**Quantum mechanics** is the body of scientific principles that explains the behavior of

matterMatter is a general term for the substance of which all physical objects consist. Typically, matter includes atoms and other particles which have mass. A common way of defining matter is as anything that has mass and occupies volume...

and its interactions with

energyIn physics, energy is an indirectly observed quantity. It is often understood as the ability a physical system has to do work on other physical systems...

on the

scale of atoms and

atomic particlesIn particle physics, an elementary particle or fundamental particle is a particle not known to have substructure; that is, it is not known to be made up of smaller particles. If an elementary particle truly has no substructure, then it is one of the basic building blocks of the universe from which...

.

Classical physicsWhat "classical physics" refers to depends on the context. When discussing special relativity, it refers to the Newtonian physics which preceded relativity, i.e. the branches of physics based on principles developed before the rise of relativity and quantum mechanics...

explains matter and energy at the macroscopic level of the scale familiar to human experience, including the behavior of astronomical bodies. It remains the key to

measurementThe measure in quantum physics is the integration measure used for performing a path integral.In quantum field theory, one must sum over all possible histories of a system....

for much of

modern scienceScience is a systematic enterprise that builds and organizes knowledge in the form of testable explanations and predictions about the universe...

and technology; but at the end of the 19th Century observers discovered phenomena in both the large (macro) and the small (micro) worlds that classical physics could not explain. Coming to terms with these limitations led to the development of quantum mechanics, a major revolution in physics. This article describes how physicists discovered the limitations of classical physics and developed the main concepts of the quantum theory that replaced them in the early decades of the 20th century.

[Classical physics also does not accurately describe the universe on the largest scales or at speeds close to that of light. An accurate description requires general relativity]General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...

. These concepts are described in roughly the order they were first discovered; for a more complete history of the subject, see

History of quantum mechanicsThe history of quantum mechanics, as it interlaces with the history of quantum chemistry, began essentially with a number of different scientific discoveries: the 1838 discovery of cathode rays by Michael Faraday; the 1859-1860 winter statement of the black body radiation problem by Gustav...

.

Some aspects of quantum mechanics can seem counter-intuitive, because they describe behavior quite different than that seen at larger length scales, where classical physics is an excellent approximation. In the words of

Richard FeynmanRichard Phillips Feynman was an American physicist known for his work in the path integral formulation of quantum mechanics, the theory of quantum electrodynamics and the physics of the superfluidity of supercooled liquid helium, as well as in particle physics...

, quantum mechanics deals with "nature as she is — absurd."

Many types of energy, such as

photonIn physics, a photon is an elementary particle, the quantum of the electromagnetic interaction and the basic unit of light and all other forms of electromagnetic radiation. It is also the force carrier for the electromagnetic force...

s (discrete units of

lightLight or visible light is electromagnetic radiation that is visible to the human eye, and is responsible for the sense of sight. Visible light has wavelength in a range from about 380 nanometres to about 740 nm, with a frequency range of about 405 THz to 790 THz...

), behave in some respects like particles and in other respects like waves. Radiators of photons (such as

neon lightNeon lighting is created by brightly glowing, electrified glass tubes or bulbs that contain rarefied neon or other gases. Georges Claude, a French engineer and inventor, presented neon tube lighting in essentially its modern form at the Paris Motor Show from December 3–18, 1910...

s) have emission

spectraA spectrum is a condition that is not limited to a specific set of values but can vary infinitely within a continuum. The word saw its first scientific use within the field of optics to describe the rainbow of colors in visible light when separated using a prism; it has since been applied by...

that are discontinuous, in that only certain frequencies of light are present. Quantum mechanics predicts the energies, the colours, and the spectral

intensitiesIn physics, intensity is a measure of the energy flux, averaged over the period of the wave. The word "intensity" here is not synonymous with "strength", "amplitude", or "level", as it sometimes is in colloquial speech...

of all forms of

electromagnetic radiationElectromagnetic radiation is a form of energy that exhibits wave-like behavior as it travels through space...

.

But quantum mechanics theory ordains that the more closely one pins down one measure (such as the position of a particle), the less precise another measurement pertaining to the same particle (such as its

momentumIn classical mechanics, linear momentum or translational momentum is the product of the mass and velocity of an object...

) must become. Put another way, measuring position first and then measuring momentum does

*not* have the same outcome as measuring momentum first and then measuring position; the act of measuring the first property necessarily introduces additional energy into the micro-system being studied, thereby perturbing that system.

Even more disconcerting, pairs of particles can be created as

entangled twinsQuantum entanglement occurs when electrons, molecules even as large as "buckyballs", photons, etc., interact physically and then become separated; the type of interaction is such that each resulting member of a pair is properly described by the same quantum mechanical description , which is...

— which means that a measurement which pins down one property of one of the particles will instantaneously pin down the same or another property of its entangled twin, regardless of the distance separating them — though this may be regarded as merely a mathematical anomaly, rather than a real one.

## The first quantum theory: Max Planck and black body radiation

Thermal radiationThermal radiation is electromagnetic radiation generated by the thermal motion of charged particles in matter. All matter with a temperature greater than absolute zero emits thermal radiation....

is

electromagnetic radiationElectromagnetic radiation is a form of energy that exhibits wave-like behavior as it travels through space...

emitted from the surface of an object due to the object's

temperatureTemperature is a physical property of matter that quantitatively expresses the common notions of hot and cold. Objects of low temperature are cold, while various degrees of higher temperatures are referred to as warm or hot...

. If an object is heated sufficiently, it starts to emit light at the red end of the

spectrumA spectrum is a condition that is not limited to a specific set of values but can vary infinitely within a continuum. The word saw its first scientific use within the field of optics to describe the rainbow of colors in visible light when separated using a prism; it has since been applied by...

— it is

*red hot*. Heating it further causes the colour to change, as light at shorter wavelengths (higher frequencies) begins to be emitted. It turns out that 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 a

black bodyA black body is an idealized physical body that absorbs all incident electromagnetic radiation. Because of this perfect absorptivity at all wavelengths, a black body is also the best possible emitter of thermal radiation, which it radiates incandescently in a characteristic, continuous spectrum...

, and the radiation it emits is called black body radiation.

In the late 19th century, thermal radiation had been fairly well-characterized experimentally. The wavelength at which the radiation is strongest is given by

Wien's displacement lawWien's displacement law states that the wavelength distribution of thermal radiation from a black body at any temperature has essentially the same shape as the distribution at any other temperature, except that each wavelength is displaced on the graph...

, and the overall power emitted per unit area is given by the Stefan–Boltzmann law. Therefore, as temperature increases, the glow colour changes from red to yellow to blue to white. Even as the peak wavelength moves into the ultra-violet, enough radiation continues to be emitted in the blue wavelengths that the body continues to appear blue. It never becomes invisible—indeed, the radiation of visible light increases

monotonicallyIn mathematics, a monotonic function is a function that preserves the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order theory....

with temperature.

Physicists were searching for a theoretical explanation for these experimental results.

The "answer" found using classical physics is the Rayleigh–Jeans law. This law agrees with experimental results at long wavelengths. At short wavelengths, however, classical physics predicts that energy will be emitted by a hot body at an infinite rate. This result, which is clearly wrong, is known as the

ultraviolet catastropheThe ultraviolet catastrophe, also called the Rayleigh–Jeans catastrophe, was a prediction of late 19th century/early 20th century classical physics that an ideal black body at thermal equilibrium will emit radiation with infinite power....

.

The first model that was able to explain the full spectrum of thermal radiation was put forward by

Max PlanckMax Karl Ernst Ludwig Planck, ForMemRS, was a German physicist who actualized the quantum physics, initiating a revolution in natural science and philosophy. He is regarded as the founder of the quantum theory, for which he received the Nobel Prize in Physics in 1918.-Life and career:Planck came...

in 1900. He modeled the thermal radiation as being in equilibrium, using a set of harmonic oscillators. To reproduce the experimental results he had to assume that each oscillator produced an integral 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 of each oscillator was "quantized."

[The word "quantum]In physics, a quantum is the minimum amount of any physical entity involved in an interaction. Behind this, one finds the fundamental notion that a physical property may be "quantized," referred to as "the hypothesis of quantization". This means that the magnitude can take on only certain discrete...

" comes from the Latin word for "how much" (as does "quantity"). Something which is "quantized," like the energy of Planck's harmonic oscillators, can only take specific values. For example, in most countries money is effectively quantized, with the "quantum 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. The

quantumIn physics, a quantum is the minimum amount of any physical entity involved in an interaction. Behind this, one finds the fundamental notion that a physical property may be "quantized," referred to as "the hypothesis of quantization". This means that the magnitude can take on only certain discrete...

of energy for each oscillator, according to Planck, was proportional to the frequency of the oscillator; the constant of proportionality is now known as the

Planck constantThe Planck constant , also called Planck's constant, is a physical constant reflecting the sizes of energy quanta in quantum mechanics. It is named after Max Planck, one of the founders of quantum theory, who discovered it in 1899...

