The history ofquantum mechanics is a fundamental part of thehistory of modern physics. The major chapters of this history begin with the emergence of quantum ideas to explain individual phenomena—blackbody radiation, the photoelectric effect, solar emission spectra—an era called the Old or Older quantum theories.^[1]

Building on the technologydeveloped in classical mechanics, the invention of wave mechanics byErwin Schrödinger and expansion by many others triggers the "modern" era beginning around 1925.Paul Dirac's relativistic quantum theory work led him to explore quantum theories of radiation, culminating inquantum electrodynamics, the firstquantum field theory. The history of quantum mechanics continues in thehistory of quantum field theory. The history ofquantum chemistry, theoretical basis ofchemical structure,reactivity, andbonding, interlaces with the events discussed in this article.

The phrase "quantum mechanics" was coined (in German,Quantenmechanik) by the group of physicists includingMax Born,Werner Heisenberg, andWolfgang Pauli, at theUniversity of Göttingen in the early 1920s, and was first used in Born and P. Jordan's September 1925 paper"Zur Quantenmechanik".^[2][3][4]

The wordquantum comes from theLatin word for "how much" (as doesquantity). Something that isquantized, as the energy of Planck's harmonic oscillators, can only take specific values. For example, in most countries, money is effectively quantized, with thequantum of money being the lowest-value coin in circulation. Mechanics is the branch of science that deals with the action of forces on objects. So, quantum mechanics is the part of mechanics that deals with objects for which particular properties are quantized.

Thediscoveries of the 19th century, both the successes and failures, set the stage for the emergence of quantum mechanics.

Beginning in 1670 and progressing over three decades,Isaac Newton developed and championed hiscorpuscular theory, arguing that the perfectly straight lines of reflection demonstrated light's particle nature, as at that time no wave theory demonstrated travel in straight lines.^[1]: 19 He explained refraction by positing that particles of light accelerated laterally upon entering a denser medium.

Around the same time, Newton's contemporariesRobert Hooke andChristiaan Huygens, and laterAugustin-Jean Fresnel, mathematically refined the wave viewpoint, showing that if light traveled at different speeds in different media, refraction could be easily explained as the medium-dependent propagation of light waves. The resultingHuygens–Fresnel principle was extremely successful at reproducing light's behaviour and was consistent withThomas Young's discovery ofwave interference of light by hisdouble-slit experiment in 1801.^[5] The wave view did not immediately displace the ray and particle view, but began to dominate scientific thinking about light in the mid 19th century, since it could explain polarization phenomena that the alternatives could not.^[6]

James Clerk Maxwell found that he could apply his previously discoveredequations, along with a slight modification, to describe self-propagating waves of oscillating electric and magnetic fields. It quickly became apparent that visible light, ultraviolet light, and infrared light were all electromagnetic waves of differing frequency.^[1]: 272 This theory became a critical ingredient in the beginning of quantum mechanics.

During the early 19th century,chemical research byJohn Dalton andAmedeo Avogadro lent weight to theatomic theory of matter, an idea thatJames Clerk Maxwell,Ludwig Boltzmann and others built upon to establish thekinetic theory of gases. The successes of kinetic theory gave further credence to the idea that matter is composed of atoms, yet the theory also had shortcomings that would only be resolved by the development of quantum mechanics.^[7] The existence of atoms was not universally accepted among physicists or chemists;Ernst Mach, for example, was a staunch anti-atomist.^[8]

The earliest hints of problems in classical mechanics were raised in relation to the temperature dependence of the properties of gases.^[9]Ludwig Boltzmann suggested in 1877 that the energy levels of a physical system, such as amolecule, could be discrete (rather than continuous). Boltzmann's rationale for the presence of discrete energy levels in molecules such as those of iodine gas had its origins in hisstatistical thermodynamics andstatistical mechanics theories and was backed up bymathematical arguments, as would also be the case twenty years later with the firstquantum theory put forward byMax Planck.

