Quantum mechanics

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

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

Quantum mechanics differs from classical physics in that energy, momentum, angular momentum, and other quantities of a bound system are restricted to discrete values (quantization), objects have characteristics of both particles and waves (wave-particle duality), and there are limits to how accurately the value of a physical quantity can be predicted prior to its measurement, given a complete set of initial conditions (the uncertainty principle).[note 1]

Quantum mechanics arose gradually, from theories to explain observations which could not be reconciled with classical physics, such as Max Planck's solution in 1900 to the black-body radiation problem, and the correspondence between energy and frequency in Albert Einstein's 1905 paper which explained the photoelectric effect. Early quantum theory was profoundly re-conceived in the mid-1920s by Niels Bohr, Erwin Schrödinger, Werner Heisenberg, Max Born and others. The original interpretation of quantum mechanics is the Copenhagen interpretation, developed by Niels Bohr and Werner Heisenberg in Copenhagen during the 1920s. The modern theory is formulated in various specially developed mathematical formalisms. In one of them, a mathematical function, the wave function, provides information about the probability amplitude of energy, momentum, and other physical properties of a particle.

History

Main article: History of quantum mechanics

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.[5] In 1803 English polymath Thomas Young described the famous double-slit experiment.[6] This experiment played a major role in the general acceptance of the wave theory of light.

In 1838 Michael Faraday discovered cathode rays. These studies were followed by the 1859 statement of the black-body radiation problem by Gustav Kirchhoff, the 1877 suggestion by Ludwig Boltzmann that the energy states of a physical system can be discrete, and the 1900 quantum hypothesis of Max Planck.[7] Planck's hypothesis that energy is radiated and absorbed in discrete "quanta" (or energy packets) precisely matched the observed patterns of black-body radiation.

In 1896 Wilhelm Wien empirically determined a distribution law of black-body radiation,[8] called Wien's law. Ludwig Boltzmann independently arrived at this result by considerations of Maxwell's equations. However, it was valid only at high frequencies and underestimated the radiance at low frequencies.

The foundations of quantum mechanics were established during the first half of the 20th century by Max Planck, Niels Bohr, Werner Heisenberg, Louis de Broglie, Arthur Compton, Albert Einstein, Richard Feynman, Erwin Schrödinger, Max Born, John von Neumann, Paul Dirac, Enrico Fermi, Wolfgang Pauli, Max von Laue, Freeman Dyson, David Hilbert, Wilhelm Wien, Satyendra Nath Bose, Arnold Sommerfeld, and others. The Copenhagen interpretation of Niels Bohr became widely accepted.

Max Planck corrected this model using Boltzmann's statistical interpretation of thermodynamics and proposed what is now called Planck's law, which led to the development of quantum mechanics. After Planck's solution in 1900 to the black-body radiation problem (reported 1859), Albert Einstein offered a quantum-based explanation of the photoelectric effect (1905, reported 1887). Around 1900–1910, the atomic theory but not the corpuscular theory of light[9] first came to be widely accepted as scientific fact; these latter theories can be considered quantum theories of matter and Electromagnetic radiation, respectively. However, the photon theory was not widely accepted until about 1915. Even until Einstein's Nobel Prize, Niels Bohr did not believe in the photon.[10]

Among the first to study quantum phenomena were Arthur Compton, C. V. Raman, and Pieter Zeeman, each of whom has a quantum effect named after him. Robert Andrews Millikan studied the photoelectric effect experimentally, and Albert Einstein developed a theory for it. At the same time, Ernest Rutherford experimentally discovered the nuclear model of the atom, and Niels Bohr developed a theory of atomic structure, confirmed by the experiments of Henry Moseley. In 1913 Peter Debye extended Bohr's theory by introducing elliptical orbits, a concept also introduced by Arnold Sommerfeld.[11] This phase is known as old quantum theory.

According to Planck, each energy element (E) is proportional to its frequency (ν):

\( E=h\nu \ , \)

Max Planck is considered the father of the quantum theory.

where h is Planck's constant.

Planck cautiously insisted that this was only an aspect of the processes of absorption and emission of radiation and was not the physical reality of the radiation.[12] In fact, he considered his quantum hypothesis a mathematical trick to get the right answer rather than a sizable discovery.[13] However, in 1905 Albert Einstein interpreted Planck's quantum hypothesis realistically and used it to explain the photoelectric effect, in which shining light on certain materials can eject electrons from the material. Einstein won the 1921 Nobel Prize in Physics for this work.

Einstein further developed this idea to show that an electromagnetic wave such as light could also be described as a particle (later called the photon), with a discrete amount of energy that depends on its frequency.[14] 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,[15] which became the basis of the laser.
The 1927 Solvay Conference in Brussels was the fifth world physics conference.

In the mid-1920s quantum mechanics was developed to become the standard formulation for atomic physics. In the summer of 1925, Bohr and Heisenberg published results that closed the old quantum theory. Due to their particle-like behavior in certain processes and measurements, light quanta came to be called photons (1926). In 1926 Erwin Schrödinger suggested a partial differential equation for the wave functions of particles like electrons. And when effectively restricted to a finite region, this equation allowed only certain modes, corresponding to discrete quantum states – whose properties turned out to be exactly the same as implied by matrix mechanics.[16] Einstein's simple postulation spurred a flurry of debate, theorizing, and testing. Thus, the entire field of quantum physics emerged, leading to its wider acceptance at the Fifth Solvay Conference in 1927.[17]

It was found that subatomic particles and electromagnetic waves are neither simply particle nor wave but have certain properties of each. This originated the concept of wave–particle duality.[18]

By 1930 quantum mechanics had been further unified and formalized by David Hilbert, Paul Dirac and John von Neumann[19] with greater emphasis on measurement, the statistical nature of our knowledge of reality, and philosophical speculation about the 'observer'.[20] It has since permeated many disciplines, including quantum chemistry, quantum electronics, quantum optics, and quantum information science. It also provides a useful framework for many features of the modern periodic table of elements, and describes the behaviors of atoms during chemical bonding and the flow of electrons in computer semiconductors, and therefore plays a crucial role in many modern technologies.[18] Its speculative modern developments include string theory and quantum gravity theory.

While quantum mechanics was constructed to describe the world of the very small, it is also needed to explain some macroscopic phenomena such as superconductors[21] and superfluids.[22]

The word quantum derives from the Latin, meaning "how great" or "how much".[23] In quantum mechanics, it refers to a discrete unit assigned to certain physical quantities such as the energy of an atom at rest (see Figure 1). The discovery that particles are discrete packets of energy with wave-like properties led to the branch of physics dealing with atomic and subatomic systems which is today called quantum mechanics. It underlies the mathematical framework of many fields of physics and chemistry, including condensed matter physics, solid-state physics, atomic physics, molecular physics, computational physics, computational chemistry, quantum chemistry, particle physics, nuclear chemistry, and nuclear physics.[24] Some fundamental aspects of the theory are still actively studied.[25]

Quantum mechanics is essential for understanding the behavior of systems at atomic length scales and smaller. If the physical nature of an atom were solely described by classical mechanics, electrons would not orbit the nucleus, since orbiting electrons emit radiation (due to circular motion) and so would quickly lose energy and collide with the nucleus. This framework was unable to explain the stability of atoms. Instead, electrons remain in an uncertain, non-deterministic, smeared, probabilistic wave–particle orbital about the nucleus, defying the traditional assumptions of classical mechanics and electromagnetism.[26]

Quantum mechanics was initially developed to provide a better explanation and description of the atom, especially the differences in the spectra of light emitted by different isotopes of the same chemical element, as well as subatomic particles. In short, the quantum-mechanical atomic model has succeeded spectacularly in the realm where classical mechanics and electromagnetism falter.

