Introduction to quantum mechanics
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Quantum mechanics is the study of
matter In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume. All everyday objects that can be touched are ultimately composed of atoms, which are made up of interacting subatomic part ...
and its interactions with
energy In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of hea ...
on the scale of
atom Every atom is composed of a nucleus and one or more electrons bound to the nucleus. The nucleus is made of one or more protons and a number of neutrons. Only the most common variety of hydrogen has no neutrons. Every solid, liquid, gas, ...
ic and
subatomic particles In physical sciences, a subatomic particle is a particle that composes an atom. According to the Standard Model of particle physics, a subatomic particle can be either a composite particle, which is composed of other particles (for example, a pro ...
. By contrast, classical physics explains matter and energy only on a scale familiar to human experience, including the behavior of astronomical bodies such as the moon. Classical physics is still used in much of modern science and technology. However, towards the end of the 19th century, scientists discovered phenomena in both the large ( macro) and the small ( micro) worlds that classical physics could not explain. The desire to resolve inconsistencies between observed phenomena and classical theory led to two major revolutions in physics that created a shift in the original
scientific paradigm In science and philosophy, a paradigm () is a distinct set of concepts or thought patterns, including theories, research methods, postulates, and standards for what constitute legitimate contributions to a field. Etymology ''Paradigm'' comes ...
: the ''
theory of relativity The theory of relativity usually encompasses two interrelated theories by Albert Einstein: special relativity and general relativity, proposed and published in 1905 and 1915, respectively. Special relativity applies to all physical phenomena in ...
'' and the development of ''quantum mechanics''. This article describes how physicists discovered the limitations of classical physics and developed the main concepts of the quantum theory that replaced it in the early decades of the 20th century. It describes these concepts in roughly the order in which they were first discovered. For a more complete history of the subject, see ''
History of quantum mechanics The history of quantum mechanics is a fundamental part of the history of modern physics. Quantum mechanics' history, as it interlaces with the history of quantum chemistry, began essentially with a number of different scientific discoveries: the ...
''. Light behaves in some aspects like particles and in other aspects like waves. Matter—the "stuff" of the universe consisting of particles such as
electron The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family, and are generally thought to be elementary particles because they have no ...
s and
atom Every atom is composed of a nucleus and one or more electrons bound to the nucleus. The nucleus is made of one or more protons and a number of neutrons. Only the most common variety of hydrogen has no neutrons. Every solid, liquid, gas, ...
s—exhibits wavelike behavior too. Some light sources, such as neon lights, give off only certain specific frequencies of light, a small set of distinct pure colors determined by neon's atomic structure. Quantum mechanics shows that light, along with all other forms of
electromagnetic radiation In physics, electromagnetic radiation (EMR) consists of waves of the electromagnetic (EM) field, which propagate through space and carry momentum and electromagnetic radiant energy. It includes radio waves, microwaves, infrared, (visible) li ...
, comes in discrete units, called
photon A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless, so they a ...
s, and predicts its
spectral ''Spectral'' is a 2016 3D military science fiction, supernatural horror fantasy and action-adventure thriller war film directed by Nic Mathieu. Written by himself, Ian Fried, and George Nolfi from a story by Fried and Mathieu. The film stars ...
energies (corresponding to pure colors), and the intensities of its light beams. A single photon is a '' quantum'', or smallest observable particle, of the electromagnetic field. A partial photon is never experimentally observed. More broadly, quantum mechanics shows that many properties of objects, such as position, speed, and
angular momentum In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syst ...
, that appeared continuous in the zoomed-out view of classical mechanics, turn out to be (in the very tiny, zoomed-in scale of quantum mechanics) '' quantized''. Such properties of
elementary particles In particle physics, an elementary particle or fundamental particle is a subatomic particle that is not composed of other particles. Particles currently thought to be elementary include electrons, the fundamental fermions (quarks, leptons, anti ...
are required to take on one of a set of small, discrete allowable values, and since the gap between these values is also small, the discontinuities are only apparent at very tiny (atomic) scales. Many aspects of quantum mechanics are counterintuitive and can seem
paradox A paradox is a logically self-contradictory statement or a statement that runs contrary to one's expectation. It is a statement that, despite apparently valid reasoning from true premises, leads to a seemingly self-contradictory or a logically u ...
ical because they describe behavior quite different from that seen at larger scales. In the words of quantum physicist
Richard Feynman Richard Phillips Feynman (; May 11, 1918 – February 15, 1988) was an American theoretical physicist, known for his work in the path integral formulation of quantum mechanics, the theory of quantum electrodynamics, the physics of the superfl ...
, quantum mechanics deals with "nature as She is—absurd". One principal "paradox" is the apparent inconsistency between Newton's laws and quantum mechanics which can be explained using
Ehrenfest's theorem The Ehrenfest theorem, named after Paul Ehrenfest, an Austrian theoretical physicist at Leiden University, relates the time derivative of the expectation values of the position and momentum operators ''x'' and ''p'' to the expectation value of th ...
, which shows that the average values obtained from quantum mechanics (e.g. position and momentum) obey classical laws. However, Ehrenfest's theorem is far from capable of explaining all the counterintuitive phenomena (
quantum weirdness Quantum weirdness encompasses the aspects of quantum mechanics that challenge and defy human physical intuition based on the Newtonian mechanics of classical physics. These aspects include: * quantum entanglement; * quantum nonlocality, referred ...
) that have been observed, but rather is a mathematical expression of the correspondence principle. For example, the
uncertainty principle In quantum mechanics, the uncertainty principle (also known as Heisenberg's uncertainty principle) is any of a variety of mathematical inequalities asserting a fundamental limit to the accuracy with which the values for certain pairs of physic ...
of quantum mechanics means that the more closely one pins down one measurement (such as the position of a particle), the less accurate another complementary measurement pertaining to the same particle (such as its
speed In everyday use and in kinematics, the speed (commonly referred to as ''v'') of an object is the magnitude Magnitude may refer to: Mathematics *Euclidean vector, a quantity defined by both its magnitude and its direction *Magnitude (ma ...
) must become. Another example is entanglement, in which a measurement of any two-valued state of a particle (such as light polarized up or down) made on either of two "entangled" particles that are very far apart causes a subsequent measurement on the other particle to always be the other of the two values (such as polarized in the opposite direction). A final example is
superfluidity Superfluidity is the characteristic property of a fluid with zero viscosity which therefore flows without any loss of kinetic energy. When stirred, a superfluid forms vortices that continue to rotate indefinitely. Superfluidity occurs in two ...
, in which a container of liquid helium, cooled down to near absolute zero in temperature spontaneously flows (slowly) up and over the opening of its container, against the force of gravity.