In the final days of the 1800s,J. J. Thomson established thatelectrons carry a negative charge opposite but the same as that of a hydrogen ion while having a mass over one thousand times less. Many such electrons were known to be associated with every atom.^[1]: 365 By 1904 Thomson proposed the first atomic model with subatomic constituents, using circulating electrons in a background of positive charge, the so-calledplum pudding model.^[10] Thomson's concepts were supported by earlybeta particle scattering experiments but by 1911Hans Geiger and his studentErnest Marsden demonstratedbackscattering of alpha particles whichErnest Rutherford interpreted as compelling evidence that the positive charge was concentrated in a small volume we now call thenucleus.^[11]

Throughout the 1800s many studies investigated details in thespectrum of intensity versus frequency for light emitted by flames, by the Sun, or red-hot objects.^[1]: 367 TheRydberg formula effectively summarized the dark lines seen in the spectrum, but Rydberg provided no physical model to explain them. The spectrum emitted by red-hot objects could be explained at high or low wavelengths but the two theories differed.

Quantum mechanics developed in two distinct phases. The first phase, known as theold quantum theory, began around 1900 with radically new approaches to explanations physical phenomena not understood by classical mechanics of the 1800s.^[1]

Thermal radiation is electromagnetic radiation emitted from the surface of an object due to the object's internal energy. If an object is heated sufficiently, it starts to emit light at the red end of the visiblespectrum, as it becomesred hot. Heating it further causes the color to change from red to yellow, white, and blue, as it emits light at increasingly shorter wavelengths (higher frequencies).^{[citation needed]}

A perfect emitter is also a perfect absorber: when it is cold, such an object looks perfectly black, because it absorbs all the light that falls on it and emits none. Consequently, an ideal thermal emitter is known as ablack body, and the radiation it emits is calledblack-body radiation.

By the late 19th century, thermal radiation had been fairly well characterized experimentally. Several formulas that describe certain experimental measurements of thermal radiation had been developed.Wien’s displacement law gives the relation between temperature and the wavelength at which the radiation is strongest, while theStefan–Boltzmann law describes the total power emitted per unit area. The best theoretical explanation of the experimental results was theRayleigh–Jeans law, which, as shown in the figure, agrees with experimental results well at large wavelengths (or, equivalently, low frequencies), but strongly disagrees at short wavelengths (or high frequencies). In fact, at short wavelengths, classical physics predicted that energy will be emitted by a hot body at an infinite rate. This result, which is clearly wrong, is known as theultraviolet catastrophe. Physicists searched for a single theory that explained all the experimental results.

The first model that was able to explain the full spectrum of thermal radiation^{[citation needed]} was put forward byMax Planck in 1900.^[12] He proposed a mathematical model in which the thermal radiation was in equilibrium with a set ofharmonic oscillators. To reproduce the experimental results, he had to assume that each oscillator emitted an integer number of units of energy at its single characteristic frequency, rather than being able to emit any arbitrary amount of energy. In other words, the energy emitted by an oscillator wasquantized. According to Planck, the quantum of energyE of a light quantum is proportional to its frequencyfrequencyν. As an equation is it written:$${\displaystyle E=hv.}$$This is now known as thePlanck relation and the proportionality constant,h, as thePlanck constant.^{[citation needed]}

Planck's law was the first quantum theory in physics, and Planck won the 1918Nobel Prize in Physics "in recognition of the services he rendered to the advancement of Physics by his discovery of energy quanta".^[13] At the time, however, Planck's view was that quantization was purely aheuristic mathematical construct, rather than (as is now believed) a fundamental change in our understanding of the world.^[14]

In 1887,Heinrich Hertz observed that when light with sufficient frequency hits a metallic surface, the surface emitscathode rays.^[1]: I:362 Ten years later, J. J. Thomson showed that the many reports of cathode rays were actually "corpuscles" and they quickly came to be calledelectrons. In 1902,Philipp Lenard discovered that the maximum possible energy of an ejected electron is unrelated to theintensity of the monochromatic light.^[15] This observation is at odds with classical electromagnetism, which predicts that the electron's energy should be proportional to the intensity of the incident radiation.^[16]: 24

In 1905,Albert Einstein suggested that even though continuous models of light worked extremely well for time-averaged optical phenomena, for instantaneous transitions the energy in light may occur a finite number of energy quanta.

