molecular ion and the hydrogen atom are the most important representatives. Much insight in Quantum mechanics can be gained from understanding the solutions to the time-dependent non-relativistic Schrödinger equation in an appropriate Configuration The quantum harmonic oscillator is the quantum mechanical analogue of the classical harmonic oscillator. In Physics, the particle in a box (also known as the infinite potential well or the infinite square well) is a problem consisting of a single particle inside A hydrogen atom is an atom of the chemical element Hydrogen. The electrically neutral Even the helium atom, which contains just one more electron than hydrogen, defies all attempts at a fully analytic treatment. Helium ( He) is a colorless odorless tasteless non-toxic Inert Monatomic Chemical There exist several techniques for generating approximate solutions. For instance, in the method known as perturbation theory one uses the analytic results for a simple quantum mechanical model to generate results for a more complicated model related to the simple model by, for example, the addition of a weak potential energy. In Quantum mechanics, perturbation theory is a set of approximation schemes directly related to mathematical perturbation for describing a complicated quantum system Potential energy can be thought of as Energy stored within a physical system Another method is the "semi-classical equation of motion" approach, which applies to systems for which quantum mechanics produces weak deviations from classical behavior. The deviations can be calculated based on the classical motion. This approach is important for the field of quantum chaos. Quantum chaos is a branch of Physics which studies how chaotic classical systems (see Dynamical systems and Chaos theory) can be shown to be limits of

An alternative formulation of quantum mechanics is Feynman's path integral formulation, in which a quantum-mechanical amplitude is considered as a sum over histories between initial and final states; this is the quantum-mechanical counterpart of action principles in classical mechanics. This article is about a formulation of quantum mechanics For integrals along a path also known as line or contour integrals see Line integral. In Physics, the action is a particular quantity in a Physical system that can be used to describe its operation

### Interactions with other scientific theories

The fundamental rules of quantum mechanics are very broad. They assert that the state space of a system is a Hilbert space and the observables are Hermitian operators acting on that space, but do not tell us which Hilbert space or which operators, or if it even exists. This article assumes some familiarity with Analytic geometry and the concept of a limit. In Mathematics, on a finite-dimensional Inner product space, a self-adjoint operator is one that is its own adjoint, or equivalently one whose matrix These must be chosen appropriately in order to obtain a quantitative description of a quantum system. An important guide for making these choices is the correspondence principle, which states that the predictions of quantum mechanics reduce to those of classical physics when a system moves to higher energies or equivalently, larger quantum numbers. This article discusses quantum theory For other uses see Correspondence principle (disambiguation. In other words, classic mechanics is simply a quantum mechanics of large systems. This "high energy" limit is known as the classical or correspondence limit. One can therefore start from an established classical model of a particular system, and attempt to guess the underlying quantum model that gives rise to the classical model in the correspondence limit

Unsolved problems in physics: In the correspondence limit of quantum mechanics: Is there a preferred interpretation of quantum mechanics? How does the quantum description of reality, which includes elements such as the superposition of states and wavefunction collapse, give rise to the reality we perceive?

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. Special relativity (SR (also known as the special theory of relativity or STR) is the Physical theory of Measurement in Inertial The Klein–Gordon equation ( Klein–Fock–Gordon equation or sometimes Klein–Gordon–Fock equation) is a relativistic version of the Schrödinger In Physics, the Dirac equation is a relativistic quantum mechanical wave equation formulated by British physicist Paul Dirac in 1928 and provides 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. In quantum field theory (QFT the forces between particles are mediated by other particles The first complete quantum field theory, quantum electrodynamics, provides a fully quantum description of the electromagnetic interaction. Quantum electrodynamics ( QED) is a relativistic Quantum field theory of Electrodynamics. Electromagnetism is the Physics of the Electromagnetic field: a field which exerts a Force on particles that possess the property of

The full apparatus of quantum field theory is often unnecessary for describing electrodynamic systems. A simpler approach, one employed since the inception of quantum mechanics, is to treat charged particles as quantum mechanical objects being acted on by a classical electromagnetic field. Electric charge is a fundamental conserved property of some Subatomic particles which determines their Electromagnetic interaction. For example, the elementary quantum model of the hydrogen atom describes the electric field of the hydrogen atom using a classical $-\frac{e^2}{4 \pi\ \epsilon_0\ } \frac{1}{r}$ Coulomb potential. A hydrogen atom is an atom of the chemical element Hydrogen. The electrically neutral This "semi-classical" approach fails if quantum fluctuations in the electromagnetic field play an important role, such as in the emission of photons by charged particles. In Physics, the photon is the Elementary particle responsible for electromagnetic phenomena