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

Thermal radiation Thermal radiation is electromagnetic radiation generated by the thermal motion of particles in matter. Thermal radiation is generated when heat from the movement of charges in the material (electrons and protons in common forms of matter) i ...
is electromagnetic radiation emitted from the surface of an object due to the object's internal energy. If an object is heated sufficiently, it starts to emit light at the red end of the
spectrum A spectrum (plural ''spectra'' or ''spectrums'') is a condition that is not limited to a specific set of values but can vary, without gaps, across a continuum. The word was first used scientifically in optics to describe the rainbow of colors ...
, as it becomes red hot. Heating it further causes the color to change from red to yellow, white, and blue, as it emits light at increasingly shorter wavelengths (higher frequencies). 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 body A black body or blackbody is an idealized physical body that absorbs all incident electromagnetic radiation, regardless of frequency or angle of incidence. The name "black body" is given because it absorbs all colors of light. A black body ...
, and the radiation it emits is called black-body radiation. By the late 19th century, thermal radiation had been fairly well characterized experimentally.Several formulas had been created that could describe some of the experimental measurements of thermal radiation: how the wavelength at which the radiation is strongest changes with temperature is given by
Wien's displacement law Wien's displacement law states that the black-body radiation curve for different temperatures will peak at different wavelengths that are inversely proportional to the temperature. The shift of that peak is a direct consequence of the Planck r ...
, the overall power emitted per unit area is given by the
Stefan–Boltzmann law The Stefan–Boltzmann law describes the power radiated from a black body in terms of its temperature. Specifically, the Stefan–Boltzmann law states that the total energy radiated per unit surface area of a black body across all wavelengths ...
. The best theoretical explanation of the experimental results was the
Rayleigh–Jeans law In physics, the Rayleigh–Jeans law is an approximation to the spectral radiance of electromagnetic radiation as a function of wavelength from a black body at a given temperature through classical arguments. For wavelength λ, it is: B_ (T) = \ ...
, which agrees with experimental results well at large wavelengths (or, equivalently, low frequencies), but strongly disagrees at short wavelengths (or high frequencies). In fact, at short wavelengths, classical physics predicted that energy will be emitted by a hot body at an infinite rate. This result, which is clearly wrong, is known as the
ultraviolet catastrophe The ultraviolet catastrophe, also called the Rayleigh–Jeans catastrophe, was the prediction of late 19th century/early 20th century classical physics that an ideal black body at thermal equilibrium would emit an unbounded quantity of energy ...
.
However, classical physics led to the
Rayleigh–Jeans law In physics, the Rayleigh–Jeans law is an approximation to the spectral radiance of electromagnetic radiation as a function of wavelength from a black body at a given temperature through classical arguments. For wavelength λ, it is: B_ (T) = \ ...
, which, as shown in the figure, agrees with experimental results well at low frequencies, but strongly disagrees at high frequencies. Physicists searched for a single theory that explained all the experimental results. The first model that was able to explain the full spectrum of thermal radiation was put forward by
Max Planck Max Karl Ernst Ludwig Planck (, ; 23 April 1858 – 4 October 1947) was a German theoretical physicist whose discovery of energy quanta won him the Nobel Prize in Physics in 1918. Planck made many substantial contributions to theoretical p ...
in 1900. He proposed a mathematical model in which the thermal radiation was in equilibrium with a set of harmonic oscillators. To reproduce the experimental results, he had to assume that each oscillator emitted an integer number of units of energy at its single characteristic frequency, rather than being able to emit any arbitrary amount of energy. In other words, the energy emitted by an oscillator was ''quantized''.The word '' quantum'' comes from the Latin word for "how much" (as does ''quantity''). Something that is ''quantized'', as 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 quantum 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 constant The Planck constant, or Planck's constant, is a fundamental physical constant of foundational importance in quantum mechanics. The constant gives the relationship between the energy of a photon and its frequency, and by the mass-energy equivale ...
. The Planck constant, usually written as , has the value of . So, the energy of an oscillator of frequency is given by :E = nhf,\quad \text\quad n = 1,2,3,\ldots To change the color of such a radiating body, it is necessary to change its temperature. Planck's law explains why: increasing the temperature of a body allows it to emit more energy overall, and means that a larger proportion of the energy is towards the violet end of the spectrum. 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 heuristic mathematical construct, rather than (as is now believed) a fundamental change in our understanding of the world.