In the introduction section of his March 1905 paper "On a Heuristic Viewpoint Concerning the Emission and Transformation of Light", Einstein states:

According to the assumption to be contemplated here, when a light ray is spreading from a point, the energy is not distributed continuously over ever-increasing spaces, but consists of a finite number of "energy quanta" that are localized in points in space, move without dividing, and can be absorbed or generated only as a whole.^[17]

This has been called the most "revolutionary" sentence written by a twentieth century physicist, meaning that it proposed an idea which altered mainstream thinking.^[18]: 143The energy of a single quantum of light of frequency${\displaystyle f}$ is given by the frequency multiplied by the Planck constant${\displaystyle h}$:

${\displaystyle E=hf}$

Einstein assumed a light quanta transfers all of its energy to a single electron imparting at most an energyhf to the electron. Therefore, only the light frequency determines the maximum energy that can be imparted to the electron; the intensity of the photoemission is proportional to the light beam intensity.^[17]

Einstein argued that it takes a certain amount of energy, called thework function and denoted byφ, to remove an electron from the metal.^[19] This amount of energy is different for each metal. If the energy of the light quanta is less than the work function, then it does not carry sufficient energy to remove the electron from the metal. The threshold frequency,f₀, is the frequency of a light quanta whose energy is equal to the work function:

${\displaystyle \phi =h{f}_0.}$

Iff is greater thanf₀, the energyhf is enough to remove an electron. The ejected electron has akinetic energy,E_k, which is, at most, equal to the light energy minus the energy needed to dislodge the electron from the metal:

${\displaystyle E_{\text{k}}=hf-\phi =h(f-f_0).}$

Einstein's description of light as being composed of energy quanta extended Planck's notion of quantized energy, which is that a single quantum of a given frequency,f, delivers an invariant amount of energy,hf. Einstein was awarded the 1921 Nobel Prize in Physics "for his services to Theoretical Physics, and especially for his discovery of the law of the photoelectric effect".^[20]

In nature, single quanta are rarely encountered. The Sun and emission sources available in the 19th century emit a vast amount of energy every second. ThePlanck constant,h, is so tiny that the amount of energy in each quantum,hf is very very small. Light we see includes many trillions of such quanta.Arthur Compton's demonstration of thescattering of light by electrons scattering convinced physicists of the reality of photons.^[21] Compton won the 1927 Nobel Prize in Physics for his discovery.^[22] The term "photon" was introduced in 1926 byGilbert N. Lewis.^[23]

Ernest Rutherford's discovery of the atomic nucleus in 1911 did not immediately cause atomic models to be revised. Mechanical models with circulating electrons had been proposed for many years but they were known to be unstable.^[11]

A second, related puzzle was theemission spectrum of atoms. When a gas is heated, it gives off light only at discrete frequencies. For example, the visible light given off byhydrogen consists of four different colors, as shown in the picture below. The intensity of the light at different frequencies is also different. By contrast, white light consists of a continuous emission across the whole range of visible frequencies. By the end of the nineteenth century, a simple rule known asBalmer's formula showed how the frequencies of the different lines related to each other, though without explaining why this was, or making any prediction about the intensities. The formula also predicted some additional spectral lines in ultraviolet and infrared light that had not been observed at the time. These lines were later observed experimentally, raising confidence in the value of the formula.

Emission spectrum ofhydrogen. When excited, hydrogen gas gives off light in four distinct colors (spectral lines) in the visible spectrum, as well as a number of lines in the infrared and ultraviolet.

In 1913, H. M. Hansen askedNiels Bohr about Balmer's formula. Bohr recalled that "As soon as I saw Balmer's formula, the whole thing was immediately clear to me."^[25] In "On the Constitution of Atoms and Molecules", he proposeda new model of the atom that included quantized electron orbits. In the Bohr model, the hydrogen atom is pictured as a heavy, positively charged nucleus orbited by a light, negatively charged electron. The electron can only exist in certain, discretely separated orbits, labeled by theirangular momentum, which is restricted to be an integer multiple of thereduced Planck constant.^[26] The electron moves to a higher or lower orbital by absorbing or emitting a photon of corresponding frequency.

These orbits had fixed or quantized values of angular momentum, a concept proposed byJohn William Nicholson in a nuclear atom model and adopted by Bohr in his model.^[27]: 169 When an atom emitted (or absorbed) energy, the electron did not move in a continuous trajectory from one orbit around the nucleus to another, as might be expected classically. Instead, the electron would jump instantaneously from one orbit to another, giving off the emitted light in the form of a photon.^[28] The possible energies of photons given off by each element were determined by the differences in energy between the orbits, and so the emission spectrum for each element would contain a number of lines.^[29]

The model's key success lay in explaining the Rydberg formula for the spectralemission lines of atomic hydrogen by using the transitions of electrons between orbits.^[30]: 276 While the Rydberg formula had been known experimentally, it did not gain a theoretical underpinning until the Bohr model was introduced. Not only did the Bohr model explain the reasons for the structure of the Rydberg formula, it also provided a justification for the fundamental physical constants that make up the formula's empirical results.