Quantum field theories for the strong nuclear force and the weak nuclear force have been developed. In particle physics the strong interaction, or strong force, or color force, holds Quarks and Gluons together to form Protons and The weak interaction (often called the weak force or sometimes the weak nuclear force) is one of the four Fundamental interactions of nature The quantum field theory of the strong nuclear force is called quantum chromodynamics, and describes the interactions of the subnuclear particles: quarks and gluons. Quantum chromodynamics (abbreviated as QCD is a theory of the Strong interaction ( color force a Fundamental force describing the interactions of the In Physics, a quark (kwɔrk kwɑːk or kwɑːrk is a type of Subatomic particle. Gluons ( Glue and the suffix -on) are Elementary particles that cause Quarks to interact and are indirectly responsible for the The weak nuclear force and the electromagnetic force were unified, in their quantized forms, into a single quantum field theory known as electroweak theory. The weak interaction (often called the weak force or sometimes the weak nuclear force) is one of the four Fundamental interactions of nature In Particle physics, the electroweak interaction is the unified description of two of the four Fundamental interactions of nature Electromagnetism and the

It has proven difficult to construct quantum models of gravity, the remaining fundamental force. Gravitation is a natural Phenomenon by which objects with Mass attract one another In Physics, a fundamental interaction or fundamental force is a mechanism by which particles interact with each other and which cannot be explained in terms Semi-classical approximations are workable, and have led to predictions such as Hawking radiation. Hawking radiation (also known as Bekenstein-Hawking radiation) is a Thermal radiation with a black body spectrum predicted to be emitted by Black holes However, the formulation of a complete theory of quantum gravity is hindered by apparent incompatibilities between general relativity, the most accurate theory of gravity currently known, and some of the fundamental assumptions of quantum theory. Quantum gravity is the field of Theoretical physics attempting to unify Quantum mechanics, which describes three of the fundamental forces of nature General relativity or the general theory of relativity is the geometric theory of Gravitation published by Albert Einstein in 1916 The resolution of these incompatibilities is an area of active research, and theories such as string theory are among the possible candidates for a future theory of quantum gravity. String theory is a still-developing scientific approach to Theoretical physics, whose original building blocks are one-dimensional extended objects called strings

## Derivation of quantization

The particle in a 1-dimensional potential energy box is the most simple example where restraints lead to the quantization of energy levels. The box is defined as zero potential energy inside a certain interval and infinite everywhere outside that interval. For the 1-dimensional case in the x direction, the time-independent Schrödinger equation can be written as[3]:

$- \frac {\hbar ^2}{2m} \frac {d ^2 \psi}{dx^2} = E \psi.$

The general solutions are:

$\psi = A e^{ikx} + B e ^{-ikx} \;\;\;\;\;\; E = \frac{k^2 \hbar^2}{2m}$
$\psi = C \sin kx + D \cos kx \;$ (exponential rewrite)

The presence of the walls of the box restricts the acceptable solutions to the wavefunction. At each wall :

$\psi = 0 \; \mathrm{at} \;\; x = 0,\; x = L$

Consider x = 0

• sin 0 = 0, cos 0 = 1. To satisfy $\psi = 0 \;$ D = 0 (cos term is removed)

Now Consider: $\psi = C \sin kx \;$

• at X = L, $\psi = C \sin kL \;$
• If C = 0 then $\psi =0 \;$ for all x and would conflict with Born interpretation
• therefore sin kL must be satisfied by
$kL = n \pi \;\;\;\; n = 1,2,3,4,5 \;$

In this situation, n must be an integer showing the quantization of the energy levels.

## Applications

Quantum mechanics has had enormous success in explaining many of the features of our world. The individual behaviour of the subatomic particles that make up all forms of matterelectrons, protons, neutrons, photons and others—can often only be satisfactorily described using quantum mechanics. Matter is commonly defined as being anything that has mass and that takes up space. The electron is a fundamental Subatomic particle that was identified and assigned the negative charge in 1897 by J The proton ( Greek πρῶτον / proton "first" is a Subatomic particle with an Electric charge of one positive This article is a discussion of neutrons in general For the specific case of a neutron found outside the nucleus see Free neutron. In Physics, the photon is the Elementary particle responsible for electromagnetic phenomena Quantum mechanics has strongly influenced string theory, a candidate for a theory of everything (see reductionism). String theory is a still-developing scientific approach to Theoretical physics, whose original building blocks are one-dimensional extended objects called strings A theory of everything ( TOE) is a putative Theory of Theoretical physics that fully explains and links together all known physical phenomena Reductionism can either mean (a an approach to understanding the nature of complex things by reducing them to the interactions of their parts or to simpler or more fundamental things It is also related to statistical mechanics. Statistical mechanics is the application of Probability theory, which includes mathematical tools for dealing with large populations to the field of Mechanics