Broadly speaking, quantum mechanics incorporates four classes of phenomena for which classical physics cannot account[20]:

quantization of certain physical properties
quantum entanglement
principle of uncertainty
wave–particle duality

Mathematical formulations
Main article: Mathematical formulation of quantum mechanics
See also: Quantum logic

In the mathematically rigorous formulation of quantum mechanics developed by Paul Dirac,[27] David Hilbert,[28] John von Neumann,[29] and Hermann Weyl,[30] the possible states of a quantum mechanical system are symbolized[31] as unit vectors (called state vectors). Formally, these vectors are elements of a complex separable Hilbert space – variously called the state space or the associated Hilbert space of the system – that is well defined up to a complex number of norm 1 (the phase factor). In other words, the possible states are points in the projective space of a Hilbert space, usually called the complex projective space. The exact nature of this Hilbert space is dependent on the system – for example, 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. Each observable is represented by a maximally Hermitian (precisely: by a self-adjoint) linear operator acting on the state space. 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. If the operator's spectrum is discrete, the observable can attain only those discrete eigenvalues.

In the formalism of quantum mechanics, the state of a system at a given time is described by a complex wave function, also referred to as state vector in a complex vector space.[32] This abstract mathematical object allows for the calculation of probabilities of outcomes of concrete experiments. 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, to arbitrary precision. For instance, electrons may be considered (to a certain probability) to be located somewhere within a given region of space, but with their exact positions unknown. Contours of constant probability density, 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. Heisenberg's uncertainty principle quantifies the inability to precisely locate the particle given its conjugate momentum.[33]

According to one interpretation, as the result of a measurement, the wave function containing the probability information for a system collapses from a given initial state to a particular eigenstate. The possible results of a measurement are the eigenvalues of the operator representing the observable – which explains the choice of Hermitian operators, for which all the eigenvalues are real. The probability distribution of an observable in a given state can be found by computing the spectral decomposition of the corresponding operator. Heisenberg's uncertainty principle is represented by the statement that the operators corresponding to certain observables do not commute.

The probabilistic nature of quantum mechanics thus stems from the act of measurement. This is one of the most difficult aspects of quantum systems to understand. It was the central topic in the famous 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. Newer interpretations of quantum mechanics have been formulated that do away with the concept of "wave function collapse" (see, for example, the relative state interpretation). The basic idea is that when a quantum system interacts with a measuring apparatus, their respective wave functions become entangled, so that the original quantum system ceases to exist as an independent entity. For details, see the article on measurement in quantum mechanics.[34]

Generally, quantum mechanics does not assign definite values. Instead, it makes a prediction using a probability distribution; that is, it describes the probability of obtaining the possible outcomes from measuring an observable. Often these results are skewed by many causes, such as dense probability clouds. Probability clouds are approximate (but better than the Bohr model) whereby electron location is given by a probability function, the wave function eigenvalue, such that the probability is the squared modulus of the complex amplitude, or quantum state nuclear attraction.[35][36] Naturally, these probabilities will depend on the quantum state at the "instant" of the measurement. Hence, uncertainty is involved in the value. 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 translated from German as meaning "inherent" or "characteristic").[37]

In the everyday world, it is natural and intuitive to think of everything (every observable) as being in an eigenstate. Everything appears to have a definite position, a definite momentum, a definite energy, and a definite time of occurrence. However, quantum mechanics does not pinpoint the exact values of a particle's position and momentum (since they are conjugate pairs) or its energy and time (since they too are conjugate pairs). Rather, it provides only a range of probabilities in which that particle might be given its momentum and momentum probability. Therefore, it is helpful to use different words to describe states having uncertain values and states having definite values (eigenstates).

Usually, a system will not be in an eigenstate of the observable (particle) we are interested in. However, if one measures the observable, the wave function will instantaneously be an eigenstate (or "generalized" eigenstate) of that observable. This process is known as wave function collapse, a controversial and much-debated process[38] that involves expanding the system under study to include the measurement device. If one knows the corresponding wave function at the instant before the measurement, one will be able to compute the probability of the wave function collapsing into each of the possible eigenstates.

For example, the free particle in the previous example will usually have a wave function that is a wave packet centered around some mean position x0 (neither an eigenstate of position nor of momentum). When one measures the position of the particle, it is impossible to predict with certainty the result.[34] 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.[39]

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 the time evolution. The time evolution of wave functions is deterministic in the sense that – given a wave function at an initial time – it makes a definite prediction of what the wave function will be at any later time.[40]

During a measurement, on the other hand, the change of the initial wave function into another, later wave function is not deterministic, it is unpredictable (i.e., random). A time-evolution simulation can be seen here.[41][42]

Wave functions change as time progresses. The Schrödinger equation describes how wave functions change in time, playing a role similar to Newton's second law in classical mechanics. 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 with time. This also has the effect of turning a position eigenstate (which can be thought of as an infinitely sharp wave packet) into a broadened wave packet that no longer represents a (definite, certain) position eigenstate.[43]
Fig. 1: Probability densities corresponding to the wave functions of an electron in a hydrogen atom possessing definite energy levels (increasing from the top of the image to the bottom: n = 1, 2, 3, ...) and angular momenta (increasing across from left to right: s, p, d, ...). Denser areas correspond to higher probability density in a position measurement. Such wave functions are directly comparable to Chladni's figures of acoustic modes of vibration in classical physics and are modes of oscillation as well, possessing a sharp energy and thus, a definite frequency. The angular momentum and energy are quantized and take only discrete values like those shown (as is the case for resonant frequencies in acoustics)

Some wave functions produce probability distributions that are constant, or independent of time – such as when in a stationary state of definite energy, time vanishes in the absolute square of the wave function (this is the basis for the energy-time uncertainty principle). 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 wave function surrounding the nucleus (Fig. 1) (however, only the lowest angular momentum states, labeled s, are spherically symmetric.)[44]

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. This gives rise to the "wave-like" behavior of quantum states.

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

However, there are techniques for finding approximate solutions. One method, called perturbation theory, uses the analytic result for a simple quantum mechanical model to create a result for a related but more complicated model by (for example) the addition of a weak potential energy. Another method is called "semi-classical equation of motion", which applies to systems for which quantum mechanics produces only small deviations from classical behavior. These deviations can then be computed based on the classical motion. This approach is particularly important in the field of quantum chaos.
Mathematically equivalent 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, which unifies and generalizes the two earliest formulations of quantum mechanics – matrix mechanics (invented by Werner Heisenberg) and wave mechanics (invented by Erwin Schrödinger).[45]

Especially since Heisenberg was awarded the Nobel Prize in Physics in 1932 for the creation of quantum mechanics, the role of Max Born in the development of QM was overlooked until the 1954 Nobel award. The role is noted in a 2005 biography of Born, which recounts his role in the matrix formulation and the use of probability amplitudes. Heisenberg acknowledges having learned matrices from Born, as published in a 1940 festschrift honoring Max Planck.[46] In the matrix formulation, the instantaneous state of a quantum system encodes the probabilities of its measurable properties, or "observables". Examples of observables include energy, position, momentum, and angular momentum. Observables can be either continuous (e.g., the position of a particle) or discrete (e.g., the energy of an electron bound to a hydrogen atom).[47] An alternative formulation of quantum mechanics is Feynman's path integral formulation, in which a quantum-mechanical amplitude is considered as a sum over all possible classical and non-classical paths between the initial and final states. This is the quantum-mechanical counterpart of the action principle in classical mechanics.
Relation to other scientific theories