. The Planck constant, usually written as , has the value , and so the energy of an oscillator of frequency is given by

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

## Photons: the quantisation of light

In 1905,

Albert EinsteinAlbert Einstein was a German-born theoretical physicist who developed the theory of general relativity, effecting a revolution in physics. For this achievement, Einstein is often regarded as the father of modern physics and one of the most prolific intellects in human history...

took an extra step. He suggested that quantisation was not just a mathematical trick: the energy in a beam of light occurs in individual packets, which are now called

photonIn physics, a photon is an elementary particle, the quantum of the electromagnetic interaction and the basic unit of light and all other forms of electromagnetic radiation. It is also the force carrier for the electromagnetic force...

s. The energy of a single photon is given by its frequency multiplied by Planck's constant:

For centuries, scientists had debated between two possible theories of

lightLight or visible light is electromagnetic radiation that is visible to the human eye, and is responsible for the sense of sight. Visible light has wavelength in a range from about 380 nanometres to about 740 nm, with a frequency range of about 405 THz to 790 THz...

: was it a

waveIn physics, a wave is a disturbance that travels through space and time, accompanied by the transfer of energy.Waves travel and the wave motion transfers energy from one point to another, often with no permanent displacement of the particles of the medium—that is, with little or no associated mass...

or did it instead comprise a stream of tiny particles? By the 19th century, the debate was generally considered to have been settled in favour of the wave theory, as it was able to explain observed effects such as

refractionRefraction is the change in direction of a wave due to a change in its speed. It is essentially a surface phenomenon . The phenomenon is mainly in governance to the law of conservation of energy. The proper explanation would be that due to change of medium, the phase velocity of the wave is changed...

,

diffractionDiffraction refers to various phenomena which occur when a wave encounters an obstacle. Italian scientist Francesco Maria Grimaldi coined the word "diffraction" and was the first to record accurate observations of the phenomenon in 1665...

and polarization.

James Clerk MaxwellJames Clerk Maxwell of Glenlair was a Scottish physicist and mathematician. His most prominent achievement was formulating classical electromagnetic theory. This united all previously unrelated observations, experiments and equations of electricity, magnetism and optics into a consistent theory...

had shown that electricity, magnetism and light are all manifestations of the same phenomenon: the

electromagnetic fieldAn electromagnetic field is a physical field produced by moving electrically charged objects. It affects the behavior of charged objects in the vicinity of the field. The electromagnetic field extends indefinitely throughout space and describes the electromagnetic interaction...

.

Maxwell's equationsMaxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electrodynamics, classical optics, and electric circuits. These fields in turn underlie modern electrical and communications technologies.Maxwell's equations...

, which are the complete set of laws of

classical electromagnetismClassical electromagnetism is a branch of theoretical physics that studies consequences of the electromagnetic forces between electric charges and currents...

, describe light as waves: a combination of oscillating electric and magnetic fields. Because of the preponderance of evidence in favour of the wave theory, Einstein's ideas were met initially with great scepticism. Eventually, however, the photon model became favoured; one of the most significant pieces of evidence in its favour was its ability to explain several puzzling properties of the

photoelectric effectIn the photoelectric effect, electrons are emitted from matter as a consequence of their absorption of energy from electromagnetic radiation of very short wavelength, such as visible or ultraviolet light. Electrons emitted in this manner may be referred to as photoelectrons...

, described in the following section. Nonetheless, the wave analogy remained indispensable for helping to understand other characteristics of light, such as

diffractionDiffraction refers to various phenomena which occur when a wave encounters an obstacle. Italian scientist Francesco Maria Grimaldi coined the word "diffraction" and was the first to record accurate observations of the phenomenon in 1665...

.

### The photoelectric effect

In 1887 Heinrich Hertz observed that light can eject electrons from metal. In 1902

Philipp LenardPhilipp Eduard Anton von Lenard , known in Hungarian as Lénárd Fülöp Eduárd Antal, was a Hungarian - German physicist and the winner of the Nobel Prize for Physics in 1905 for his research on cathode rays and the discovery of many of their properties...

discovered that the maximum possible energy of an ejected electron is related to the

frequencyFrequency is the number of occurrences of a repeating event per unit time. It is also referred to as temporal frequency.The period is the duration of one cycle in a repeating event, so the period is the reciprocal of the frequency...

of the light, not to its

*intensity*In physics, intensity is a measure of the energy flux, averaged over the period of the wave. The word "intensity" here is not synonymous with "strength", "amplitude", or "level", as it sometimes is in colloquial speech...

; if the frequency is too low, no electrons are ejected regardless of the intensity. The lowest frequency of light that causes electrons to be emitted, called the threshold frequency, is different for every metal. This observation is at odds with classical electromagnetism, which predicts that the electron's energy should be proportional to the

*intensity* of the radiation.

Einstein explained the effect by postulating that a beam of light is a stream of particles (

*photon*In physics, a photon is an elementary particle, the quantum of the electromagnetic interaction and the basic unit of light and all other forms of electromagnetic radiation. It is also the force carrier for the electromagnetic force...

s), and that if the beam is of frequency then each photon has an energy equal to . An electron is likely to be struck only by a single photon, which imparts at most an energy to the electron. Therefore, the intensity of the beam has no effect; only its frequency determines the maximum energy that can be imparted to the electron.

To explain the threshold effect, Einstein argued that it takes a certain amount of energy, called the

*work function*In solid-state physics, the work function is the minimum energy needed to remove an electron from a solid to a point immediately outside the solid surface...

, denoted by , to remove an electron from the metal. This amount of energy is different for each metal. If the energy of the photon is less than the work function then it does not carry sufficient energy to remove the electron from the metal. The threshold frequency, , is the frequency of a photon whose energy is equal to the work function:

If is greater than , the energy is enough to remove an electron. The ejected electron has a

kinetic energyThe kinetic energy of an object is the energy which it possesses due to its motion.It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acceleration, the body maintains this kinetic energy unless its speed changes...

which is, at most, equal to the photon's energy minus the energy needed to dislodge the electron from the metal:

Einstein's description of light as being composed of particles

*extended* Planck's notion of quantised energy: a single photon of a given frequency delivers an invariant amount of energy . In other words, individual photons can deliver more or less energy, but only depending on their frequencies. However, although the photon is a

*particle* it was still being described as having the wave-like property of frequency. Once again, the particle account of light was being "compromised".

The relationship between the frequency of electromagnetic radiation and the energy of each individual photon is why

ultravioletUltraviolet light is electromagnetic radiation with a wavelength shorter than that of visible light, but longer than X-rays, in the range 10 nm to 400 nm, and energies from 3 eV to 124 eV...

light can cause sunburn, but visible or

infraredInfrared light is electromagnetic radiation with a wavelength longer than that of visible light, measured from the nominal edge of visible red light at 0.74 micrometres , and extending conventionally to 300 µm...

light cannot. A photon of ultraviolet light will deliver a high amount of

energyIn physics, energy is an indirectly observed quantity. It is often understood as the ability a physical system has to do work on other physical systems...

—enough to contribute to cellular damage such as occurs in a sunburn. A photon of infrared light will deliver a lower amount of energy—only enough to warm one's skin. So an infrared lamp can warm a large surface, perhaps large enough to keep people comfortable in a cold room, but it cannot give anyone a sunburn.

If each individual photon had identical energy, it would not be correct to talk of a "high energy" photon. Light of high frequency could carry more energy only because of flooding a surface with more photons arriving

*per second*. Light of low frequency could carry more energy only for the same reason. If it were true that all photons carry the same energy, then if you doubled the rate of photon delivery, you would double the number of energy units arriving each second. Einstein rejected that wave-dependent classical approach in favour of a particle-based analysis where the energy of the particle must be absolute and varies with frequency in discrete steps (i.e. is quantised). All photons of the same frequency have identical energy, and all photons of different frequencies have proportionally different energies.