Starting from only one simple assumption about the rule that the orbits must obey, the Bohr model was able to relate the observed spectral lines in the emission spectrum of hydrogen to previously known constants. In Bohr's model, the electron was not allowed to emit energy continuously and crash into the nucleus: once it was in the closest permitted orbit, it was stable forever. Bohr's model did not explain why the orbits should be quantized in that way, nor was it able to make accurate predictions for atoms with more than one electron, or to explain why some spectral lines are brighter than others.

Some fundamental assumptions of the Bohr model were soon proven wrong—but the key result that the discrete lines in emission spectra are due to some property of the electrons in atoms being quantized is correct. The way that the electrons actually behave is strikingly different from Bohr's atom, and from what we see in the world of our everyday experience; this modern quantum mechanical model of the atom is discussedbelow.

Moreover, the application of Planck's quantum theory to the electron allowedȘtefan Procopiu in 1911–1913, and subsequently Niels Bohr in 1913, to calculate themagnetic moment of theelectron, which was later called the "magneton"; similar quantum computations, but with numerically quite different values, were subsequently made possible for both the magnetic moments of theproton and theneutron that are threeorders of magnitude smaller than that of the electron.

These theories, though successful, were strictlyphenomenological: during this time, there was no rigorous justification forquantization, aside, perhaps, fromHenri Poincaré's discussion of Planck's theory in his 1912 paperSur la théorie des quanta.^[31][32] They are collectively known as theold quantum theory. Bohr was awarded the 1922 Nobel Prize in Physics "for his services in the investigation of the structure of atoms and of the radiation emanating from them".^[33]

Quantization of the orbital angular momentum of the electron combined with the magnetic moment of the electron suggested that atoms with a magnetic moment should show quantized behavior in a magnetic field. In 1922,Otto Stern andWalther Gerlach set out to test this theory. They heated silver in a vacuum tube equipped with a series of narrow aligned slits, creating a molecular beam of silver atoms. They shot this beam through aninhomogeneousmagnetic field. Rather than a continuous pattern of silver atoms, they found two bunches.^[34]

Relative to its northern pole, pointing up, down, or somewhere in between, in classical mechanics, a magnet thrown through a magnetic field may be deflected a small or large distance upwards or downwards. The atoms that Stern and Gerlach shot through the magnetic field acted similarly. However, while the magnets could be deflected variable distances, the atoms would always be deflected a constant distance either up or down. This implied that the property of the atom that corresponds to the magnet's orientation must be quantized, taking one of two values (either up or down), as opposed to being chosen freely from any angle.

The choice of the orientation of the magnetic field used in the Stern–Gerlach experiment is arbitrary. In the animation shown here, the field is vertical and so the atoms are deflected either up or down. If the magnet is rotated a quarter turn, the atoms are deflected either left or right. Using a vertical field shows that the spin along the vertical axis is quantized, and using a horizontal field shows that the spin along the horizontal axis is quantized.

The results of the Stern-Gerlach experiment caused a sensation, most especially because leading scientists, including Einstein andPaul Ehrenfest argued that the silver atoms should have random orientations in the conditions of the experiment: quantization should not have been observable.^[34] At least five years would elapse before this mystery was resolved: quantization was observed but it was not due to orbital angular momentum. PerWolfgang Pauli'sexclusion principle, only two electrons can occupy the same quantum state at the same time, and their spins must be opposite. Pauli received the 1945 Nobel Prize in Physics for his "decisive contribution through his discovery of a new law of Nature, the exclusion principle or Pauli principle".^[35]

In 1925,Ralph Kronig proposed that electrons behave as if they self-rotate, or "spin", about an axis.^[36]: 56 Spin would generate a tiny magnetic moment that would split the energy levels responsible for spectral lines, in agreement with existing measurements. Two electrons in the same orbital would occupy distinctquantum states if they "spun" in opposite directions thus satisfying the exclusion principle. Unfortunately, the theory had two significant flaws: two values computed by Kronig were off by a factor of two. Kronig's senior colleagues discouraged his work and it was never published.