The rules of quantum mechanics are fundamental. They assert that the state space of a system is a Hilbert space (crucially, that the space has an inner product) and that observables of the system are Hermitian operators acting on vectors in that space – although they do not tell us which Hilbert space or which operators. These can be chosen appropriately in order to obtain a quantitative description of a quantum system. An important guide for making these choices is the correspondence principle, which states that the predictions of quantum mechanics reduce to those of classical mechanics when a system moves to higher energies or, equivalently, larger quantum numbers, i.e. whereas a single particle exhibits a degree of randomness, in systems incorporating millions of particles averaging takes over and, at the high energy limit, the statistical probability of random behaviour approaches zero. In other words, classical mechanics is simply a quantum mechanics of large systems. This "high energy" limit is known as the classical or correspondence limit. One can even start from an established classical model of a particular system, then try to guess the underlying quantum model that would give rise to the classical model in the correspondence limit.
Question, Web Fundamentals.svg Unsolved problem 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 "wave function collapse", give rise to the reality we perceive?
(more unsolved problems in physics)

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

Early attempts to merge quantum mechanics with special relativity involved the replacement of the Schrödinger equation with a covariant equation such as the Klein–Gordon equation or the Dirac equation. While these theories were successful in explaining many experimental results, they had certain unsatisfactory qualities stemming from their neglect of the relativistic creation and annihilation of particles. A fully relativistic quantum theory required the development of quantum field theory, which applies quantization to a field (rather than a fixed set of particles). The first complete quantum field theory, quantum electrodynamics, provides a fully quantum description of the electromagnetic interaction. The full apparatus of quantum field theory is often unnecessary for describing electrodynamic systems. A simpler approach, one that has been used since the inception of quantum mechanics, is to treat charged particles as quantum mechanical objects being acted on by a classical electromagnetic field. For example, the elementary quantum model of the hydrogen atom describes the electric field of the hydrogen atom using a classical \( {\displaystyle \textstyle -e^{2}/(4\pi \epsilon _{_{0}}r)} \) Coulomb potential. This "semi-classical" approach fails if quantum fluctuations in the electromagnetic field play an important role, such as in the emission of photons by charged particles.

Quantum field theories for the strong nuclear force and the weak nuclear force have also been developed. The quantum field theory of the strong nuclear force is called quantum chromodynamics, and describes the interactions of subnuclear particles such as quarks and gluons. The weak nuclear force and the electromagnetic force were unified, in their quantized forms, into a single quantum field theory (known as electroweak theory), by the physicists Abdus Salam, Sheldon Glashow and Steven Weinberg. These three men shared the Nobel Prize in Physics in 1979 for this work.[48]

It has proven difficult to construct quantum models of gravity, the remaining fundamental force. Semi-classical approximations are workable, and have led to predictions such as Hawking radiation. 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. The resolution of these incompatibilities is an area of active research. Candidates for a future theory of quantum gravity include string theory.

Classical mechanics has also been extended into the complex domain, with complex classical mechanics exhibiting behaviors similar to quantum mechanics.[49]
Relation to classical physics

Predictions of quantum mechanics have been verified experimentally to an extremely high degree of accuracy.[50] According to the correspondence principle between classical and quantum mechanics, all objects obey the laws of quantum mechanics, and classical mechanics is just an approximation for large systems of objects (or a statistical quantum mechanics of a large collection of particles).[51] The laws of classical mechanics thus follow from the laws of quantum mechanics as a statistical average at the limit of large systems or large quantum numbers (Ehrenfest theorem).[52][53] However, chaotic systems do not have good quantum numbers, and quantum chaos studies the relationship between classical and quantum descriptions in these systems.

Quantum coherence is an essential difference between classical and quantum theories as illustrated by the Einstein–Podolsky–Rosen (EPR) paradox – an attack on a certain philosophical interpretation of quantum mechanics by an appeal to local realism.[54] Quantum interference involves adding together probability amplitudes, whereas classical "waves" infer that there is an adding together of intensities. For microscopic bodies, the extension of the system is much smaller than the coherence length, which gives rise to long-range entanglement and other nonlocal phenomena characteristic of quantum systems.[55] Quantum coherence is not typically evident at macroscopic scales, except maybe at temperatures approaching absolute zero at which quantum behavior may manifest macroscopically.[56] This is in accordance with the following observations:

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.[57]
While the seemingly "exotic" behavior of matter posited by quantum mechanics and relativity theory become more apparent for extremely small particles or for velocities approaching the speed of light, the laws of classical, often considered "Newtonian", physics remain accurate in predicting the behavior of the vast majority of "large" objects (on the order of the size of large molecules or bigger) at velocities much smaller than the velocity of light.[58]

Copenhagen interpretation of quantum versus classical kinematics

A big difference between classical and quantum mechanics is that they use very different kinematic descriptions.[59]

In Niels Bohr's mature view, quantum mechanical phenomena are required to be experiments, with complete descriptions of all the devices for the system, preparative, intermediary, and finally measuring. The descriptions are in macroscopic terms, expressed in ordinary language, supplemented with the concepts of classical mechanics.[60][61][62][63] The initial condition and the final condition of the system are respectively described by values in a configuration space, for example a position space, or some equivalent space such as a momentum space. Quantum mechanics does not admit a completely precise description, in terms of both position and momentum, of an initial condition or "state" (in the classical sense of the word) that would support a precisely deterministic and causal prediction of a final condition.[64][65] In this sense, a quantum phenomenon is a process, a passage from initial to final condition, not an instantaneous "state" in the classical sense of that word.[66][67] Thus there are two kinds of processes in quantum mechanics: stationary and transitional. For a stationary process, the initial and final condition are the same. For a transition, they are different. Obviously by definition, if only the initial condition is given, the process is not determined.[64] Given its initial condition, prediction of its final condition is possible, causally but only probabilistically, because the Schrödinger equation is deterministic for wave function evolution, but the wave function describes the system only probabilistically.[68][69]

For many experiments, it is possible to think of the initial and final conditions of the system as being a particle. In some cases it appears that there are potentially several spatially distinct pathways or trajectories by which a particle might pass from initial to final condition. It is an important feature of the quantum kinematic description that it does not permit a unique definite statement of which of those pathways is actually followed. Only the initial and final conditions are definite, and, as stated in the foregoing paragraph, they are defined only as precisely as allowed by the configuration space description or its equivalent. In every case for which a quantum kinematic description is needed, there is always a compelling reason for this restriction of kinematic precision. An example of such a reason is that for a particle to be experimentally found in a definite position, it must be held motionless; for it to be experimentally found to have a definite momentum, it must have free motion; these two are logically incompatible.[70][71]

Classical kinematics does not primarily demand experimental description of its phenomena. It allows completely precise description of an instantaneous state by a value in phase space, the Cartesian product of configuration and momentum spaces. This description simply assumes or imagines a state as a physically existing entity without concern about its experimental measurability. Such a description of an initial condition, together with Newton's laws of motion, allows a precise deterministic and causal prediction of a final condition, with a definite trajectory of passage. Hamiltonian dynamics can be used for this. Classical kinematics also allows the description of a process analogous to the initial and final condition description used by quantum mechanics. Lagrangian mechanics applies to this.[72] For processes that need account to be taken of actions of a small number of Planck constants, classical kinematics is not adequate; quantum mechanics is needed.
Relation to general relativity