Photons: the quantization of light

In 1905,
Albert Einstein Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theory ...
took an extra step. He suggested that quantization was not just a mathematical construct, but that the energy in a beam of light actually occurs in individual packets, which are now called
photon A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless, so they a ...
s. The energy of a single photon of light of frequency f is given by the frequency multiplied by Planck's constant h (an extremely tiny positive number): :E = hf For centuries, scientists had debated between two possible theories of light: was it a
wave In physics, mathematics, and related fields, a wave is a propagating dynamic disturbance (change from equilibrium) of one or more quantities. Waves can be periodic, in which case those quantities oscillate repeatedly about an equilibrium (re ...
or did it instead comprise a stream of tiny particles? By the 19th century, the debate was generally considered to have been settled in favor of the wave theory, as it was able to explain observed effects such as
refraction In physics, refraction is the redirection of a wave as it passes from one medium to another. The redirection can be caused by the wave's change in speed or by a change in the medium. Refraction of light is the most commonly observed phenome ...
, diffraction,
interference Interference is the act of interfering, invading, or poaching. Interference may also refer to: Communications * Interference (communication), anything which alters, modifies, or disrupts a message * Adjacent-channel interference, caused by extr ...
, and polarization.
James Clerk Maxwell James Clerk Maxwell (13 June 1831 – 5 November 1879) was a Scottish mathematician and scientist responsible for the classical theory of electromagnetic radiation, which was the first theory to describe electricity, magnetism and li ...
had shown that electricity, magnetism, and light are all manifestations of the same phenomenon: the electromagnetic field.
Maxwell's equations Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits. ...
, which are the complete set of laws of
classical electromagnetism Classical electromagnetism or classical electrodynamics is a branch of theoretical physics that studies the interactions between electric charges and currents using an extension of the classical Newtonian model; It is, therefore, a classical fie ...
, describe light as waves: a combination of oscillating electric and magnetic fields. Because of the preponderance of evidence in favor of the wave theory, Einstein's ideas were met initially with great skepticism. Eventually, however, the photon model became favored. One of the most significant pieces of evidence in its favor was its ability to explain several puzzling properties of the
photoelectric effect The photoelectric effect is the emission of electrons when electromagnetic radiation, such as light, hits a material. Electrons emitted in this manner are called photoelectrons. The phenomenon is studied in condensed matter physics, and solid sta ...
, described in the following section. Nonetheless, the wave analogy remained indispensable for helping to understand other characteristics of light: diffraction,
refraction In physics, refraction is the redirection of a wave as it passes from one medium to another. The redirection can be caused by the wave's change in speed or by a change in the medium. Refraction of light is the most commonly observed phenome ...
, and
interference Interference is the act of interfering, invading, or poaching. Interference may also refer to: Communications * Interference (communication), anything which alters, modifies, or disrupts a message * Adjacent-channel interference, caused by extr ...
.