In nature, single photons are rarely encountered. The sun emits photons continuously at all electromagnetic frequencies, so they appear to propagate as a continuous wave, not as discrete units. The emission sources available to Hertz and Lennard in the 19th century shared that characteristic. A sun that radiates red light, or a piece of iron in a forge that glows red, may both be said to contain a great deal of energy. It might be surmised that adding continuously to the total energy of some radiating body would make it radiate red light, orange light, yellow light, green light, blue light, violet light, and so on in that order. But that is not so for otherwise larger suns and larger pieces of iron in a forge would glow with colours more toward the violet end of the spectrum. To change the color of such a radiating body it is necessary to change its temperature, and increasing its temperature changes the quanta of energy that are available to excite individual atoms to higher levels and permit them to emit photons of higher frequencies. The total energy emitted per unit of time by a sun or by a piece of iron in a forge depends on both the number of photons emitted per unit of time and also on the amount of energy carried by each of the photons involved. In other words, the characteristic frequency of a radiating body is dependent on its temperature. When physicists were looking only at beams of light containing huge numbers of individual and virtually indistinguishable photons it was difficult to understand the importance of the energy levels of individual photons. So when physicists first discovered devices exhibiting the photoelectric effect, the effect that makes the light meters of modern cameras work, they initially expected that a higher intensity of light would produce a higher voltage from the photoelectric device. They discovered that strong beams of light toward the red end of the spectrum might produce no electrical potential at all, and that weak beams of light toward the violet end of the spectrum would produce higher and higher voltages. Einstein's idea that individual units of light may contain different amounts of energy depending on their frequency made it possible to explain the experimental results that hitherto had seemed quite counter-intuitive.

Although the energy imparted by photons is invariant at any given frequency, the initial energy-state of the electrons in a photoelectric device prior to absorption of light is not necessarily uniform. Therefore anomalous results may occur in the case of individual electrons. An electron that was already excited above the equilibrium level of the photoelectric device might be ejected when it absorbed uncharacteristically low frequency illumination. Statistically, however, the characteristic behavior of a photoelectric device will reflect the behavior of the vast majority of its electrons, which will be at their equilibrium level. This point is helpful in comprehending the distinction between the study of individual particles in quantum dynamics and the study of massed particles in classical physics.

## The quantisation of matter: the Bohr model of the atom

By the dawn of the 20th century, it was known that

atomThe atom is a basic unit of matter that consists of a dense central nucleus surrounded by a cloud of negatively charged electrons. The atomic nucleus contains a mix of positively charged protons and electrically neutral neutrons...

s comprise a diffuse cloud of negatively-charged

electronThe electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...

s surrounding a small, dense, positively-charged

nucleusThe nucleus is the very dense region consisting of protons and neutrons at the center of an atom. It was discovered in 1911, as a result of Ernest Rutherford's interpretation of the famous 1909 Rutherford experiment performed by Hans Geiger and Ernest Marsden, under the direction of Rutherford. The...

. This understanding suggested a model in which the electrons circle around the nucleus like planets orbiting a sun.

[The classical model of the atom is called the planetary model, or sometimes the Rutherford model]The Rutherford model or planetary model is a model of the atom devised by Ernest Rutherford. Rutherford directed the famous Geiger-Marsden experiment in 1909, which suggested on Rutherford's 1911 analysis that the so-called "plum pudding model" of J. J. Thomson of the atom was incorrect...

after Ernest RutherfordErnest Rutherford, 1st Baron Rutherford of Nelson OM, FRS was a New Zealand-born British chemist and physicist who became known as the father of nuclear physics...

who proposed it in 1911, based on the Geiger-Marsden gold foil experimentThe Geiger–Marsden experiment was an experiment to probe the structure of the atom performed by Hans Geiger and Ernest Marsden in 1909, under the direction of Ernest Rutherford at the Physical Laboratories of the University of Manchester...

which first demonstrated the existence of the nucleus. However, it was also known that the atom in this model would be unstable: according to classical theory orbiting electrons are undergoing centripetal acceleration, and should therefore give off electromagnetic radiation, the loss of energy also causing them to spiral toward the nucleus, colliding with it in a fraction of a second.

A second, related, puzzle was the

emission spectrumThe emission spectrum of a chemical element or chemical compound is the spectrum of frequencies of electromagnetic radiation emitted by the element's atoms or the compound's molecules when they are returned to a lower energy state....

of atoms. When a gas is heated, it gives off light only at discrete frequencies. For example, the visible light given off by

hydrogenHydrogen is the chemical element with atomic number 1. It is represented by the symbol H. With an average atomic weight of , hydrogen is the lightest and most abundant chemical element, constituting roughly 75% of the Universe's chemical elemental mass. Stars in the main sequence are mainly...

consists of four different colours, as shown in the picture below. By contrast, white light consists of a continuous emission across the whole range of visible frequencies.

In 1885 the Swiss mathematician Johann Balmer discovered that each wavelength (lambda) in the visible spectrum of hydrogen is related to some integer by the equation

where is a constant which Balmer determined to be equal to 364.56 nm. Thus Balmer's constant was the basis of a system of discrete, i.e. quantised, integers.

In 1888

Johannes RydbergJohannes Robert Rydberg, , , was a Swedish physicist mainly known for devising the Rydberg formula, in 1888, which is used to predict the wavelengths of photons emitted by changes in the energy level of an electron in a hydrogen atom.The physical constant known as the...

generalized and greatly increased the explanatory utility of Balmer's formula. He predicted that is related to two integers and according to what is now known as the

Rydberg formulaThe Rydberg formula is used in atomic physics to describe the wavelengths of spectral lines of many chemical elements. It was formulated by the Swedish physicist Johannes Rydberg, and presented on November 5, 1888.-History:...

:

where

*R* is the

Rydberg constantThe Rydberg constant, symbol R∞, named after the Swedish physicist Johannes Rydberg, is a physical constant relating to atomic spectra in the science of spectroscopy. Rydberg initially determined its value empirically from spectroscopy, but Niels Bohr later showed that its value could be calculated...

, equal to 0.0110 nm

^{−1}, and

*n* must be greater than

*m*.

Rydberg's formula accounts for the four visible wavelengths of hydrogen by setting and . It also predicts additional wavelengths in the emission spectrum: for and for , the emission spectrum should contain certain ultraviolet wavelengths, and for and , it should also contain certain infrared wavelengths. Experimental observation of these wavelengths came two decades later: in 1908 Louis Paschen found some of the predicted infrared wavelengths, and in 1914

Theodore LymanTheodore Lyman was a U.S. physicist and spectroscopist, born in Boston. He graduated from Harvard in 1897, from which he also received his Ph.D. in 1900. He became an assistant professor in physics at Harvard, where he remained, becoming full professor in 1917, and where he was also director of...

found some of the predicted ultraviolet wavelengths.

### Bohr's model

In 1913

Niels BohrNiels Henrik David Bohr was a Danish physicist who made foundational contributions to understanding atomic structure and quantum mechanics, for which he received the Nobel Prize in Physics in 1922. Bohr mentored and collaborated with many of the top physicists of the century at his institute in...

proposed a new model of the atom that included quantized electron orbits. In Bohr's model, electrons could inhabit only certain orbits around the atomic nucleus. 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. 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.

Bohr theorised that the

angular momentumIn physics, angular momentum, moment of momentum, or rotational momentum is a conserved vector quantity that can be used to describe the overall state of a physical system...

, , of an electron is quantised:

where is an integer and is the Planck constant. Starting from this assumption,

Coulomb's lawCoulomb's law or Coulomb's inverse-square law, is a law of physics describing the electrostatic interaction between electrically charged particles. It was first published in 1785 by French physicist Charles Augustin de Coulomb and was essential to the development of the theory of electromagnetism...

and the equations of

circular motionIn physics, uniform circular motion describes the motion of a body traversing a circular path at constant speed. The distance of the body from the axis of rotation remains constant at all times. Though the body's speed is constant, its velocity is not constant: velocity, a vector quantity, depends...

show that an electron with units of angular momentum will orbit a proton at a distance given by

,

where is the Coulomb constant, is the mass of an electron, and is the

charge on an electronThe elementary charge, usually denoted as e, is the electric charge carried by a single proton, or equivalently, the absolute value of the electric charge carried by a single electron. This elementary charge is a fundamental physical constant. To avoid confusion over its sign, e is sometimes called...

.

For simplicity this is written as

where , called the

Bohr radiusThe Bohr radius is a physical constant, approximately equal to the most probable distance between the proton and electron in a hydrogen atom in its ground state. It is named after Niels Bohr, due to its role in the Bohr model of an atom...

, is equal to 0.0529 nm.

The Bohr radius is the radius of the smallest allowed orbit.

The energy of the electron

[In this case, the energy of the electron is the sum of its kinetic]The kinetic energy of an object is the energy which it possesses due to its motion.It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acceleration, the body maintains this kinetic energy unless its speed changes...

and potentialIn physics, potential energy is the energy stored in a body or in a system due to its position in a force field or due to its configuration. The SI unit of measure for energy and work is the Joule...

energies. The electron has kinetic energy by virtue of its actual motion around the nucleus, and potential energy because of its electromagnetic interaction with the nucleus. can also be calculated, and is given by

.