Even with the defining postulates of both Einstein's theory of general relativity and quantum theory being indisputably supported by rigorous and repeated empirical evidence, and while they do not directly contradict each other theoretically (at least with regard to their primary claims), they have proven extremely difficult to incorporate into one consistent, cohesive model.[73]

Gravity is negligible in many areas of particle physics, so that unification between general relativity and quantum mechanics is not an urgent issue in those particular applications. However, the lack of a correct theory of quantum gravity is an important issue in physical cosmology and the search by physicists for an elegant "Theory of Everything" (TOE). Consequently, resolving the inconsistencies between both theories has been a major goal of 20th- and 21st-century physics. Many prominent physicists, including Stephen Hawking, worked for many years to create a theory underlying everything. This TOE would combine not only the models of subatomic physics, but also derive the four fundamental forces of nature – the strong force, electromagnetism, the weak force, and gravity – from a single force or phenomenon. However, after considering Gödel's Incompleteness Theorem, Hawking concluded that a theory of everything is not possible, and stated so publicly in his lecture "Gödel and the End of Physics" (2002).[74]
Attempts at a unified field theory
Main article: Grand unified theory

The quest to unify the fundamental forces through quantum mechanics is ongoing. Quantum electrodynamics (or "quantum electromagnetism"), which is (at least in the perturbative regime) the most accurately tested physical theory in competition with general relativity,[75][76] has been merged with the weak nuclear force into the electroweak force; work continues, to merge it with the strong force into the electrostrong force. Current predictions state that at around 1014 GeV these three forces fuse into a single field.[77] Beyond this "grand unification", it is speculated that it may be possible to merge gravity with the other three gauge symmetries, expected to occur at roughly 1019 GeV. However – and while special relativity is parsimoniously incorporated into quantum electrodynamics – the expanded general relativity, currently the best theory describing the gravitation force, has not been fully incorporated into quantum theory. One of those searching for a coherent TOE is Edward Witten, a theoretical physicist who formulated the M-theory, which is an attempt at describing the supersymmetrical based string theory. M-theory posits that our apparent 4-dimensional spacetime is, in reality, actually an 11-dimensional spacetime containing 10 spatial dimensions and 1 time dimension, although 7 of the spatial dimensions are – at lower energies – completely "compactified" (or infinitely curved) and not readily amenable to measurement or probing.

Another popular theory is loop quantum gravity (LQG) proposed by Carlo Rovelli, that describes quantum properties of gravity. It is also a theory of quantum spacetime and quantum time, because in general relativity the geometry of spacetime is a manifestation of gravity. LQG is an attempt to merge and adapt standard quantum mechanics and standard general relativity. This theory describes space as granular analogous to the granularity of photons in the quantum theory of electromagnetism and the discrete energy levels of atoms. More precisely, space is an extremely fine fabric or networks "woven" of finite loops called spin networks. The evolution of a spin network over time is called a spin foam. The predicted size of this structure is the Planck length, which is approximately 1.616×10⁻³⁵ m. According to this theory, there is no meaning to length shorter than this (cf. Planck scale energy).

Philosophical implications
Main article: Interpretations of quantum mechanics

Since its inception, the many counter-intuitive aspects and results of quantum mechanics have provoked strong philosophical debates and many interpretations. Even fundamental issues, such as Max Born's basic rules about probability amplitudes and probability distributions, took decades to be appreciated by society and many leading scientists. Richard Feynman once said, "I think I can safely say that nobody understands quantum mechanics."[78] According to Steven Weinberg, "There is now in my opinion no entirely satisfactory interpretation of quantum mechanics."[79]

The Copenhagen interpretation – due largely to Niels Bohr and Werner Heisenberg – remains most widely accepted some 75 years after its enunciation. According to this interpretation, 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". It also states that any well-defined application of the quantum mechanical formalism must always make reference to the experimental arrangement, due to the conjugate nature of evidence obtained under different experimental situations.

Albert Einstein, himself one of the founders of quantum theory, did not accept some of the more philosophical or metaphysical interpretations of quantum mechanics, such as rejection of determinism and of causality. He famously said about this, "God does not play with dice".[80] He rejected the concept that the state of a physical system depends on the experimental arrangement for its measurement. He held that a state of nature occurs in its own right, regardless of whether or how it might be observed. That view is supported by the currently accepted definition of a quantum state, which does not depend on the configuration space for its representation, that is to say, manner of observation. Einstein also believed that underlying quantum mechanics must be a theory that thoroughly and directly expresses the rule against action at a distance; in other words, he insisted on the principle of locality. He considered, but rejected on theoretical grounds, a particular proposal for hidden variables to obviate the indeterminism or acausality of quantum mechanical measurement. He believed that quantum mechanics was a currently valid but not a permanently definitive theory for quantum phenomena. He thought its future replacement would require profound conceptual advances, and would not come quickly or easily. The Bohr-Einstein debates provide a vibrant critique of the Copenhagen interpretation from an epistemological point of view. In arguing for his views, he produced a series of objections, of which the most famous has become known as the Einstein–Podolsky–Rosen paradox.

John Bell showed that this EPR paradox led to experimentally testable differences between quantum mechanics and theories that rely on local hidden variables. Experiments confirmed the accuracy of quantum mechanics, thereby showing that quantum mechanics cannot be improved upon by addition of local hidden variables.[81] Alain Aspect's experiments in 1982 and many later experiments definitively verified quantum entanglement. Entanglement, as demonstrated in Bell-type experiments, does not violate causality, since it does not involve transfer of information. By the early 1980s, experiments had shown that such inequalities were indeed violated in practice – so that there were in fact correlations of the kind suggested by quantum mechanics. At first these just seemed like isolated esoteric effects, but by the mid-1990s, they were being codified in the field of quantum information theory, and led to constructions with names like quantum cryptography and quantum teleportation.[82] Quantum cryptography is proposed for use in high-security applications in banking and government.

The Everett many-worlds interpretation, formulated in 1956, holds that all the possibilities described by quantum theory simultaneously occur in a multiverse composed of mostly independent parallel universes.[83] This is not accomplished by introducing a "new axiom" to quantum mechanics, but by removing the axiom of the collapse of the wave packet. All possible consistent states of the measured system and the measuring apparatus (including the observer) are present in a real physical – not just formally mathematical, as in other interpretations – quantum superposition. Such a superposition of consistent state combinations of different systems is called an entangled state. While the multiverse is deterministic, we perceive non-deterministic behavior governed by probabilities, because we can only observe the universe (i.e., the consistent state contribution to the aforementioned superposition) that we, as observers, inhabit. Everett's interpretation is perfectly consistent with John Bell's experiments and makes them intuitively understandable. However, according to the theory of quantum decoherence, these "parallel universes" will never be accessible to us. The 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 at the speed of light 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 originally measured. Not only is this completely impractical, but even if one could theoretically do this, it would have to destroy any evidence that the original measurement took place (including the physicist's memory).

In light of the Bell tests, Cramer in 1986 formulated his transactional interpretation[84] which is unique in providing a physical explanation for the Born rule.[85] Relational quantum mechanics appeared in the late 1990s as the modern derivative of the Copenhagen interpretation.
Applications
A working mechanism of a resonant tunneling diode device, based on the phenomenon of quantum tunneling through potential barriers. (Left: band diagram; Center: transmission coefficient; Right: current-voltage characteristics) As shown in the band diagram(left), although there are two barriers, electrons still tunnel through via the confined states between two barriers(center), conducting current.