Quantum mechanics is important for understanding how individual atoms combine covalently to form chemicals or molecules. The application of quantum mechanics to chemistry is known as quantum chemistry. Chemistry (from Egyptian kēme (chem meaning "earth") is the Science concerned with the composition structure and properties Quantum chemistry is a branch of Theoretical chemistry, which applies Quantum mechanics and Quantum field theory to address issues and problems in (Relativistic) quantum mechanics can in principle mathematically describe most of chemistry. Quantum mechanics can provide quantitative insight into ionic and covalent bonding processes by explicitly showing which molecules are energetically favorable to which others, and by approximately how much. Most of the calculations performed in computational chemistry rely on quantum mechanics. Computational chemistry is a branch of Chemistry that uses computers to assist in solving chemical problems

Much of modern technology operates at a scale where quantum effects are significant. Technology is a broad concept that deals with a Species ' usage and knowledge of Tools and Crafts and how it affects a species' ability to control and adapt Examples include the laser, the transistor, the electron microscope, and magnetic resonance imaging. A laser is a device that emits Light ( Electromagnetic radiation) through a process called Stimulated emission. In Electronics, a transistor is a Semiconductor device commonly used to amplify or switch electronic signals An electron microscope is a type of Microscope that uses Electrons to illuminate a specimen and create an enlarged image The study of semiconductors led to the invention of the diode and the transistor, which are indispensable for modern electronics. Dioden2jpg|thumb|right|150px|Figure 2 Various semiconductor diodes In Electronics, a transistor is a Semiconductor device commonly used to amplify or switch electronic signals Electronics refers to the flow of charge (moving Electrons through Nonmetal conductors (mainly Semiconductors, whereas electrical

Researchers are currently seeking robust methods of directly manipulating quantum states. Efforts are being made to develop quantum cryptography, which will allow guaranteed secure transmission of information. Quantum cryptography, or quantum key distribution (QKD uses Quantum mechanics to guarantee secure communication Information as a concept has a diversity of meanings from everyday usage to technical settings A more distant goal is the development of quantum computers, which are expected to perform certain computational tasks exponentially faster than classical computers. A quantum computer is a device for Computation that makes direct use of distinctively Quantum mechanical Phenomena, such as superposition A computer is a Machine that manipulates data according to a list of instructions. Another active research topic is quantum teleportation, which deals with techniques to transmit quantum states over arbitrary distances. Quantum teleportation, or entanglement-assisted teleportation, is a technique used to transfer information on a Quantum level usually from one Particle

In many devices, even the simple light switch, quantum tunneling is vital, as otherwise the electrons in the electric current could not penetrate the potential barrier made up, in the case of the light switch, of a layer of oxide. A light switch is a Switch, most commonly used to operate Electric lights permanently connected equipment or Electrical outlets In modern homes In Quantum mechanics, quantum tunnelling is a nanoscopic phenomenon in which a particle violates the principles of Classical mechanics by penetrating a Flash memory chips found in USB drives also use quantum tunneling to erase their memory cells. Flash memory is non-volatile computer memory that can be electrically erased and reprogrammed

## Philosophical consequences

Main article: Interpretation of quantum mechanics

Since its inception, the many counter-intuitive results of quantum mechanics have provoked strong philosophical debate and many interpretations. An interpretation of quantum mechanics is a statement which attempts to explain how Quantum mechanics informs our Understanding of Nature. Philosophy is the study of general problems concerning matters such as existence knowledge truth beauty justice validity mind and language An interpretation of quantum mechanics is a statement which attempts to explain how Quantum mechanics informs our Understanding of Nature. Even fundamental issues such as Max Born's basic rules concerning probability amplitudes and probability distributions took decades to be appreciated. Max Born (11 December 1882 &ndash 5 January 1970 was a German Physicist and Mathematician who was instrumental in the development of Quantum In Quantum mechanics, a probability amplitude is a complex -valued function that describes an uncertain or unknown quantity In Probability theory and Statistics, a probability distribution identifies either the probability of each value of an unidentified Random variable