Thus Bohr's assumption that angular momentum is quantised means that an electron can only inhabit certain orbits around the nucleus, and that it can have only certain energies. A consequence of these constraints is that the electron will not crash into the nucleus: it cannot continuously emit energy, and it cannot come closer to the nucleus than

*a*_{0} (the Bohr radius).

An electron loses energy by jumping instantaneously from its original orbit to a lower orbit; the extra energy is emitted in the form of a photon. Conversely, an electron that absorbs a photon gains energy, hence it jumps to an orbit that is farther from the nucleus.

Each photon from glowing atomic hydrogen is due to an electron moving from a higher orbit, with radius , to a lower orbit, . The energy of this photon is the difference in the energies and of the electron:

Since Planck's equation shows that the photon's energy is related to its wavelength by , the wavelengths of light that can be emitted are given by

This equation has the same form as the Rydberg formula, and predicts that the constant should be given by

Therefore the Bohr model of the atom can predict the emission spectrum of hydrogen in terms of fundamental constants.

[The model can be easily modified to account of the emission spectrum of any system consisting of a nucleus and a single electron (that is, ion]An ion is an atom or molecule in which the total number of electrons is not equal to the total number of protons, giving it a net positive or negative electrical charge. The name was given by physicist Michael Faraday for the substances that allow a current to pass between electrodes in a...

s such as He^{+} or O^{7+} which contain only one electron). However, it was not able to make accurate predictions for multi-electron atoms, or to explain why some spectral lines are brighter than others.

## Wave-particle duality

In 1924, Louis de Broglie proposed the idea that just as light has both wave-like and particle-like properties,

matter also has wave-like propertiesIn quantum mechanics, a matter wave or de Broglie wave is the wave of matter. The de Broglie relations show that the wavelength is inversely proportional to the momentum of a particle and that the frequency is directly proportional to the particle's kinetic energy...

.

The wavelength,

*λ*, associated with a particle is related to its momentum,

*p*:

The relationship, called the de Broglie hypothesis, holds for all types of matter. Thus all matter exhibits properties of both particles and waves.

Three years later, the wave-like nature of electrons was demonstrated by showing that a beam of electrons could exhibit

diffractionDiffraction refers to various phenomena which occur when a wave encounters an obstacle. Italian scientist Francesco Maria Grimaldi coined the word "diffraction" and was the first to record accurate observations of the phenomenon in 1665...

, just like a beam of light. At the

University of AberdeenThe University of Aberdeen, an ancient university founded in 1495, in Aberdeen, Scotland, is a British university. It is the third oldest university in Scotland, and the fifth oldest in the United Kingdom and wider English-speaking world...

,

George ThomsonSir George Paget Thomson, FRS was an English physicist and Nobel laureate in physics recognised for his discovery with Clinton Davisson of the wave properties of the electron by electron diffraction.-Biography:...

passed a beam of electrons through a thin metal film and observed the predicted diffraction patterns. At

Bell LabsBell Laboratories is the research and development subsidiary of the French-owned Alcatel-Lucent and previously of the American Telephone & Telegraph Company , half-owned through its Western Electric manufacturing subsidiary.Bell Laboratories operates its...

, Davisson and Germer guided their beam through a crystalline grid. Similar wave-like phenomena were later shown for atoms and even small molecules. De Broglie was awarded the Nobel Prize for Physics in 1929 for his hypothesis; Thomson and Davisson shared the Nobel Prize for Physics in 1937 for their experimental work.

The concept of wave-particle duality says that neither the classical concept of "particle" nor of "wave" can fully describe the behavior of quantum-scale objects, either photons or matter. Indeed, astrophysicist

A.S. EddingtonSir Arthur Stanley Eddington, OM, FRS was a British astrophysicist of the early 20th century. He was also a philosopher of science and a popularizer of science...

proposed in 1927 that "We can scarcely describe such an entity as a wave or as a particle; perhaps as a compromise we had better call it a 'wavicle' ". (This term was later popularised by mathematician

Banesh HoffmannBanesh Hoffmann was a British mathematician and physicist known for his association with Albert Einstein.-Life:Banesh Hoffmann was born in Richmond, England, on 6 September 1906...

.) Wave-particle duality is an example of the

principle of complementarityIn physics, complementarity is a basic principle of quantum theory proposed by Niels Bohr, closely identified with the Copenhagen interpretation, and refers to effects such as the wave–particle duality...

in quantum physics. An elegant example of wave-particle duality, the double slit experiment, is discussed in the section below.

De Broglie's treatment of quantum events served as a jumping off point for Schrödinger when he set about to construct a wave equation to describe quantum theoretical events.

### The double-slit experiment

In the double-slit experiment as originally performed by

Thomas YoungThomas Young was an English polymath. He is famous for having partly deciphered Egyptian hieroglyphics before Jean-François Champollion eventually expanded on his work...

and Augustin Fresnel in 1827, a beam of light is directed through two narrow, closely spaced slits, producing an interference pattern of light and dark bands on a screen. If one of the slits is covered up, one might naively expect that the intensity of the fringes due to interference would be halved everywhere. In fact, a much simpler pattern is seen, a simple

diffraction patternDiffraction refers to various phenomena which occur when a wave encounters an obstacle. Italian scientist Francesco Maria Grimaldi coined the word "diffraction" and was the first to record accurate observations of the phenomenon in 1665...

. Closing one slit results in a much simpler pattern diametrically opposite the open slit. Exactly the same behaviour can be demonstrated in water waves, and so the double-slit experiment was seen as a demonstration of the wave nature of light.

The double-slit experiment has also been performed using electrons, atoms, and even molecules, and the same type of interference pattern is seen. Thus it has been demonstrated that all matter possesses both particle and wave characteristics.

Even if the source intensity is turned down so that only one particle (e.g. photon or electron) is passing through the apparatus at a time, the same interference pattern develops over time. The quantum particle acts as a wave when passing through the double slits, but as a particle when it is detected. This is a typical feature of quantum complementarity: a quantum particle will act as a wave when we do an experiment to measure its wave-like properties, and like a particle when we do an experiment to measure its particle-like properties. Where on the detector screen any individual particle shows up will be the result of an entirely random process.

### Application to the Bohr model

De Broglie expanded the Bohr 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

electronThe electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...

will be observed only in situations that permit a

standing waveIn physics, a standing wave – also known as a stationary wave – is a wave that remains in a constant position.This phenomenon can occur because the medium is moving in the opposite direction to the wave, or it can arise in a stationary medium as a result of interference between two waves traveling...

around a

nucleusThe nucleus is the very dense region consisting of protons and neutrons at the center of an atom. It was discovered in 1911, as a result of Ernest Rutherford's interpretation of the famous 1909 Rutherford experiment performed by Hans Geiger and Ernest Marsden, under the direction of Rutherford. The...

. 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 2

*l*/

*n*, where

*l* is the length and

*n* 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.

## Development of modern quantum mechanics

In 1925, building on de Broglie's hypothesis,

Erwin SchrödingerErwin Rudolf Josef Alexander Schrödinger was an Austrian physicist and theoretical biologist who was one of the fathers of quantum mechanics, and is famed for a number of important contributions to physics, especially the Schrödinger equation, for which he received the Nobel Prize in Physics in 1933...

developed the equation that describes the behaviour of a quantum mechanical wave. The equation, called the

Schrödinger equationThe Schrödinger equation was formulated in 1926 by Austrian physicist Erwin Schrödinger. Used in physics , it is an equation that describes how the quantum state of a physical system changes in time....

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. In the paper that introduced

Schrödinger's catSchrödinger's cat is a thought experiment, usually described as a paradox, devised by Austrian physicist Erwin Schrödinger in 1935. It illustrates what he saw as the problem of the Copenhagen interpretation of quantum mechanics applied to everyday objects. The scenario presents a cat that might be...

, he says that the psi-function featured in his equation provides the "means for predicting probability of measurement results," and that it therefore provides "future expectation[s] , somewhat as laid down in a

*catalog*."

Schrödinger was able to calculate the energy levels of hydrogen by treating a hydrogen atom's

electronThe electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...

as a classical wave, moving in a well of electrical potential created by the proton. This calculation accurately reproduced the energy levels of the Bohr model.