Main article: Quantum physics

Quantum mechanics has had enormous[18] success in explaining many of the features of our universe, with regards to small-scale and discrete quantities and interactions which cannot be explained by classical methods. Quantum mechanics is often the only theory that can reveal the individual behaviors of the subatomic particles that make up all forms of matter (electrons, protons, neutrons, photons, and others). Quantum mechanics has strongly influenced string theories, candidates for a Theory of Everything (see reductionism).

In many aspects modern technology operates at a scale where quantum effects are significant. Important applications of quantum theory include quantum chemistry, quantum optics, quantum computing, superconducting magnets, light-emitting diodes, the optical amplifier and the laser, the transistor and semiconductors such as the microprocessor, medical and research imaging such as magnetic resonance imaging and electron microscopy.[86] Explanations for many biological and physical phenomena are rooted in the nature of the chemical bond, most notably the macro-molecule DNA.[87]
Examples
Free particle

For example, consider a free particle. In quantum mechanics, a free matter is described by a wave function. The particle properties of the matter become apparent when we measure its position and velocity. The wave properties of the matter become apparent when we measure its wave properties like interference. The wave–particle duality feature is incorporated in the relations of coordinates and operators in the formulation of quantum mechanics. Since the matter is free (not subject to any interactions), its quantum state can be represented as a wave of arbitrary shape and extending over space as a wave function. The position and momentum of the particle are observables. The Uncertainty Principle states that both the position and the momentum cannot simultaneously be measured with complete precision. However, one can measure the position (alone) of a moving free particle, creating an eigenstate of position with a wave function that is very large (a Dirac delta) at a particular position x, and zero everywhere else. If one performs a position measurement on such a wave function, the resultant x will be obtained with 100% probability (i.e., with full certainty, or complete precision). This is called an eigenstate of position – or, stated in mathematical terms, a generalized position eigenstate (eigendistribution). If the particle is in an eigenstate of position, then its momentum is completely unknown. On the other hand, if the particle is in an eigenstate of momentum, then its position is completely unknown.[88] In an eigenstate of momentum having a plane wave form, 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.[89]
Particle in a box
1-dimensional potential energy box (or infinite potential well)
Main article: 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[90]

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

With the differential operator defined by

\( {\hat {p}}_{x}=-i\hbar {\frac {d}{dx}} \)

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

\( {\frac {1}{2m}}{\hat {p}}_{x}^{2}=E, \)

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

\( \psi (x)=Ae^{ikx}+Be^{-ikx}\qquad \qquad E={\frac {\hbar ^{2}k^{2}}{2m}} \)

or, from Euler's formula,

\( {\displaystyle \psi (x)=C\sin(kx)+D\cos(kx).\!} \)

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

\( {\displaystyle \psi (0)=0=C\sin(0)+D\cos(0)=D} \)

and D=0. At x=L,

\( {\displaystyle \psi (L)=0=C\sin(kL),} \)

in which C cannot be zero as this would conflict with the Born interpretation. Therefore, since \( {\displaystyle \sin(kL)=0} \) , \( {\displaystyle kL} \) must be an integer multiple of \( \pi , \)

\( k={\frac {n\pi }{L}}\qquad \qquad n=1,2,3,\ldots .

The quantization of energy levels follows from this constraint on k, since

\( E={\frac {\hbar ^{2}\pi ^{2}n^{2}}{2mL^{2}}}={\frac {n^{2}h^{2}}{8mL^{2}}}. \)

The ground state energy of the particles is \( E_{1} \) for \( {\displaystyle n=1.} \)

The energy of the particle in the nth state is \( {\displaystyle E_{n}=n^{2}E_{1},\;n=2,3,4,\dots } \)

Particle in a box with boundary condition \( {\displaystyle V(x)=0,\;-a/2<x<+a/2} \)
A particle in a box with a little change in the boundary condition.

In this condition the general solution will be same, there will little change to the final result, since the boundary conditions are changed only slightly:

\( {\displaystyle \psi (x)=C\sin(kx)+D\cos(kx).} \)

At x=0, the wave function is not actually zero at all values of n.

Clearly, from the wave function variation graph we have, At \( {\displaystyle n=1,3,5,\dots ,} \) the wave function follows a cosine curve with x=0 as the origin.

At \( {\displaystyle n=2,4,6,\dots ,} \) the wave function follows a sine curve with x=0 as the origin.
Variation of wave function with x and n.
Wave Function Variation with x and n.

From this observation we can conclude that the wave function is alternatively sine and cosine. So in this case the resultant wave equation is

\( {\displaystyle \psi _{n}(x)={\begin{cases}A\cos(k_{n}x),&n=1,3,5,\dots \\B\sin(k_{n}x),&n=2,4,6,\dots \end{cases}}} \)
Finite potential well
Main article: Finite potential well

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.
Rectangular potential barrier
Main article: Rectangular potential barrier

This is a model for the quantum tunneling effect which plays an important role in the performance of modern technologies such as flash memory and scanning tunneling microscopy. Quantum tunneling is central to physical phenomena involved in superlattices.
Harmonic oscillator
Main article: Quantum harmonic oscillator
Some trajectories of a harmonic oscillator (i.e. a ball attached to a spring) in classical mechanics (A-B) and quantum mechanics (C-H). In quantum mechanics, the position of the ball is represented by a wave (called the wave function), with the real part shown in blue and the imaginary part shown in red. Some of the trajectories (such as C, D, E, and F) are standing waves (or "stationary states"). Each standing-wave frequency is proportional to a possible energy level of the oscillator. This "energy quantization" does not occur in classical physics, where the oscillator can have any energy.

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

\( {\displaystyle V(x)={\frac {1}{2}}m\omega ^{2}x^{2}.} \)

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 eigenstates are given by

\( {\displaystyle \psi _{n}(x)={\sqrt {\frac {1}{2^{n}\,n!}}}\cdot \left({\frac {m\omega }{\pi \hbar }}\right)^{1/4}\cdot e^{-{\frac {m\omega x^{2}}{2\hbar }}}\cdot H_{n}\left({\sqrt {\frac {m\omega }{\hbar }}}x\right),\qquad } \)

\( {\displaystyle n=0,1,2,\ldots .} \)

where Hn are the Hermite polynomials

\( {\displaystyle H_{n}(x)=(-1)^{n}e^{x^{2}}{\frac {d^{n}}{dx^{n}}}\left(e^{-x^{2}}\right),} \)

and the corresponding energy levels are

\( {\displaystyle E_{n}=\hbar \omega \left(n+{1 \over 2}\right).} \)

This is another example illustrating the quantification of energy for bound states.
Step potential
Main article: Solution of Schrödinger equation for a step potential
Scattering at a finite potential step of height V0, shown in green. The amplitudes and direction of left- and right-moving waves are indicated. Yellow is the incident wave, blue are reflected and transmitted waves, red does not occur. E > V0 for this figure.

The potential in this case is given by:

\( V(x)={\begin{cases}0,&x<0,\\V_{0},&x\geq 0.\end{cases}} \)

The solutions are superpositions of left- and right-moving waves:

\( {\displaystyle \psi _{1}(x)={\frac {1}{\sqrt {k_{1}}}}\left(A_{\rightarrow }e^{ik_{1}x}+A_{\leftarrow }e^{-ik_{1}x}\right)\qquad x<0} \)

and

\( {\displaystyle \psi _{2}(x)={\frac {1}{\sqrt {k_{2}}}}\left(B_{\rightarrow }e^{ik_{2}x}+B_{\leftarrow }e^{-ik_{2}x}\right)\qquad x>0}, \)

with coefficients A and B determined from the boundary conditions and by imposing a continuous derivative on the solution, and where the wave vectors are related to the energy via

\( k_{1}={\sqrt {2mE/\hbar ^{2}}} \)

and

\( k_{2}={\sqrt {2m(E-V_{0})/\hbar ^{2}}}.