Ten months later, Dutch physicistsGeorge Uhlenbeck andSamuel Goudsmit atLeiden University published their theory of electron self rotation.^[37] The model, like Kronig's was essentially classical but resulted in a quantum prediction.

In 1924,Louis de Broglie published a breakthrough hypothesis:matter has wave properties. Building on Einstein's proposal that the photoelectric effect can be described using quantized energy transfers and by Einstein's separate proposal, from special relativity, that mass at rest is equivalent to energy via${\displaystyle E=m_0{c}^2}$, de Broglie proposed that matter in motion appears to have an associated wave with wavelength${\displaystyle \lambda =h/p}$ where${\displaystyle p}$ is the matter momentum from the motion.^[38][39] Requiring his wavelength to encircle an atom, he explained quantization of Bohr's orbits.^[1]: 217 Simultaneously this showed that the wave behavior of light was essentially a quantum effect.^[1]: 216

De Broglie expanded Bohr's model of the atom by showing that an electron in orbit around a nucleus could be thought of as having wave-like properties. In particular, an electron is observed only in situations that permit astanding wave around anucleus. An example of a standing wave is a violin string, which is fixed at both ends and can be made to vibrate. The waves created by a stringed instrument appear to oscillate in place, moving from crest to trough in an up-and-down motion. The wavelength of a standing wave is related to the length of the vibrating object and the boundary conditions. For example, because the violin string is fixed at both ends, it can carry standing waves of wavelengths$\frac{2l}{n}$, wherel is the length andn is a positive integer.

De Broglie suggested that the allowed electron orbits were those for which the circumference of the orbit would be an integer number of wavelengths. The electron's wavelength, therefore, determines that only Bohr orbits of certain distances from the nucleus are possible. In turn, at any distance from the nucleus smaller than a certain value, it would be impossible to establish an orbit. The minimum possible distance from the nucleus is called the Bohr radius.^[40] De Broglie's treatment of the Bohr atom was ultimately unsuccessful, but his hypothesis served as a starting point for Schrödinger's wave equation.

Matter behaving as a wave was first demonstrated experimentally: a beam of electrons can exhibitdiffraction, just like a beam of light or a water wave. Three years after de Broglie published his hypothesis two different groups demonstrated electron diffraction. At theUniversity of Aberdeen,George Paget Thomson and Alexander Reid passed a beam of electrons through a thin celluloid film, then later metal films, and observed the predicted interference patterns. (Alexander Reid, who was Thomson's graduate student, performed the first experiments but he died soon after in a motorcycle accident^[41] and is rarely mentioned.) AtBell Labs,Clinton Joseph Davisson andLester Halbert Germer reflected an electron beam from a nickel sample in their experiment, observing well-defined beams predicted by wave models returning from the crystal.^{[1]: II:218} De Broglie was awarded the Nobel Prize in Physics in 1929 for his hypothesis;^[42] Thomson and Davisson shared the Nobel Prize for Physics in 1937 for their experimental work.^[43]

The end of the first era of quantum mechanics was triggered by de Broglie's publication of his hypothesis ofmatter waves^[1]: 268 leading to not just one new quantum theory but two. Almost simultaneously, the German physicistsWerner Heisenberg,Max Born, andPascual Jordan^[44][45] developedmatrix mechanics and the Austrian physicist Erwin Schrödinger inventedwave mechanics.^[46] Schrödinger subsequently showed that the two approaches were equivalent.

The first applications of quantum mechanics to physical systems were the algebraic determination of the hydrogen spectrum byWolfgang Pauli^[47] and the treatment of diatomic molecules byLucy Mensing.^[48]Accurate predictions of the absorption spectrum of hydrogen ensured wide acceptance of the new quantum theory.^[1]: 275

In 1925, Werner Heisenberg attempted to solve one of the problems that the Bohr model left unanswered, explaining the intensities of the different lines in the hydrogen emission spectrum. Through a series of mathematical analogies, he wrote out the quantum-mechanical analog for the classical computation of intensities.^[49] Shortly afterward, Heisenberg's colleague Max Born realized that Heisenberg's method of calculating the probabilities for transitions between the different energy levels could best be expressed by using the mathematical concept ofmatrices.