At a somewhat earlier time,

Werner HeisenbergWerner Karl Heisenberg was a German theoretical physicist who made foundational contributions to quantum mechanics and is best known for asserting the uncertainty principle of quantum theory...

was trying to find an explanation for the intensities of the different lines in the hydrogen emission spectrum. By means of a series of mathematical analogies, Heisenberg wrote out the quantum mechanical analogue for the classical computation of intensities. Shortly afterwards, Heisenberg's colleague

Max BornMax Born was a German-born physicist and mathematician who was instrumental in the development of quantum mechanics. He also made contributions to solid-state physics and optics and supervised the work of a number of notable physicists in the 1920s and 30s...

realised that Heisenberg's method of calculating the probabilities for transitions between the different energy levels could best be expressed by using the mathematical concept of

matricesIn mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...

.

[For a somewhat more sophisticated look at how Heisenberg transitioned from the old quantum theory and classical physics to the new quantum mechanics, see Heisenberg's entryway to matrix mechanics]Werner Heisenberg contributed to science at the point that the old quantum physics was discovering the field littered with more and more stumbling blocks. He decided that quantum physics had to be re-thought from the ground up. In so doing he excised several items that were grounded in classical...

.
In May 1926, Schrödinger proved that Heisenberg's

matrix mechanicsMatrix mechanics is a formulation of quantum mechanics created by Werner Heisenberg, Max Born, and Pascual Jordan in 1925.Matrix mechanics was the first conceptually autonomous and logically consistent formulation of quantum mechanics. It extended the Bohr Model by describing how the quantum jumps...

and his own

wave mechanicsThe Schrödinger equation was formulated in 1926 by Austrian physicist Erwin Schrödinger. Used in physics , it is an equation that describes how the quantum state of a physical system changes in time....

made the same predictions about the properties and behaviour of the electron; mathematically, the two theories were identical. Yet the two men disagreed on the interpretation of their mutual theory. For instance, Heisenberg saw no problem in the theoretical prediction of instantaneous transitions of electrons between orbits in an atom, but Schrödinger hoped that a theory based on continuous wave-like properties could avoid what he called (in the words of

Wilhelm WienWilhelm Carl Werner Otto Fritz Franz Wien was a German physicist who, in 1893, used theories about heat and electromagnetism to deduce Wien's displacement law, which calculates the emission of a blackbody at any temperature from the emission at any one reference temperature.He also formulated an...

) "this nonsense about quantum jumps."

## Copenhagen interpretation

Bohr, Heisenberg and others tried to explain what these experimental results and mathematical models really mean. Their description, known as the Copenhagen interpretation of quantum mechanics, aimed to describe the nature of reality that was being probed by the measurements and described by the mathematical formulations of quantum mechanics.

The main principles of the Copenhagen interpretation are:

- A system is completely described by a wave function, . (Heisenberg)
- How changes over time is given by the Schrödinger equation.
- The description of nature is essentially probabilistic. The probability of an event — for example, where on the screen a particle will show up in the two slit experiment — is related to the square of the amplitude of its wave function. (Born rule
The Born rule is a law of quantum mechanics which gives the probability that a measurement on a quantum system will yield a given result. It is named after its originator, the physicist Max Born. The Born rule is one of the key principles of quantum mechanics...

, due to Max BornMax Born was a German-born physicist and mathematician who was instrumental in the development of quantum mechanics. He also made contributions to solid-state physics and optics and supervised the work of a number of notable physicists in the 1920s and 30s...

, which gives a physical meaning to the wavefunction in the Copenhagen interpretation: the probability amplitudeIn quantum mechanics, a probability amplitude is a complex number whose modulus squared represents a probability or probability density.For example, if the probability amplitude of a quantum state is \alpha, the probability of measuring that state is |\alpha|^2...

)
- It is not possible to know the values of all of the properties of the system at the same time; those properties that are not known with precision must be described by probabilities. (Heisenberg's uncertainty principle
In quantum mechanics, the Heisenberg uncertainty principle states a fundamental limit on the accuracy with which certain pairs of physical properties of a particle, such as position and momentum, can be simultaneously known...

)
- Matter, like energy, 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)
- Measuring devices are essentially classical devices, and measure classical properties such as position and momentum.
- The quantum mechanical description of large systems should closely approximate the classical description. (Correspondence principle
In physics, the correspondence principle states that the behavior of systems described by the theory of quantum mechanics reproduces classical physics in the limit of large quantum numbers....

of Bohr and Heisenberg)

Various consequences of these principles are discussed in more detail in the following subsections.

### Uncertainty principle

Suppose that we want to measure the position and speed of an object — for example a car going through a radar speed trap. Naively, we assume that the car has a definite position and speed at a particular moment in time, and how accurately we can measure these values depends on the quality of our measuring equipment — if we improve the precision of our measuring equipment, we will get a result that is closer to the true value. In particular, we would assume that how precisely we measure the speed of the car does not affect its position, and vice versa.

In 1927, Heisenberg proved that these assumptions are not correct. Quantum mechanics shows that certain pairs of physical properties, like position and speed, cannot both be known to arbitrary precision: the more precisely one property is known, the less precisely the other can be known. This statement is known as the

uncertainty principleIn quantum mechanics, the Heisenberg uncertainty principle states a fundamental limit on the accuracy with which certain pairs of physical properties of a particle, such as position and momentum, can be simultaneously known...

. The uncertainty principle isn't a statement about the accuracy of our measuring equipment, but about the nature of the system itself — our naive assumption that the car had a definite position and speed was incorrect. On a scale of cars and people, these uncertainties are too small to notice, but when dealing with atoms and electrons they become critical.

Heisenberg gave, as an illustration, the measurement of the position and momentum of an electron using a photon of light. In measuring the electron's position, the higher the frequency of the photon the more accurate is the measurement of the position of the impact, but the greater is the disturbance of the electron, which absorbs a random amount of energy, rendering the measurement obtained of its

momentumIn classical mechanics, linear momentum or translational momentum is the product of the mass and velocity of an object...

increasingly uncertain (momentum is velocity multiplied by mass), for one is necessarily measuring its post-impact disturbed momentum, from the collision products, not its original momentum. With a photon of lower frequency the disturbance - hence uncertainty - in the momentum is less, but so is the accuracy of the measurement of the position of the impact.

The uncertainty principle shows mathematically that the product of the uncertainty in the position and

momentumIn classical mechanics, linear momentum or translational momentum is the product of the mass and velocity of an object...

of a particle (momentum is velocity multiplied by mass) could never be less than a certain value, and that this value is related to Planck's constant.

### Wave function collapse

Wave function collapse is a forced term for whatever happened when it becomes appropriate to replace the description of an uncertain state of a system by a description of the system in a definite state. Explanations for the nature of the process of becoming certain are controversial. At any time before a photon "shows up" on a detection screen it can only be described by a set of probabilities for where it might show up. When it does show up, for instance in the

CCDA charge-coupled device is a device for the movement of electrical charge, usually from within the device to an area where the charge can be manipulated, for example conversion into a digital value. This is achieved by "shifting" the signals between stages within the device one at a time...

of an electronic camera, the time and the space where it interacted with the device are known within very tight limits. However, the photon has disappeared, and the wave function has disappeared with it. In its place some physical change in the detection screen has appeared, e.g., an exposed spot in a sheet of photographic film.

### Eigenstates and eigenvalues

*For a more detailed introduction to this subject, see: Introduction to eigenstates*Because of the uncertainty principle, statements about both the position and momentum of particles can only assign a probability that the position or momentum will have some numerical value. The uncertainty principle also says that eliminating uncertainty about position maximizes uncertainty about...

Because of the

uncertainty principleIn quantum mechanics, the Heisenberg uncertainty principle states a fundamental limit on the accuracy with which certain pairs of physical properties of a particle, such as position and momentum, can be simultaneously known...

, statements about both the position and momentum of particles can only assign a

probabilityProbability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...

that the position or momentum will have some numerical value. Therefore it is necessary to formulate clearly the difference between the state of something that is indeterminate, such as an electron in a probability cloud, and the state of something having a definite value. When an object can definitely be "pinned-down" in some respect, it is said to possess an eigenstate.

### The Pauli exclusion principle

In 1924,

Wolfgang PauliWolfgang Ernst Pauli was an Austrian theoretical physicist and one of the pioneers of quantum physics. In 1945, after being nominated by Albert Einstein, he received the Nobel Prize in Physics for his "decisive contribution through his discovery of a new law of Nature, the exclusion principle or...

proposed a new quantum degree of freedom (or

quantum numberQuantum numbers describe values of conserved quantities in the dynamics of the quantum system. Perhaps the most peculiar aspect of quantum mechanics is the quantization of observable quantities. This is distinguished from classical mechanics where the values can range continuously...

), with two possible values, to resolve inconsistencies between observed molecular spectra and the predictions of quantum mechanics. In particular, the spectrum of atomic hydrogen had a

doubletDoublet may refer to:*Doublet , a man's snug-fitting buttoned jacket that was worn from the late 14th century to the mid 17th century*Doublet , an assembled gem composed in two sections, such as a garnet overlaying green glass...