Each term of the solution can be interpreted as an incident, reflected, or transmitted component of the wave, allowing the calculation of transmission and reflection coefficients. Notably, in contrast to classical mechanics, incident particles with energies greater than the potential step are partially reflected.
See also

Angular momentum diagrams (quantum mechanics)
Einstein's thought experiments
Hamiltonian (quantum mechanics)
Two-state quantum system
Fractional quantum mechanics
List of quantum-mechanical systems with analytical solutions
List of textbooks on classical and quantum mechanics
Macroscopic quantum phenomena
Phase space formulation
Quantum dynamics
Regularization (physics)
Spherical basis

Notes

Born, M. (1926). "Zur Quantenmechanik der Stoßvorgänge". Zeitschrift für Physik. 37 (12): 863–867. Bibcode:1926ZPhy...37..863B. doi:10.1007/BF01397477. S2CID 119896026.
Feynman, Richard; Leighton, Robert; Sands, Matthew (1964). The Feynman Lectures on Physics, Vol. 3. California Institute of Technology. p. 1.1. ISBN 978-0201500646. Archived from the original on 2018-11-26. Retrieved 2017-01-03.
Jaeger, Gregg (September 2014). "What in the (quantum) world is macroscopic?". American Journal of Physics. 82 (9): 896–905. Bibcode:2014AmJPh..82..896J. doi:10.1119/1.4878358.
Section 3.2 of Ballentine, Leslie E. (1970), "The Statistical Interpretation of Quantum Mechanics", Reviews of Modern Physics, 42 (4): 358–381, Bibcode:1970RvMP...42..358B, doi:10.1103/RevModPhys.42.358. This fact is experimentally well-known for example in quantum optics (see e.g. chap. 2 and Fig. 2.1 Leonhardt, Ulf (1997), Measuring the Quantum State of Light, Cambridge: Cambridge University Press, ISBN 0-521-49730-2
Max Born & Emil Wolf, Principles of Optics, 1999, Cambridge University Press
"Thomas Young's experiment". www.cavendishscience.org. Retrieved 2017-07-23.
Mehra, J.; Rechenberg, H. (1982). The historical development of quantum theory. New York: Springer-Verlag. ISBN 978-0387906423.
Kragh, Helge (2002). Quantum Generations: A History of Physics in the Twentieth Century. Princeton University Press. ISBN 978-0-691-09552-3. Extract of p. 58
Ben-Menahem, Ari (2009). Historical Encyclopedia of Natural and Mathematical Sciences, Volume 1. Springer. ISBN 978-3540688310. Extract of p, 3678
Stachel, John (2009) “Bohr and the Photon” Quantum Reality, Relativistic Causality and the Closing of the Epistemically Circle. Dordrecht, Springer p. 79.
E Arunan (2010). "Peter Debye" (PDF). Resonance. 15 (12): 1056–1059. doi:10.1007/s12045-010-0117-2. S2CID 195299361.
Kuhn, T. S. (1978). Black-body theory and the quantum discontinuity 1894–1912. Oxford: Clarendon Press. ISBN 978-0195023831.
Kragh, Helge (1 December 2000), Max Planck: the reluctant revolutionary, PhysicsWorld.com
Einstein, A. (1905). "Über einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt" [On a heuristic point of view concerning the production and transformation of light]. Annalen der Physik. 17 (6): 132–148. Bibcode:1905AnP...322..132E. doi:10.1002/andp.19053220607. Reprinted in The collected papers of Albert Einstein, John Stachel, editor, Princeton University Press, 1989, Vol. 2, pp. 149–166, in German; see also Einstein's early work on the quantum hypothesis, ibid. pp. 134–148.
EINSTEIN, A. (1967), "On the Quantum Theory of Radiation", The Old Quantum Theory, Elsevier, pp. 167–183, doi:10.1016/b978-0-08-012102-4.50018-8, ISBN 9780080121024
Wolfram, Stephen (2002). A New Kind of Science. Wolfram Media, Inc. p. 1056. ISBN 978-1-57955-008-0.
Pais, Abraham (1997). A Tale of Two Continents: A Physicist's Life in a Turbulent World. Princeton, New Jersey: Princeton University Press. ISBN 0-691-01243-1.
See, for example, the Feynman Lectures on Physics for some of the technological applications which use quantum mechanics, e.g., transistors (vol III, pp. 14–11 ff), integrated circuits, which are follow-on technology in solid-state physics (vol II, pp. 8–6), and lasers (vol III, pp. 9–13).
van Hove, Leon (1958). "Von Neumann's contributions to quantum mechanics" (PDF). Bulletin of the American Mathematical Society. 64 (3): Part 2:95–99. doi:10.1090/s0002-9904-1958-10206-2.
A New Kind of Science Note (a) for Quantum phenomena
Feynman, Richard. "The Feynman Lectures on Physics III 21-4". California Institute of Technology. Retrieved 2015-11-24. ""...it was long believed that the wave function of the Schrödinger equation would never have a macroscopic representation analogous to the macroscopic representation of the amplitude for photons. On the other hand, it is now realized that the phenomena of superconductivity presents us with just this situation."
Richard Packard (2006) "Berkeley Experiments on Superfluid Macroscopic Quantum Effects" Archived November 25, 2015, at the Wayback Machine accessdate=2015-11-24
"Quantum – Definition and More from the Free Merriam-Webster Dictionary". Merriam-webster.com. Retrieved 2012-08-18.
Thall, Edwin. "Thall's History of Quantum Mechanics". Florida Community College at Jacksonville. Archived from the original on October 7, 2009. Retrieved May 23, 2009.
"ysfine.com". Retrieved 11 September 2015.
"Quantum Mechanics". geocities.com. 2009-10-26. Archived from the original on 2009-10-26. Retrieved 2016-06-13.
P.A.M. Dirac, The Principles of Quantum Mechanics, Clarendon Press, Oxford, 1930.
D. Hilbert Lectures on Quantum Theory, 1915–1927
J. von Neumann, Mathematische Grundlagen der Quantenmechanik, Springer, Berlin, 1932 (English translation: Mathematical Foundations of Quantum Mechanics, Princeton University Press, 1955).
H.Weyl "The Theory of Groups and Quantum Mechanics", 1931 (original title: "Gruppentheorie und Quantenmechanik").
Dirac, P.A.M. (1958). The Principles of Quantum Mechanics, 4th edition, Oxford University Press, Oxford, p. ix: "For this reason I have chosen the symbolic method, introducing the representatives later merely as an aid to practical calculation."
Greiner, Walter; Müller, Berndt (1994). Quantum Mechanics Symmetries, Second edition. Springer-Verlag. p. 52. ISBN 978-3-540-58080-5., Chapter 1, p. 52
"Heisenberg – Quantum Mechanics, 1925–1927: The Uncertainty Relations". Aip.org. Retrieved 2012-08-18.
Greenstein, George; Zajonc, Arthur (2006). The Quantum Challenge: Modern Research on the Foundations of Quantum Mechanics, Second edition. Jones and Bartlett Publishers, Inc. p. 215. ISBN 978-0-7637-2470-2., Chapter 8, p. 215
Lodha, Suresh K.; Faaland, Nikolai M.; et al. (2002). "Visualization of Uncertain Particle Movement (Proceeding Computer Graphics and Imaging)" (PDF). Actapress.com. Archived (PDF) from the original on 2018-08-01. Retrieved 2018-08-01.
Hirshleifer, Jack (2001). The Dark Side of the Force: Economic Foundations of Conflict Theory. Cambridge University Press. p. 265. ISBN 978-0-521-80412-7., Chapter, p.
"dict.cc dictionary :: eigen :: German-English translation". dict.cc. Retrieved 11 September 2015.
"Topics: Wave-Function Collapse". Phy.olemiss.edu. 2012-07-27. Archived from the original on 2017-02-28. Retrieved 2012-08-18.
"Collapse of the wave-function". Farside.ph.utexas.edu. Retrieved 2012-08-18.
Weinberg, Steven (2010). Dreams Of A Final Theory: The Search for The Fundamental Laws of Nature. Random House. p. 82. ISBN 978-1-4070-6396-6. Extract of page 82
Michael Trott. "Time-Evolution of a Wavepacket in a Square Well – Wolfram Demonstrations Project". Demonstrations.wolfram.com. Retrieved 2010-10-15.
Michael Trott. "Time Evolution of a Wavepacket In a Square Well". Demonstrations.wolfram.com. Retrieved 2010-10-15.
Mathews, Piravonu Mathews; Venkatesan, K. (1976). A Textbook of Quantum Mechanics. Tata McGraw-Hill. p. 36. ISBN 978-0-07-096510-2., Chapter 2, p. 36
"Wave Functions and the Schrödinger Equation" (PDF). Retrieved 2010-10-15.[dead link]
Rechenberg, Helmut (1987). "Erwin Schrödinger and the creation of wave mechanics" (PDF). Acta Physica Polonica B. 19 (8): 683–695. Retrieved 13 June 2016.
Nancy Thorndike Greenspan, "The End of the Certain World: The Life and Science of Max Born" (Basic Books, 2005), pp. 124–128, 285–826.
"Archived copy" (PDF). Archived from the original (PDF) on 2011-07-19. Retrieved 2009-06-04.
"The Nobel Prize in Physics 1979". Nobel Foundation. Retrieved 2010-02-16.
Carl M. Bender; Daniel W. Hook; Karta Kooner (2009-12-31). "Complex Elliptic Pendulum".arXiv:1001.0131 [hep-th].
See, for example, Precision tests of QED. The relativistic refinement of quantum mechanics known as quantum electrodynamics (QED) has been shown to agree with experiment to within 1 part in 108 for some atomic properties.
Tipler, Paul; Llewellyn, Ralph (2008). Modern Physics (5 ed.). W.H. Freeman and Company. pp. 160–161. ISBN 978-0-7167-7550-8.