Heisenberg formulated an early version of theuncertainty principle in 1927, analyzing athought experiment where one attempts tomeasure an electron's position and momentum simultaneously. However, Heisenberg did not give precise mathematical definitions of what the "uncertainty" in these measurements meant, a step that would be taken soon after byEarle Hesse Kennard, Wolfgang Pauli, andHermann Weyl.^[50][51]

In the first half of 1926, building on de Broglie's hypothesis, Erwin Schrödinger developed the equation that describes the behavior of a quantum-mechanical wave.^[52] The mathematical model, called the Schrödinger equation after its creator, is central to quantum mechanics, defines the permitted stationary states of a quantum system, and describes how the quantum state of a physical system changes in time.^[53] The wave itself is described by a mathematical function known as a "wave function". Schrödinger said that the wave function provides the "means for predicting the probability of measurement results".^[54]

Schrödinger was able to calculate the energy levels of hydrogen by treating a hydrogen atom's electron as a classical wave, moving in a well of the electrical potential created by the proton. This calculation accurately reproduced the energy levels of the Bohr model.

In May 1926, Schrödinger proved that Heisenberg'smatrix mechanics and his own wave mechanics made the same predictions about the properties and behavior of the electron; mathematically, the two theories had an underlying common form. Yet the two men disagreed on the interpretation of their mutual theory. For instance, Heisenberg accepted the theoretical prediction of jumps of electrons between orbitals in an atom,^[55] but Schrödinger hoped that a theory based on continuous wave-like properties could avoid what he called (as paraphrased byWilhelm Wien) "this nonsense about quantum jumps".^[56] In the end, Heisenberg's approach won out, and quantum jumps were confirmed.^[57]

Bohr's model of the atom was essentially a planetary one, with the electrons orbiting around the nuclear "sun". However, the uncertainty principle states that an electron cannot simultaneously have an exact location and velocity in the way that a planet does. Instead of classical orbits, electrons are said to inhabitatomic orbitals.

An orbital is the "cloud" of possible locations in which an electron might be found, a distribution of probabilities rather than a precise location.^[58] Each orbital is three dimensional, rather than the two-dimensional orbit, and is often depicted as a three-dimensional region within which there is a 95 percent probability of finding the electron.^[59]

Schrödinger was able to calculate the energy levels of hydrogen by treating a hydrogen atom's electron as a wave, represented by the "wave function"Ψ, in anelectric potentialwell,V, created by the proton. The solutions to Schrödinger's equation^{[clarification needed]} are distributions of probabilities for electron positions and locations. Orbitals have a range of different shapes in three dimensions. The energies of the different orbitals can be calculated, and they accurately match the energy levels of the Bohr model.

Within Schrödinger's picture, each electron has four properties:

1. An "orbital" designation, indicating whether the particle-wave is one that is closer to the nucleus with less energy or one that is farther from the nucleus with more energy;
2. The "shape" of the orbital, spherical or otherwise;
3. The "inclination" of the orbital, determining the magnetic moment of the orbital around thez-axis.
4. The "spin" of the electron.

The collective name for these properties is thequantum state of the electron. The quantum state can be described by giving a number to each of these properties; these are known as the electron'squantum numbers. The quantum state of the electron is described by its wave function. The Pauli exclusion principle demands that no two electrons within an atom may have the same values of all four numbers.

The first property describing the orbital is theprincipal quantum number,n, which is the same as in the Bohr model.n denotes the energy level of each orbital. The possible values forn are integers:

${\displaystyle n=1,2,3\dots }$

The next quantum number, theazimuthal quantum number, denotedl, describes the shape of the orbital. The shape is a consequence of theangular momentum of the orbital. The angular momentum represents the resistance of a spinning object to speeding up or slowing down under the influence of external force. The azimuthal quantum number represents the orbital angular momentum of an electron around its nucleus. The possible values forl are integers from 0 ton − 1 (wheren is the principal quantum number of the electron):

${\displaystyle l=0,1,\dots ,n-1.}$

The shape of each orbital is usually referred to by a letter, rather than by its azimuthal quantum number. The first shape (l=0) is denoted by the letters (amnemonic being "sphere"). The next shape is denoted by the letterp and has the form of a dumbbell. The other orbitals have more complicated shapes (seeatomic orbital), and are denoted by the lettersd,f,g, etc.