, or pair of lines differing by a small amount, where only one line was expected. Pauli formulated his

*exclusion principle*, stating that "There cannot exist an atom in such a quantum state that two electrons within [it] have the same set of quantum numbers."

A year later,

UhlenbeckGeorge Eugene Uhlenbeck was a Dutch-American theoretical physicist.-Background and education:George Uhlenbeck was the son of Eugenius and Anne Beeger Uhlenbeck...

and Goudsmit identified Pauli's new degree of freedom with a property called

spinIn quantum mechanics and particle physics, spin is a fundamental characteristic property of elementary particles, composite particles , and atomic nuclei.It is worth noting that the intrinsic property of subatomic particles called spin and discussed in this article, is related in some small ways,...

. The idea, originating with

Ralph KronigRalph Kronig was a German-American physicist . He is noted for the discovery of particle spin and for his theory of x-ray absorption spectroscopy...

, was that electrons behave as if they rotate, or "spin", about an axis. Spin would account for the missing

magnetic momentThe magnetic moment of a magnet is a quantity that determines the force that the magnet can exert on electric currents and the torque that a magnetic field will exert on it...

, and allow two electrons in the same orbital to occupy distinct quantum states if they, "spun" in opposite directions, thus satisfying the exclusion principle. The quantum number represented the sense (positive or negative) of spin.

### Application to the hydrogen atom

Bohr's model of the atom was essentially two-dimensional — an electron orbiting in a plane around its nuclear "sun." However, the uncertainty principle states that an electron cannot be viewed as having an exact location at any given time. In the modern theory the orbit has been replaced by an

*atomic orbital*An atomic orbital is a mathematical function that describes the wave-like behavior of either one electron or a pair of electrons in an atom. This function can be used to calculate the probability of finding any electron of an atom in any specific region around the atom's nucleus...

, a "cloud" of possible locations. It is often depicted as a three-dimensional region within which there is a 95 percent probability of finding the electron.

Schrödinger was able to calculate the energy levels of hydrogen by treating a hydrogen atom's

electronThe electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...

as a wave, represented by the "wave function" , in a

electric potentialIn classical electromagnetism, the electric potential at a point within a defined space is equal to the electric potential energy at that location divided by the charge there...

wellA potential well is the region surrounding a local minimum of potential energy. Energy captured in a potential well is unable to convert to another type of energy because it is captured in the local minimum of a potential well...

, , created by the proton. The solutions to Schrödinger's equation 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 reproduce the energy levels of the Bohr model.

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

- 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;
- The "shape" of the orbital, spherical or otherwise;
- The "inclination" of the orbital, determining the magnetic moment
The magnetic moment of a magnet is a quantity that determines the force that the magnet can exert on electric currents and the torque that a magnetic field will exert on it...

of the orbital around the -axis.
- The "spin" of the electron.

The collective name for these properties is the

quantum 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's quantum numbers. The quantum state of the electron is described by its wavefunction. 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 the

principal quantum numberIn atomic physics, the principal quantum symbolized as n is the firstof a set of quantum numbers of an atomic orbital. The principal quantum number can only have positive integer values...

, , which is the same as in Bohr's model. denotes the energy level of each orbital. The possible values for are integers:

The next quantum number, the

azimuthal quantum numberThe azimuthal quantum number is a quantum number for an atomic orbital that determines its orbital angular momentum and describes the shape of the orbital...

, denoted , describes the shape of the orbital. The shape is a consequence of the

angular momentumIn physics, angular momentum, moment of momentum, or rotational momentum is a conserved vector quantity that can be used to describe the overall state of a physical system...

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 for are integers from 0 to :

The shape of each orbital has its own letter as well. The first shape is denoted by the letter (a

mnemonicA mnemonic , or mnemonic device, is any learning technique that aids memory. To improve long term memory, mnemonic systems are used to make memorization easier. Commonly encountered mnemonics are often verbal, such as a very short poem or a special word used to help a person remember something,...

being "

*s*phere"). The next shape is denoted by the letter and has the form of a dumbbell. The other orbitals have more complicated shapes (see

atomic orbitalAn atomic orbital is a mathematical function that describes the wave-like behavior of either one electron or a pair of electrons in an atom. This function can be used to calculate the probability of finding any electron of an atom in any specific region around the atom's nucleus...

), and are denoted by the letters , , and .

The third quantum number, the

magnetic quantum numberIn atomic physics, the magnetic quantum number is the third of a set of quantum numbers which describe the unique quantum state of an electron and is designated by the letter m...

, describes the

magnetic momentThe magnetic moment of a magnet is a quantity that determines the force that the magnet can exert on electric currents and the torque that a magnetic field will exert on it...

of the electron, and is denoted by (or simply

*m*). The possible values for are integers from to :

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, the

spin quantum numberIn atomic physics, the spin quantum number is a quantum number that parameterizes the intrinsic angular momentum of a given particle...

(pertaining to the "orientation" of the electron's spin) is denoted , with values + or −.

The chemist

Linus PaulingLinus Carl Pauling was an American chemist, biochemist, peace activist, author, and educator. He was one of the most influential chemists in history and ranks among the most important scientists of the 20th century...

wrote, by way of example:

It is the underlying structure and symmetry of atomic orbitals, and the way that electrons fill them, that determines the organisation of the

periodic tableThe periodic table of the chemical elements is a tabular display of the 118 known chemical elements organized by selected properties of their atomic structures. Elements are presented by increasing atomic number, the number of protons in an atom's atomic nucleus...

and the structure and strength of chemical bonds between atoms.

## Dirac wave equation

In 1928,

Paul DiracPaul Adrien Maurice Dirac, OM, FRS was an English theoretical physicist who made fundamental contributions to the early development of both quantum mechanics and quantum electrodynamics...

extended the

Pauli equationThe Pauli equation, also known as the Schrödinger–Pauli equation, is the formulation of the Schrödinger equation for spin- particles which takes into account the interaction of the particle's spin with the electromagnetic field...

, which described spinning electrons, to account for

special relativitySpecial relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...

. The result was a theory that dealt properly with events, such as the speed at which an electron orbits the nucleus, occurring at a substantial fraction of the

speed of lightThe speed of light in vacuum, usually denoted by c, is a physical constant important in many areas of physics. Its value is 299,792,458 metres per second, a figure that is exact since the length of the metre is defined from this constant and the international standard for time...

. By using the simplest electromagnetic interaction, Dirac was able to predict the value of the magnetic moment associated with the electron's spin, and found the experimentally observed value, which was too large to be that of a spinning charged sphere governed by

classical physicsWhat "classical physics" refers to depends on the context. When discussing special relativity, it refers to the Newtonian physics which preceded relativity, i.e. the branches of physics based on principles developed before the rise of relativity and quantum mechanics...

. He was able to solve for the spectral lines of the hydrogen atom, and to reproduce from physical first principles

SommerfeldArnold Johannes Wilhelm Sommerfeld was a German theoretical physicist who pioneered developments in atomic and quantum physics, and also educated and groomed a large number of students for the new era of theoretical physics...

's successful formula for the

fine structureIn atomic physics, the fine structure describes the splitting of the spectral lines of atoms due to first order relativistic corrections.The gross structure of line spectra is the line spectra predicted by non-relativistic electrons with no spin. For a hydrogenic atom, the gross structure energy...

of the hydrogen spectrum.

Dirac's equations sometimes yielded a negative value for energy, for which he proposed a novel solution: he posited the existence of an antielectron and of a dynamical vacuum. This led to the many-particle

quantum field theoryQuantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...

.

## Quantum entanglement

The Pauli exclusion principle says that two electrons in one system cannot be in the same state. Nature leaves open the possibility, however, that two electrons can have both states "superimposed" over them. Recall that the wave functions that emerge simultaneously from the double slits arrive at the detection screen in a state of superposition. Nothing is certain until the superimposed waveforms "collapse," At that instant an electron shows up somewhere in accordance with the probabilities that are the squares of the amplitudes of the two superimposed waveforms. The situation there is already very abstract. A concrete way of thinking about entangled photons, photons in which two contrary states are superimposed on each of them in the same event, is as follows:

Imagine that the superposition of a state that can be mentally

**labeled** as blue and another state that can be mentally labeled as red will then appear (in imagination, of course) as a purple state. Two photons are produced as the result of the same atomic event. Perhaps they are produced by the excitation of a crystal that characteristically absorbs a photon of a certain frequency and emits two photons of half the original frequency. So the two photons come out "purple." If the experimenter now performs some experiment that will determine whether one of the photons is either blue or red, then that experiment changes the photon involved from one having a superposition of "blue" and "red" characteristics to a photon that has only one of those characteristics. The problem that Einstein had with such an imagined situation was that if one of these photons had been kept bouncing between mirrors in a laboratory on earth, and the other one had traveled halfway to the nearest star, when its twin was made to reveal itself as either blue or red, that meant that the distant photon now had to lose its "purple" status too. So whenever it might be investigated, it would necessarily show up, instantaneously, in the opposite state to whatever its twin had revealed.