Ehrenfest, P. (1927). "Bemerkung über die angenäherte Gültigkeit der klassischen Mechanik innerhalb der Quantenmechanik". Zeitschrift für Physik. 45 (7–8): 455–457. Bibcode:1927ZPhy...45..455E. doi:10.1007/BF01329203. S2CID 123011242.
Smith, Henrik (1991). Introduction to Quantum Mechanics. World Scientific Pub Co Inc. pp. 108–109. ISBN 978-9810204754.
Einstein, A.; Podolsky, B.; Rosen, N. (1935). "Can quantum-mechanical description of physical reality be considered complete?". Phys. Rev. 47 (10): 777. Bibcode:1935PhRv...47..777E. doi:10.1103/physrev.47.777.
N.P. Landsman (June 13, 2005). "Between classical and quantum" (PDF). Retrieved 2012-08-19. Handbook of the Philosophy of Science Vol. 2: Philosophy of Physics (eds. John Earman & Jeremy Butterfield).
(see macroscopic quantum phenomena, Bose–Einstein condensate, and Quantum machine)
"Atomic Properties". Academic.brooklyn.cuny.edu. Retrieved 2012-08-18.
http://assets.cambridge.org/97805218/29526/excerpt/9780521829526_excerpt.pdf
Born, M., Heisenberg, W., Jordan, P. (1926). Z. Phys. 35: 557–615. Translated as 'On quantum mechanics II', pp. 321–385 in Van der Waerden, B.L. (1967), Sources of Quantum Mechanics, North-Holland, Amsterdam, "The basic difference between the theory proposed here and that used hitherto ... lies in the characteristic kinematics ...", p. 385.
Dirac, P.A.M. (1930/1958). The Principles of Quantum Mechanics, fourth edition, Oxford University Press, Oxford UK, p. 5: "A question about what will happen to a particular photon under certain conditions is not really very precise. To make it precise one must imagine some experiment performed having a bearing on the question, and enquire what will be the result of the experiment. Only questions about the results of experiments have a real significance and it is only such questions that theoretical physics has to consider."
Bohr, N. (1939). The Causality Problem in Atomic Physics, in New Theories in Physics, Conference organized in collaboration with the International Union of Physics and the Polish Intellectual Co-operation Committee, Warsaw, May 30th – June 3rd 1938, International Institute of Intellectual Co-operation, Paris, 1939, pp. 11–30, reprinted in Niels Bohr, Collected Works, volume 7 (1933–1958) edited by J. Kalckar, Elsevier, Amsterdam, ISBN 0-444-89892-1, pp. 303–322. "The essential lesson of the analysis of measurements in quantum theory is thus the emphasis on the necessity, in the account of the phenomena, of taking the whole experimental arrangement into consideration, in complete conformity with the fact that all unambiguous interpretation of the quantum mechanical formalism involves the fixation of the external conditions, defining the initial state of the atomic system and the character of the possible predictions as regards subsequent observable properties of that system. Any measurement in quantum theory can in fact only refer either to a fixation of the initial state or to the test of such predictions, and it is first the combination of both kinds which constitutes a well-defined phenomenon."
Bohr, N. (1948). On the notions of complementarity and causality, Dialectica 2: 312–319. "As a more appropriate way of expression, one may advocate limitation of the use of the word phenomenon to refer to observations obtained under specified circumstances, including an account of the whole experiment."
Ludwig, G. (1987). An Axiomatic Basis for Quantum Mechanics, volume 2, Quantum Mechanics and Macrosystems, translated by K. Just, Springer, Berlin, ISBN 978-3-642-71899-1, Chapter XIII, Special Structures in Preparation and Registration Devices, §1, Measurement chains, p. 132.
Heisenberg, W. (1927). Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik, Z. Phys. 43: 172–198. Translation as 'The actual content of quantum theoretical kinematics and mechanics' here [1], "But in the rigorous formulation of the law of causality, – "If we know the present precisely, we can calculate the future" – it is not the conclusion that is faulty, but the premise."
Green, H.S. (1965). Matrix Mechanics, with a foreword by Max Born, P. Noordhoff Ltd, Groningen. "It is not possible, therefore, to provide 'initial conditions' for the prediction of the behaviour of atomic systems, in the way contemplated by classical physics. This is accepted by quantum theory, not merely as an experimental difficulty, but as a fundamental law of nature", p. 32.
Rosenfeld, L. (1957). Misunderstandings about the foundations of quantum theory, pp. 41–45 in Observation and Interpretation, edited by S. Körner, Butterworths, London. "A phenomenon is therefore a process (endowed with the characteristic quantal wholeness) involving a definite type of interaction between the system and the apparatus."
Dirac, P.A.M. (1973). Development of the physicist's conception of nature, pp. 1–55 in The Physicist's Conception of Nature, edited by J. Mehra, D. Reidel, Dordrecht, ISBN 90-277-0345-0, p. 5: "That led Heisenberg to his really masterful step forward, resulting in the new quantum mechanics. His idea was to build up a theory entirely in terms of quantities referring to two states."
Born, M. (1927). Physical aspects of quantum mechanics, Nature 119: 354–357, "These probabilities are thus dynamically determined. But what the system actually does is not determined ..."
Messiah, A. (1961). Quantum Mechanics, volume 1, translated by G.M. Temmer from the French Mécanique Quantique, North-Holland, Amsterdam, p. 157.
Bohr, N. (1928). "The Quantum postulate and the recent development of atomic theory". Nature. 121 (3050): 580–590. Bibcode:1928Natur.121..580B. doi:10.1038/121580a0.
Heisenberg, W. (1930). The Physical Principles of the Quantum Theory, translated by C. Eckart and F.C. Hoyt, University of Chicago Press.
Goldstein, H. (1950). Classical Mechanics, Addison-Wesley, ISBN 0-201-02510-8.
"There is as yet no logically consistent and complete relativistic quantum field theory.", p. 4. – V.B. Berestetskii, E.M. Lifshitz, L.P. Pitaevskii (1971). J.B. Sykes, J.S. Bell (translators). Relativistic Quantum Theory 4, part I. Course of Theoretical Physics (Landau and Lifshitz) ISBN 0-08-016025-5
"Stephen Hawking; Gödel and the end of physics". cam.ac.uk. Archived from the original on 21 May 2011. Retrieved 11 September 2015.
Hawking, Stephen; Penrose, Roger (2010). The Nature of Space and Time. ISBN 978-1400834747.
Tatsumi Aoyama; Masashi Hayakawa; Toichiro Kinoshita; Makiko Nio (2012). "Tenth-Order QED Contribution to the Electron g-2 and an Improved Value of the Fine Structure Constant". Physical Review Letters. 109 (11): 111807.arXiv:1205.5368. Bibcode:2012PhRvL.109k1807A. doi:10.1103/PhysRevLett.109.111807. PMID 23005618. S2CID 14712017.
Parker, B. (1993). Overcoming some of the problems. pp. 259–279.
The Character of Physical Law (1965) Ch. 6; also quoted in The New Quantum Universe (2003), by Tony Hey and Patrick Walters
Weinberg, Steven (2012). "Collapse of the state vector". Physical Review A. 85 (6): 062116.arXiv:1109.6462. Bibcode:2012PhRvA..85f2116W. doi:10.1103/PhysRevA.85.062116. S2CID 119273840.
Harrison, Edward (2000). Cosmology: The Science of the Universe. Cambridge University Press. p. 239. ISBN 978-0-521-66148-5.
"Action at a Distance in Quantum Mechanics (Stanford Encyclopedia of Philosophy)". Plato.stanford.edu. 2007-01-26. Retrieved 2012-08-18.
Wolfram, Stephen (2002). A New Kind of Science. Wolfram Media, Inc. p. 1058. ISBN 978-1-57955-008-0.
"Everett's Relative-State Formulation of Quantum Mechanics (Stanford Encyclopedia of Philosophy)". Plato.stanford.edu. Retrieved 2012-08-18.
Cramer, John G. (1986). "The transactional interpretation of quantum mechanics". Reviews of Modern Physics. 58 (3): 647–687. Bibcode:1986RvMP...58..647C. doi:10.1103/RevModPhys.58.647.
The Transactional Interpretation of quantum mechanics. R.E. Kastner. Cambridge University Press. 2013. ISBN 978-0-521-76415-5. p. 35.
Matson, John. "What Is Quantum Mechanics Good for?" . Scientific American. Retrieved 18 May 2016.
The Nobel laureates Watson and Crick cited Pauling, Linus (1939). The Nature of the Chemical Bond and the Structure of Molecules and Crystals. Cornell University Press. for chemical bond lengths, angles, and orientations.
Davies, P.C.W.; Betts, David S. (1984). Quantum Mechanics, Second edition. Chapman and Hall. ISBN 978-0-7487-4446-6., [https://books.google.com/books?id=XRyHCrGNstoC&pg=PA79 Chapter 6, p. 79
Baofu, Peter (2007). The Future of Complexity: Conceiving a Better Way to Understand Order and Chaos. Bibcode:2007fccb.book.....B. ISBN 9789812708991. Retrieved 2012-08-18.