The third quantum number, themagnetic quantum number, describes the magnetic moment of the electron, and is denoted bym_l (or simplym). The possible values form_l are integers from−l tol (wherel is the azimuthal quantum number of the electron):

${\displaystyle m_l=-l,-(l-1),\dots ,0,\dots ,(l-1),l.}$

The magnetic quantum number measures the component of the angular momentum in a particular direction. The choice of direction is arbitrary; conventionally the z-direction is chosen.

The fourth quantum number, thespin quantum number (pertaining to the "orientation" of the electron's spin) is denotedm_s, with values +1⁄2 or −1⁄2.

The chemistLinus Pauling wrote, by way of example:

In the case of ahelium atom with two electrons in the 1s orbital, the Pauli Exclusion Principle requires that the two electrons differ in the value of one quantum number. Their values ofn,l, andm_l are the same. Accordingly they must differ in the value ofm_s, which can have the value of +1⁄2 for one electron and −1⁄2 for the other."^[58]

It is the underlying structure and symmetry of atomic orbitals, and the way that electrons fill them, that leads to the organization of theperiodic table. The way the atomic orbitals on different atoms combine to formmolecular orbitals determines the structure and strength of chemical bonds between atoms.

The field ofquantum chemistry was pioneered by physicistsWalter Heitler andFritz London, who published a study of thecovalent bond of thehydrogen molecule in 1927. Quantum chemistry was subsequently developed by a large number of workers, including the American theoretical chemistLinus Pauling atCaltech, andJohn C. Slater into various theories such as Molecular Orbital Theory or Valence Theory.

Starting around 1927, Paul Dirac began the process of unifying quantum mechanics withspecial relativity by proposing theDirac equation for the electron. The Dirac equation achieves the relativistic description of the wavefunction of an electron that Schrödinger failed to obtain. It predicts electron spin and led Dirac to predict the existence of thepositron. He also pioneered the use of operator theory, including the influentialbra–ket notation, as described in his famous 1930 textbook.

During the same period, Hungarian polymathJohn von Neumann formulated a mathematical basis for quantum mechanics as the theory of linear operators on Hilbert spaces, as described in hisMathematical Foundations of Quantum Mechanics textbook. These, like many other works from the founding period, still stand, and remain widely used.^{[citation needed]}

Bohr, Heisenberg, and others tried to explain what these experimental results and mathematical models really mean. The termCopenhagen interpretation has been applied to their views in retrospect, glossing over differences among them.^{[60][61][62][63][64][65]} While no definitive statement of "the" Copenhagen interpretation exists, the following ideas are widely seen as characteristic of it.

1. A system is completely described by a quantum state (Heisenberg)
2. How the quantum state changes over time is given by a wave equation, theSchrödinger equation imparting wave characteristics to light and matter.
3. Atomic interactions are discontinuous (Planck referred to a "quantum of action").
4. The description of nature is essentially probabilistic. The probability of an event—for example, where on the screen a particle shows up in the double-slit experiment—is related to the square of the absolute value of the amplitude of its wave function. (Born rule, due toMax Born, which gives a physical meaning to the wave function in the Copenhagen interpretation: theprobability amplitude)
5. The values of incompatible pairs of properties of the system cannot be known at the same time. (Heisenberg'suncertainty principle)
6. Matter, like light, exhibits a wave-particle duality. An experiment can demonstrate the particle-like properties of matter, or its wave-like properties; but not both at the same time. (Complementarity principle due to Bohr^[66])
7. Measuring devices are essentially classical devices and measure classical properties such as position and momentum.
8. The quantum mechanical description of large systems should closely approximate the classical description. (Correspondence principle of Bohr and Heisenberg)

Beginning in 1927, researchers attempted to apply quantum mechanics to fields instead of single particles, resulting in quantum field theories. Early workers in this area include Pauli,Paul Dirac,Victor Weisskopf, andPaul Jordan. This area of research culminated in the formulation ofquantum electrodynamics byRichard Feynman,Freeman Dyson,Julian Schwinger, andShin'ichiro Tomonaga during the 1940s. Quantum electrodynamics describes a quantum theory of electrons,positrons, and theelectromagnetic field, and served as a model for subsequentquantum field theories.^[44][45][67]

The theory ofquantum chromodynamics was formulated beginning in the early 1960s. The theory as we know it today was formulated byHugh David Politzer,David Gross andFrank Wilczek in 1975.