The Copenhagen interpretation, due largely to the Danish theoretical physicist Niels Bohr, is the interpretation of quantum mechanics most widely accepted amongst physicists. The Copenhagen interpretation is an interpretation of Quantum mechanics. Niels Henrik David Bohr (nels ˈb̥oɐ̯ˀ in Danish 7 October 1885 – 18 November 1962 was a Danish Physicist who made fundamental contributions to understanding According to it, the probabilistic nature of quantum mechanics predictions cannot be explained in terms of some other deterministic theory, and does not simply reflect our limited knowledge. Quantum mechanics provides probabilistic results because the physical universe is itself probabilistic rather than deterministic. Probability is the likelihood or chance that something is the case or will happen Determinism is the philosophical Proposition that every event including human cognition and behaviour decision and action is causally determined

Albert Einstein, himself one of the founders of quantum theory, disliked this loss of determinism in measurement. Albert Einstein ( German: ˈalbɐt ˈaɪ̯nʃtaɪ̯n; English: ˈælbɝt ˈaɪnstaɪn (14 March 1879 – 18 April 1955 was a German -born theoretical (Hence his famous quote "God does not play dice with the universe. ") He held that there should be a local hidden variable theory underlying quantum mechanics and consequently the present theory was incomplete. In Quantum mechanics, a local Hidden variable theory is one in which distant events are assumed to have no instantaneous (or at least Faster-than-light He produced a series of objections to the theory, the most famous of which has become known as the EPR paradox. In Quantum mechanics, the EPR paradox is a Thought experiment which challenged long-held ideas about the relation between the observed values of physical quantities John Bell showed that the EPR paradox led to experimentally testable differences between quantum mechanics and local theories. John Stewart Bell ( June 28 1928 &ndash October 1 1990) was a Physicist, and the originator of Bell's Theorem, one of the Bell's theorem is a theorem that shows that the predictions of Quantum mechanics (QM are not intuitive and touches upon fundamental philosophical issues that relate to modern Experiments have been taken as confirming that quantum mechanics is correct and the real world must be described in terms of nonlocal theories. The Bell test experiments serve to investigate the validity of the entanglement effect in Quantum mechanics by using some kind of Bell inequality.

The writer C. S. Lewis viewed quantum mechanics as incomplete, because notions of indeterminism did not agree with his religious beliefs. Clive Staples Lewis (29 November 1898 – 22 November 1963 [4] Lewis, a professor of English, was of the opinion that the Heisenberg uncertainty principle was more of an epistemic limitation than an indication of ontological indeterminacy, and in this respect believed similarly to many advocates of hidden variables theories. In Quantum physics, the Heisenberg uncertainty principle states that locating a particle in a small region of space makes the Momentum of the particle uncertain Epistemology (from Greek επιστήμη - episteme, "knowledge" + λόγος, " Logos " or theory of knowledge In Philosophy, ontology (from the Greek, genitive: of being (part The Bohr-Einstein debates provide a vibrant critique of the Copenhagen Interpretation from an epistemological point of view.

The Everett many-worlds interpretation, formulated in 1956, holds that all the possibilities described by quantum theory simultaneously occur in a "multiverse" composed of mostly independent parallel universes. The many-worlds interpretation or MWI (also known as relative state formulation, theory of the universal wavefunction, parallel universes, The multiverse (or meta-universe) is the hypothetical set of multiple possible Universes (including our universe that together comprise all of Reality. This is not accomplished by introducing some new axiom to quantum mechanics, but on the contrary by removing the axiom of the collapse of the wave packet: All the possible consistent states of the measured system and the measuring apparatus (including the observer) are present in a real physical (not just formally mathematical, as in other interpretations) quantum superposition. Quantum superposition is the fundamental law of Quantum mechanics. (Such a superposition of consistent state combinations of different systems is called an entangled state. Quantum entanglement is a quantum mechanical Phenomenon in which the Quantum states of two or more objects are linked together so that one object ) While the multiverse is deterministic, we perceive non-deterministic behavior governed by probabilities, because we can observe only the universe, i. e. the consistent state contribution to the mentioned superposition, we inhabit. Everett's interpretation is perfectly consistent with John Bell's experiments and makes them intuitively understandable. John Stewart Bell ( June 28 1928 &ndash October 1 1990) was a Physicist, and the originator of Bell's Theorem, one of the However, according to the theory of quantum decoherence, the parallel universes will never be accessible for us, making them physically meaningless. In Quantum mechanics, quantum decoherence is the mechanism by which quantum systems interact with their environments to exhibit probabilistically additive behavior This inaccessibility can be understood as follows: once a measurement is done, the measured system becomes entangled with both the physicist who measured it and a huge number of other particles, some of which are photons flying away towards the other end of the universe; in order to prove that the wave function did not collapse one would have to bring all these particles back and measure them again, together with the system that was measured originally. In Physics, the photon is the Elementary particle responsible for electromagnetic phenomena This is completely impractical, but even if one can theoretically do this, it would destroy any evidence that the original measurement took place (including the physicist's memory).