In trying to show that quantum mechanics was not a complete theory, Einstein started with the theory's prediction that two or more particles that have interacted in the past can appear strongly correlated when their various properties are later measured. He sought to explain this seeming interaction in a classical way, through their common past, and preferably not by some "spooky action at a distance." The argument is worked out in a famous paper, Einstein, Podolsky, and Rosen (1935; abbreviated EPR), setting out what is now called the

EPR paradoxThe EPR paradox is a topic in quantum physics and the philosophy of science concerning the measurement and description of microscopic systems by the methods of quantum physics...

. Assuming what is now usually called local realism, EPR attempted to show from quantum theory that a particle has both position and momentum simultaneously, while according to the

Copenhagen interpretationThe Copenhagen interpretation is one of the earliest and most commonly taught interpretations of quantum mechanics. It holds that quantum mechanics does not yield a description of an objective reality but deals only with probabilities of observing, or measuring, various aspects of energy quanta,...

, only one of those two properties actually exists and only at the moment that it is being measured. EPR concluded that quantum theory is incomplete in that it refuses to consider physical properties which objectively exist in nature. (Einstein, Podolsky, & Rosen 1935 is currently Einstein's most cited publication in physics journals.) In the same year,

Erwin SchrödingerErwin Rudolf Josef Alexander Schrödinger was an Austrian physicist and theoretical biologist who was one of the fathers of quantum mechanics, and is famed for a number of important contributions to physics, especially the Schrödinger equation, for which he received the Nobel Prize in Physics in 1933...

used the word "entanglement" and declared: "I would not call that

*one* but rather

*the* characteristic trait of quantum mechanics." The question of whether entanglement is a real condition is still in dispute. The Bell inequalities are the most powerful challenge to Einstein's claims.

## Quantum field theory

The idea of quantum field theory began in the late 1920s with British physicist

Paul DiracPaul Adrien Maurice Dirac, OM, FRS was an English theoretical physicist who made fundamental contributions to the early development of both quantum mechanics and quantum electrodynamics...

, when he attempted to

quantiseIn physics, quantization is the process of explaining a classical understanding of physical phenomena in terms of a newer understanding known as "quantum mechanics". It is a procedure for constructing a quantum field theory starting from a classical field theory. This is a generalization of the...

the

electromagnetic fieldAn electromagnetic field is a physical field produced by moving electrically charged objects. It affects the behavior of charged objects in the vicinity of the field. The electromagnetic field extends indefinitely throughout space and describes the electromagnetic interaction...

— a procedure for constructing a quantum theory starting from a classical theory.

A

*field* in physics is "a region or space in which a given effect (such as

magnetismMagnetism is a property of materials that respond at an atomic or subatomic level to an applied magnetic field. Ferromagnetism is the strongest and most familiar type of magnetism. It is responsible for the behavior of permanent magnets, which produce their own persistent magnetic fields, as well...

) exists." Other effects that manifest themselves as fields are

gravitationGravitation, or gravity, is a natural phenomenon by which physical bodies attract with a force proportional to their mass. Gravitation is most familiar as the agent that gives weight to objects with mass and causes them to fall to the ground when dropped...

and

static electricityStatic electricity refers to the build-up of electric charge on the surface of objects. The static charges remain on an object until they either bleed off to ground or are quickly neutralized by a discharge. Static electricity can be contrasted with current electricity, which can be delivered...

. In 2008, physicist

Richard HammondRichard Hammond is an Adjunct Professor at the University of North Carolina at Chapel Hill and the author of the book "The Unknown Universe: The Origin of the Universe, Quantum Gravity, Wormholes, and Other Things Science Still Can't Explain". He also works for the United States Army Research...

wrote that

Sometimes we distinguish between quantum mechanics (QM) and quantum field theory (QFT). QM refers to a system in which the number of particles is fixed, and the fields (such as the electromechanical field) are continuous classical entities. QFT . . . goes a step further and allows for the creation and annihilation of particles . . . .

He added, however, that

*quantum mechanics* is often used to refer to "the entire notion of quantum view."

In 1931, Dirac proposed the existence of particles that later became known as anti-matter. Dirac shared the

Nobel Prize in physicsThe Nobel Prize in Physics is awarded once a year by the Royal Swedish Academy of Sciences. It is one of the five Nobel Prizes established by the will of Alfred Nobel in 1895 and awarded since 1901; the others are the Nobel Prize in Chemistry, Nobel Prize in Literature, Nobel Peace Prize, and...

for 1933 with

SchrödingerErwin Rudolf Josef Alexander Schrödinger was an Austrian physicist and theoretical biologist who was one of the fathers of quantum mechanics, and is famed for a number of important contributions to physics, especially the Schrödinger equation, for which he received the Nobel Prize in Physics in 1933...

, "for the discovery of new productive forms of

atomic theoryIn chemistry and physics, atomic theory is a theory of the nature of matter, which states that matter is composed of discrete units called atoms, as opposed to the obsolete notion that matter could be divided into any arbitrarily small quantity...

."

## Quantum electrodynamics

Quantum electrodynamics (QED) is the name of the quantum theory of the electromagnetic force. Understanding QED begins with understanding

electromagnetismElectromagnetism is one of the four fundamental interactions in nature. The other three are the strong interaction, the weak interaction and gravitation...

. Electromagnetism can be called "electrodynamics" because it is a dynamic interaction between electrical and magnetic forces. Electromagnetism begins with the

electric chargeElectric charge is a physical property of matter that causes it to experience a force when near other electrically charged matter. Electric charge comes in two types, called positive and negative. Two positively charged substances, or objects, experience a mutual repulsive force, as do two...

.

Electric charges are the sources of, and create, electric fields. An electric field is a field which exerts a force on any particles that carry electric charges, at any point in space. This includes the electron, proton, and even quarks, among others. As a force is exerted, electric charges move, a current flows and a magnetic field is produced. The magnetic field, in turn causes

electric currentElectric current is a flow of electric charge through a medium.This charge is typically carried by moving electrons in a conductor such as wire...

(moving electrons). The interacting electric and magnetic field is called an

electromagnetic fieldAn electromagnetic field is a physical field produced by moving electrically charged objects. It affects the behavior of charged objects in the vicinity of the field. The electromagnetic field extends indefinitely throughout space and describes the electromagnetic interaction...

.

The physical description of interacting

charged particleIn physics, a charged particle is a particle with an electric charge. It may be either a subatomic particle or an ion. A collection of charged particles, or even a gas containing a proportion of charged particles, is called a plasma, which is called the fourth state of matter because its...

s, electrical currents, electrical fields, and magnetic fields is called electromagnetism.

In 1928

Paul DiracPaul Adrien Maurice Dirac, OM, FRS was an English theoretical physicist who made fundamental contributions to the early development of both quantum mechanics and quantum electrodynamics...

produced a relativistic quantum theory of electromagnetism. This was the progenitor to modern quantum electrodynamics, in that it had essential ingredients of the modern theory. However, the problem of unsolvable infinities developed in this relativistic quantum theory. Years later,

renormalizationIn quantum field theory, the statistical mechanics of fields, and the theory of self-similar geometric structures, renormalization is any of a collection of techniques used to treat infinities arising in calculated quantities....

solved this problem. Initially viewed as a suspect, provisional procedure by some of its originators, renormalization eventually was embraced as an important and self-consistent tool in QED and other fields of physics. Also, in the late 1940s

Feynman's diagramsFeynman diagrams are a pictorial representation scheme for the mathematical expressions governing the behavior of subatomic particles, first developed by the Nobel Prize-winning American physicist Richard Feynman, and first introduced in 1948...

depicted all possible interactions pertaining to a given event. The diagrams showed that the electromagnetic force is the interactions of photons between interacting particles.

An example of a prediction of quantum electrodynamics which has been verified experimentally is the

Lamb shift. This refers to an effect whereby the quantum nature of the electromagnetic field causes the energy levels in an atom or ion to deviate slightly from what they would otherwise be. As a result, spectral lines may shift or split.