Building on pioneering work by Schwinger,Peter Higgs andJeffrey Goldstone, the physicistsSheldon Glashow,Steven Weinberg andAbdus Salam independently showed how the weak nuclear force and quantum electrodynamics could be merged into a singleelectroweak force, for which they received the 1979 Nobel Prize in Physics.

Quantum information science developed in the latter decades of the 20th century, beginning with theoretical results likeHolevo's theorem, the concept of generalized measurements orPOVMs, the proposal ofquantum key distribution byBennett and Brassard in 1984, andShor's algorithm in 1994.

• Thomas Young'sdouble-slit experiment demonstrating the wave nature of light. (c. 1801)
• Henri Becquerel discoversradioactivity. (1896)
• J. J. Thomson's cathode ray tube experiments (discovers the electron and its negative charge). (1897)
• The study ofblack-body radiation between 1850 and 1900, which could not be explained without quantum concepts.
• Thephotoelectric effect: Einstein explained this in 1905 (and later received a Nobel prize for it) using the concept of photons, particles of light with quantized energy.
• Robert Millikan'soil-drop experiment, which showed thatelectric charge occurs asquanta (whole units). (1909)
• Ernest Rutherford'sgold foil experiment disproved the plum pudding model of theatom which suggested that the mass and positive charge of the atom are almost uniformly distributed. This led to the planetary model of the atom (1911).
• James Franck andGustav Hertz'selectron collision experiment shows that energy absorption by mercury atoms is quantized. (1914)
• Otto Stern andWalther Gerlach conduct theStern–Gerlach experiment, which demonstrates the quantized nature of particlespin. (1920)
• Arthur Compton withCompton scattering experiment (1923)
• Clinton Davisson andLester Germer demonstrate the wave nature of the electron^[68] in theelectron diffraction experiment. (1927)
• Carl David Anderson with the discovery positron (1932), validated Paul Dirac's theoretical prediction of this particle (1928)
• Lamb–Retherford experiment discoveredLamb shift (1947), which led to the development of quantum electrodynamics.
• Clyde L. Cowan andFrederick Reines confirm the existence of theneutrino in theneutrino experiment. (1955)
• Claus Jönsson's double-slit experiment with electrons. (1961)
• Thequantum Hall effect, discovered in 1980 byKlaus von Klitzing. The quantized version of theHall effect has allowed for the definition of a new practical standard forelectrical resistance and for an extremely precise independent determination of thefine-structure constant.
• Theexperimental verification ofquantum entanglement byJohn Clauser andStuart Freedman. (1972)
• TheMach–Zehnder interferometer experiment conducted byPaul Kwiat, Harold Wienfurter, Thomas Herzog,Anton Zeilinger, and Mark Kasevich, providingexperimental verification of the Elitzur–Vaidman bomb tester, provinginteraction-free measurement is possible. (1994)

• Bacciagaluppi, Guido;Valentini, Antony (2009),Quantum theory at the crossroads: reconsidering the 1927 Solvay conference, Cambridge, UK: Cambridge University Press, p. 9184,arXiv:quant-ph/0609184,Bibcode:2006quant.ph..9184B,ISBN 978-0-521-81421-8,OCLC 227191829
• Bernstein, Jeremy (2009),Quantum Leaps, Cambridge, Massachusetts: Belknap Press of Harvard University Press,ISBN 978-0-674-03541-6
• Greenberger, Daniel,Hentschel, Klaus, Weinert, Friedel (Eds.)Compendium of Quantum Physics. Concepts, Experiments, History and Philosophy, New York: Springer, 2009.ISBN 978-3-540-70626-7.
• Jammer, Max (1966),The conceptual development of quantum mechanics, New York: McGraw-Hill,OCLC 534562
• Jammer, Max (1974),The philosophy of quantum mechanics: The interpretations of quantum mechanics in historical perspective, New York: Wiley,ISBN 0-471-43958-4,OCLC 969760
• A. Whitaker.The New Quantum Age: From Bell's Theorem to Quantum Computation and Teleportation, Oxford University Press, 2011,ISBN 978-0-19-958913-5
• Stephen Hawking.The Dreams that Stuff is Made of, Running Press, 2011,ISBN 978-0-76-243434-3
• A. Douglas Stone.Einstein and the Quantum, the Quest of the Valiant Swabian, Princeton University Press, 2006.

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