## Notes

1. ^ Life on the lattice: The most accurate theory we have
2. ^ Especially since Werner Heisenberg was awarded the Nobel Prize in Physics in 1932 for the creation of quantum mechanics, the role of Max Born has been obfuscated. An interpretation of quantum mechanics is a statement which attempts to explain how Quantum mechanics informs our Understanding of Nature. The many-worlds interpretation or MWI (also known as relative state formulation, theory of the universal wavefunction, parallel universes, The framework of Quantum mechanics requires a careful definition of measurement, and a thorough discussion of its practical and philosophical implications The dynamics of photons in the double-slit experiment describes the relationship between classical electromagnetic waves and Photons the quantum counterpart of classical electromagnetic Photon polarization is the quantum mechanical description of the classical polarized sinusoidal plane electromagnetic wave The scientific school of Quantum electrochemistry began to form in the 1960s under Revaz Dogonadze. Quantum chromodynamics (abbreviated as QCD is a theory of the Strong interaction ( color force a Fundamental force describing the interactions of the Quantum chemistry is a branch of Theoretical chemistry, which applies Quantum mechanics and Quantum field theory to address issues and problems in A quantum computer is a device for Computation that makes direct use of distinctively Quantum mechanical Phenomena, such as superposition Quantum electronics is the area of Physics dealing with the effects of Quantum mechanics on the behaviour of Electrons in matter and their interactions In quantum field theory (QFT the forces between particles are mediated by other particles In Quantum mechanics, quantum information is Physical information that is held in the "state" of a Quantum system. Quantum mind theories are based on the premise that Quantum mechanics is necessary to fully understand the Mind and Brain, particularly concerning an explanation In the physical sciences quantum thermodynamics is the study of Heat and work dynamics in quantum systems The theoretical and experimental justification for the Schrödinger equation motivates the discovery of the Schrödinger equation, the equation that describes the dynamics of Theoretical chemistry involves the use of physics to explain or predict chemical phenomena Quasi-set theory is a formal mathematical theory of collections of indistinguishable objects mainly motivated by the assumption that certain objects treated in Quantum physics Quantum optics is a field of research in Physics, dealing with the application of Quantum mechanics to phenomena involving Light and its interactions Werner Heisenberg (5 December 1901 in Würzburg &ndash1 February 1976 in Munich) was a German theoretical physicist best known for enunciating the Max Born (11 December 1882 &ndash 5 January 1970 was a German Physicist and Mathematician who was instrumental in the development of Quantum A 2005 biography of Born details his role as the creator of the matrix formulation of quantum mechanics. This was recognized in a paper by Heisenberg, in 1940, honoring Max Planck. See: Nancy Thorndike Greenspan, "The End of the Certain World: The Life and Science of Max Born" (Basic Books, 2005), pp. 124 - 128, and 285 - 286.
3. ^ Derivation of particle in a box, chemistry.tidalswan.com
4. ^ Does God Play Dice? Archived Lecture by Professor Steven Hawking, Department of Applied Mathematics and Theoretical Physics (DAMTP) University of Cambridge. Retrieved on 2007-09-07. Year 2007 ( MMVII) was a Common year starting on Monday of the Gregorian calendar in the 21st century. Events 1251 BC - A Solar eclipse on this date might mark the birth of legendary Heracles at Thebes Greece.