In the 1960s

physicistA physicist is a scientist who studies or practices physics. Physicists study a wide range of physical phenomena in many branches of physics spanning all length scales: from sub-atomic particles of which all ordinary matter is made to the behavior of the material Universe as a whole...

s realized that QED broke down at extremely high energies. From this inconsistency the

Standard ModelThe Standard Model of particle physics is a theory concerning the electromagnetic, weak, and strong nuclear interactions, which mediate the dynamics of the known subatomic particles. Developed throughout the mid to late 20th century, the current formulation was finalized in the mid 1970s upon...

of particle physics was discovered, which remedied the higher energy breakdown in theory. The

Standard ModelThe Standard Model of particle physics is a theory concerning the electromagnetic, weak, and strong nuclear interactions, which mediate the dynamics of the known subatomic particles. Developed throughout the mid to late 20th century, the current formulation was finalized in the mid 1970s upon...

unifies the electromagnetic and

weak interactionWeak interaction , is one of the four fundamental forces of nature, alongside the strong nuclear force, electromagnetism, and gravity. It is responsible for the radioactive decay of subatomic particles and initiates the process known as hydrogen fusion in stars...

s into one theory. This is called the electroweak theory.

## Interpretations

The physical measurements, equations, and predictions pertinent to quantum mechanics are all consistent and hold a very high level of confirmation. However, the question of what these abstract models say about the underlying nature of the real world has received competing answers.

## Applications

Applications of quantum mechanics include the

laserA laser is a device that emits light through a process of optical amplification based on the stimulated emission of photons. The term "laser" originated as an acronym for Light Amplification by Stimulated Emission of Radiation...

, the

transistorA transistor is a semiconductor device used to amplify and switch electronic signals and power. It is composed of a semiconductor material with at least three terminals for connection to an external circuit. A voltage or current applied to one pair of the transistor's terminals changes the current...

, the

electron microscopeAn electron microscope is a type of microscope that uses a beam of electrons to illuminate the specimen and produce a magnified image. Electron microscopes have a greater resolving power than a light-powered optical microscope, because electrons have wavelengths about 100,000 times shorter than...

, and

magnetic resonance imagingMagnetic resonance imaging , nuclear magnetic resonance imaging , or magnetic resonance tomography is a medical imaging technique used in radiology to visualize detailed internal structures...

. The study of semiconductors led to the invention of the

diodeIn electronics, a diode is a type of two-terminal electronic component with a nonlinear current–voltage characteristic. A semiconductor diode, the most common type today, is a crystalline piece of semiconductor material connected to two electrical terminals...

and the

transistorA transistor is a semiconductor device used to amplify and switch electronic signals and power. It is composed of a semiconductor material with at least three terminals for connection to an external circuit. A voltage or current applied to one pair of the transistor's terminals changes the current...

, which are indispensable for modern

electronicsElectronics is the branch of science, engineering and technology that deals with electrical circuits involving active electrical components such as vacuum tubes, transistors, diodes and integrated circuits, and associated passive interconnection technologies...

.

In even the simple

light switchA light switch is a switch, most commonly used to operate electric lights, permanently connected equipment, or electrical outlets. In torches the switch is often near the bulb, but may be in the tail, or even the entire head itself may constitute the switch .-Wall-mounted switches:Switches for...

, quantum tunnelling is vital, as otherwise the electrons in the

electric currentElectric current is a flow of electric charge through a medium.This charge is typically carried by moving electrons in a conductor such as wire...

could not penetrate the potential barrier made up of a layer of oxide.

Flash memoryFlash memory is a non-volatile computer storage chip that can be electrically erased and reprogrammed. It was developed from EEPROM and must be erased in fairly large blocks before these can be rewritten with new data...

chips found in USB drives also use quantum tunnelling, to erase their memory cells.

## See also

- Heisenberg's entryway to matrix mechanics
Werner Heisenberg contributed to science at the point that the old quantum physics was discovering the field littered with more and more stumbling blocks. He decided that quantum physics had to be re-thought from the ground up. In so doing he excised several items that were grounded in classical...

- Orbital:
- Atomic
An atomic orbital is a mathematical function that describes the wave-like behavior of either one electron or a pair of electrons in an atom. This function can be used to calculate the probability of finding any electron of an atom in any specific region around the atom's nucleus...

- Molecular
In chemistry, a molecular orbital is a mathematical function describing the wave-like behavior of an electron in a molecule. This function can be used to calculate chemical and physical properties such as the probability of finding an electron in any specific region. The term "orbital" was first...

- P-adic quantum mechanics
P-adic quantum mechanics is a relatively recent approach to understanding the nature of fundamental physics. It is the application of p-adic analysis to quantum mechanics. The p-adic numbers are a counterintuitive arithmetic system that was discovered by the German mathematician Kurt Hensel in...

- Philosophy of physics
In philosophy, the philosophy of physics studies the fundamental philosophical questions underlying modern physics, the study of matter and energy and how they interact. The philosophy of physics begins by reflecting on the basic metaphysical and epistemological questions posed by physics:...

- Physicists:
- Markus Fierz
Markus Eduard Fierz was a Swiss physicist, particularly remembered for his formulation of Spin-statistics theorem, and for his contributions to the development of quantum theory, particle physics, and statistical mechanics...

- Quantum computer
A quantum computer is a device for computation that makes direct use of quantum mechanical phenomena, such as superposition and entanglement, to perform operations on data. Quantum computers are different from traditional computers based on transistors...

- Quantum pseudo-telepathy
Quantum pseudo-telepathy is a phenomenon in quantum game theory resulting in anomalously high success rates in coordination games between separated players. These high success rates would require communication between the players in a purely classical world; however, the game is set up such that...

- Quantum Zeno effect
The quantum Zeno effect is a name coined by George Sudarshan and Baidyanath Misra of the University of Texas in 1977 in their analysis of the situation in which an unstable particle, if observed continuously, will never decay. One can nearly "freeze" the evolution of the system by measuring it...

- Virtual particle
In physics, a virtual particle is a particle that exists for a limited time and space. The energy and momentum of a virtual particle are uncertain according to the uncertainty principle...

## Further reading

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

- Jim Al-Khalili
Jim Al-Khalili OBE is an Iraqi-born British theoretical physicist, author and science communicator. He is Professor of Theoretical Physics and Chair in the Public Engagement in Science at the University of Surrey...

(2003) *Quantum: A Guide for the Perplexed*. Weidenfield & Nicholson.
- Richard Feynman
Richard Phillips Feynman was an American physicist known for his work in the path integral formulation of quantum mechanics, the theory of quantum electrodynamics and the physics of the superfluidity of supercooled liquid helium, as well as in particle physics...

(1985) *QED: The Strange Theory of Light and Matter*. Princeton University Press. ISBN 0-691-08388-6
- Ford, Kenneth (2005)
*The Quantum World*. Harvard Univ. Press. Includes elementary particle physics.
- 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 algebraAlgebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures...

, trigonometryTrigonometry is a branch of mathematics that studies triangles and the relationships between their sides and the angles between these sides. Trigonometry defines the trigonometric functions, which describe those relationships and have applicability to cyclical phenomena, such as waves...

, and bra-ket notationBra-ket notation is a standard notation for describing quantum states in the theory of quantum mechanics composed of angle brackets and vertical bars. It can also be used to denote abstract vectors and linear functionals in mathematics...

can be passed over on a first reading.
- Tony Hey
Anthony John Grenville Hey CBE FREng FIET FInstP FBCS is a researcher and educator across a range of science and engineering fields....

and Walters, Patrick (2003) *The New Quantum Universe*. Cambridge Univ. Press. Includes much about the technologies quantum theory has made possible.
- Vladimir G. Ivancevic, Tijana T. Ivancevic (2008)
*Quantum leap: from Dirac and Feynman, across the universe, to human body and mind*. World Scientific Publishing Company. Provides an intuitive introduction in non-mathematical terms and an introduction in comparatively basic mathematical terms.
- N. David Mermin (1990) “Spooky actions at a distance: mysteries of the QT” in his
*Boojums all the way through*. Cambridge Univ. Press: 110–176. The author is a rare physicist who tries to communicate to philosophers and humanists.
- Roland Omnes
Roland Omnès is the author of several books which aim to close the gap between our common sense experience of the classical world and the complex, formal mathematics which is now required to accurately describe reality at its most fundamental level.- Biography :Omnès is currently Professor...

(1999) *Understanding Quantum Mechanics*. Princeton Univ. Press.
- Victor Stenger (2000)
*Timeless Reality: Symmetry, Simplicity, and Multiple Universes*. Buffalo NY: Prometheus Books. Chpts. 5–8.
- Martinus Veltman (2003)
*Facts and Mysteries in Elementary Particle Physics*. World Scientific Publishing Company.
- A website with good introduction to Quantum mechanics can be found here.

## External links