Derivation of particle in a box, chemistry.tidalswan.com

N.B. on precision: If \( \delta x \) and\( \delta p \) are the precisions of position and momentum obtained in an individual measurement and \( \sigma _{x} \), \( {\displaystyle \sigma _{p}} \) their standard deviations in an ensemble of individual measurements on similarly prepared systems, then "There are, in principle, no restrictions on the precisions of individual measurements \( \delta x \) and \( \delta p \), but the standard deviations will always satisfy \( {\displaystyle \sigma _{x}\sigma _{p}\geq \hbar /2} \)".[4]

References

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

Chester, Marvin (1987) Primer of Quantum Mechanics. John Wiley. ISBN 0-486-42878-8
Cox, Brian; Forshaw, Jeff (2011). The Quantum Universe: Everything That Can Happen Does Happen. Allen Lane. ISBN 978-1-84614-432-5.
Richard Feynman, 1985. QED: The Strange Theory of Light and Matter, Princeton University Press. ISBN 0-691-08388-6. Four elementary lectures on quantum electrodynamics and quantum field theory, yet containing many insights for the expert.
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. ISBN 0-691-08131-X
Dirac, P.A.M. (1930). The Principles of Quantum Mechanics. ISBN 978-0-19-852011-5. The beginning chapters make up a very clear and comprehensible introduction.
Everett, Hugh (1957). "Relative State Formulation of Quantum Mechanics". Reviews of Modern Physics. 29 (3): 454–462. Bibcode:1957RvMP...29..454E. doi:10.1103/RevModPhys.29.454. S2CID 17178479.
Feynman, Richard P.; Leighton, Robert B.; Sands, Matthew (1965). The Feynman Lectures on Physics. 1–3. Addison-Wesley. ISBN 978-0-7382-0008-8.
Griffiths, David J. (2004). Introduction to Quantum Mechanics (2nd ed.). Prentice Hall. ISBN 978-0-13-111892-8. OCLC 40251748. 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.
L.D. Landau, E.M. Lifshitz (1977). Quantum Mechanics: Non-Relativistic Theory. Vol. 3 (3rd ed.). Pergamon Press. ISBN 978-0-08-020940-1. Online copy
Gunther Ludwig, 1968. Wave Mechanics. London: Pergamon Press. ISBN 0-08-203204-1
George Mackey (2004). The mathematical foundations of quantum mechanics. Dover Publications. ISBN 0-486-43517-2.
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
Omnès, Roland (1999). Understanding Quantum Mechanics. Princeton University Press. ISBN 978-0-691-00435-8. OCLC 39849482.
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. ISBN 0-19-530573-6
Transnational College of Lex (1996). What is Quantum Mechanics? A Physics Adventure. Language Research Foundation, Boston. ISBN 978-0-9643504-1-0. OCLC 34661512.
von Neumann, John (1955). Mathematical Foundations of Quantum Mechanics. Princeton University Press. ISBN 978-0-691-02893-4.
Hermann Weyl, 1950. The Theory of Groups and Quantum Mechanics, Dover Publications.
D. Greenberger, K. Hentschel, F. Weinert, eds., 2009. Compendium of quantum physics, Concepts, experiments, history and philosophy, Springer-Verlag, Berlin, Heidelberg.