## References

• P. A. M. Dirac, The Principles of Quantum Mechanics (1930) -- the beginning chapters provide a very clear and comprehensible introduction
• David J. Griffiths, Introduction to Quantum Mechanics, Prentice Hall, 1995. David J Griffiths (born 1942 is a US physicist and educator He has worked at Reed College since 1978 where he is currently the Howard Vollum ISBN 0-13-124405-1 -- A standard undergraduate level text written in an accessible style.
• Richard P. Feynman, Robert B. Leighton and Matthew Sands (1965). Richard Phillips Feynman (ˈfaɪnmən May 11 1918 – February 15 1988 was an American Physicist known for the Path integral formulation of quantum Robert B Leighton ( September 10, 1919 &ndash March 9, 1997) was an American The Feynman Lectures on Physics, Addison-Wesley. The Feynman Lectures on Physics by Richard Feynman, Robert Leighton, and Matthew Sands is perhaps Feynman's most accessible technical work
• Hugh Everett, Relative State Formulation of Quantum Mechanics, Reviews of Modern Physics vol 29, (1957) pp 454-462. Hugh Everett III ( November 11, 1930 – July 19, 1982) was an American physicist who first proposed the Many-worlds interpretation
• Bryce DeWitt, R. Bryce Seligman DeWitt ( January 8, 1923 &ndash September 23, 2004) was a Theoretical physicist best known for formulating Canonical Neill Graham, eds, The Many-Worlds Interpretation of Quantum Mechanics, Princeton Series in Physics, Princeton University Press (1973), ISBN 0-691-08131-X
• Albert Messiah, Quantum Mechanics, English translation by G. The Princeton University Press is an independent publisher with close connections to Princeton University. M. Temmer of Mécanique Quantique, 1966, John Wiley and Sons, vol. I, chapter IV, section III.
• Richard P. Feynman, QED: The Strange Theory of Light and Matter -- a popular science book about quantum mechanics and quantum field theory that contains many enlightening insights that are interesting for the expert as well
• Marvin Chester, Primer of Quantum Mechanics, 1987, John Wiley, N. Richard Phillips Feynman (ˈfaɪnmən May 11 1918 – February 15 1988 was an American Physicist known for the Path integral formulation of quantum QED The Strange Theory of Light and Matter is an adaptation for the general reader of four Lectures on Quantum electrodynamics (QED by Richard Feynman In quantum field theory (QFT the forces between particles are mediated by other particles Y. ISBN 0-486-42878-8
• Hagen Kleinert, Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets, 3th edition, World Scientific (Singapore, 2004) (drafts of a forthcoming fourth edition available online here)
• George Mackey (2004). Hagen Kleinert (born 1941 is Professor of Theoretical Physics at the Free University of Berlin, Germany (since 1968 Honorary Professor at the Kyrgyz-Russian George Whitelaw Mackey (February 1 1916 in St Louis, Missouri – March 15 2006 in Belmont, Massachusetts) was an American Mathematician The mathematical foundations of quantum mechanics. Dover Publications. ISBN 0-486-43517-2.
• Griffiths, David J. (2004). Introduction to Quantum Mechanics (2nd ed. ). Prentice Hall. ISBN 0-13-805326-X.
• Omnès, Roland (1999). Roland Omnès is the author of several books which aim to close the gap between our common sense experience of the classical world and the complex formal mathematics which is now required Understanding Quantum Mechanics. Princeton University Press. ISBN 0-691-00435-8.
• Transnational College of Lex (1996). Transnational College of Lex is a research institution in Japan dedicated to the proposition that learning is highly influenced by environment What is Quantum Mechanics? A Physics Adventure. Language Research Foundation, Boston. ISBN 0-9643504-1-6.
• J. von Neumann, Mathematical Foundations of Quantum Mechanics, Princeton University Press, 1955.
• H. Weyl, The Theory of Groups and Quantum Mechanics, Dover Publications 1950. Hermann Klaus Hugo Weyl ( 9 November 1885 – 8 December 1955) was a German Mathematician.
• Max Jammer, "The Conceptual Development of Quantum Mechanics" (McGraw Hill Book Co. Max Jammer (born 1915 is an Israeli physicist and philosopher of physics. , 1966)
• Gunther Ludwig, "Wave Mechanics" (Pergamon Press, 1968) ISBN 0-08-203204-1
• Albert Messiah, Quantum Mechanics (Vol. I), English translation from French by G. M. Temmer, fourth printing 1966, North Holland, John Wiley & Sons.
• Eric R. Scerri, The Periodic Table: Its Story and Its Significance, Oxford University Press, 2006. Considers the extent to which chemistry and especially the periodic system has been reduced to quantum mechanics. ISBN 0-19-530573-6
• Gary Zukav, The Dancing Wu Li Masters: An Overview of the New Physics (1979) ISBN 0-553-26382-X