The photoelectric effect

In 1887,
Heinrich Hertz Heinrich Rudolf Hertz ( ; ; 22 February 1857 – 1 January 1894) was a German physicist who first conclusively proved the existence of the electromagnetic waves predicted by James Clerk Maxwell's equations of electromagnetism. The unit ...
observed that when light with sufficient frequency hits a metallic surface, the surface emits electrons. In 1902,
Philipp Lenard Philipp Eduard Anton von Lenard (; hu, Lénárd Fülöp Eduárd Antal; 7 June 1862 – 20 May 1947) was a Hungarian-born German physicist and the winner of the Nobel Prize for Physics in 1905 for his work on cathode rays and the discovery of ...
discovered that the maximum possible energy of an ejected electron is related to the
frequency Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...
of the light, not to its intensity: if the frequency is too low, no electrons are ejected regardless of the intensity. Strong beams of light toward the red end of the spectrum might produce no electrical potential at all, while weak beams of light toward the violet end of the spectrum would produce higher and higher voltages. The lowest frequency of light that can cause electrons to be emitted, called the threshold frequency, is different for different metals. This observation is at odds with classical electromagnetism, which predicts that the electron's energy should be proportional to the intensity of the incident radiation.Alt URL
So when physicists first discovered devices exhibiting the photoelectric effect, they initially expected that a higher intensity of light would produce a higher voltage from the photoelectric device. Einstein explained the effect by postulating that a beam of light is a stream of particles ("
photon A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless, so they a ...
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 and 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 (sometimes spelt workfunction) is the minimum thermodynamic work (i.e., energy) needed to remove an electron from a solid to a point in the vacuum immediately outside the solid surface. Here "immediately" ...
'' and 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: :\varphi = h f_0. If is greater than , the energy is enough to remove an electron. The ejected electron has a
kinetic energy In physics, the kinetic energy of an object is the energy that 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 acc ...
, , which is, at most, equal to the photon's energy minus the energy needed to dislodge the electron from the metal: :E_K = hf - \varphi = h(f - f_0). Einstein's description of light as being composed of particles extended Planck's notion of quantized energy, which is that 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. In nature, single photons are rarely encountered. The Sun and emission sources available in the 19th century emit vast numbers of photons every second, and so the importance of the energy carried by each photon was not obvious. Einstein's idea that the energy contained in individual units of light depends on their frequency made it possible to explain experimental results that had seemed counterintuitive. However, although the photon is a particle, it was still being described as having the wave-like property of frequency. Effectively, the account of light as a particle is insufficient, and its wave-like nature is still required.


Consequences of light being quantized

The relationship between the frequency of electromagnetic radiation and the energy of each photon is why
ultraviolet Ultraviolet (UV) is a form of electromagnetic radiation with wavelength from 10 nm (with a corresponding frequency around 30  PHz) to 400 nm (750  THz), shorter than that of visible light, but longer than X-rays. UV radiation ...
light can cause
sunburn Sunburn is a form of radiation burn that affects living tissue, such as skin, that results from an overexposure to ultraviolet (UV) radiation, usually from the Sun. Common symptoms in humans and animals include: red or reddish skin that is h ...
, but visible or
infrared Infrared (IR), sometimes called infrared light, is electromagnetic radiation (EMR) with wavelengths longer than those of visible light. It is therefore invisible to the human eye. IR is generally understood to encompass wavelengths from around ...
light cannot. A photon of ultraviolet light delivers a high amount of
energy In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of hea ...
—enough to contribute to cellular damage such as occurs in a sunburn. A photon of infrared light delivers less 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. All photons of the same frequency have identical energy, and all photons of different frequencies have proportionally (order 1, ) different energies. However, although the energy imparted by photons is invariant at any given frequency, the initial energy state of the electrons in a photoelectric device before absorption of light is not necessarily uniform. Anomalous results may occur in the case of individual electrons. For instance, 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 reflects the behavior of the vast majority of its electrons, which are at their equilibrium level. This point helps clarify the distinction between the study of small individual particles in quantum dynamics and the study of massive individual particles in classical physics.


The quantization of matter: the Bohr model of the atom

By the dawn of the 20th century, the evidence required a model of the atom with a diffuse cloud of negatively charged
electron The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family, and are generally thought to be elementary particles because they have no ...
s surrounding a small, dense, positively charged
nucleus Nucleus ( : nuclei) is a Latin word for the seed inside a fruit. It most often refers to: *Atomic nucleus, the very dense central region of an atom * Cell nucleus, a central organelle of a eukaryotic cell, containing most of the cell's DNA Nucl ...
. These properties suggested a model in which electrons circle the nucleus like planets orbiting a star.The classical model of the atom is called the planetary model, or sometimes the
Rutherford model The Rutherford model was devised by the New Zealand-born physicist Ernest Rutherford to describe an atom. Rutherford directed the Geiger–Marsden experiment in 1909, which suggested, upon Rutherford's 1911 analysis, that J. J. Thomson's plum ...
—after
Ernest Rutherford Ernest Rutherford, 1st Baron Rutherford of Nelson, (30 August 1871 – 19 October 1937) was a New Zealand physicist who came to be known as the father of nuclear physics. ''Encyclopædia Britannica'' considers him to be the greatest ...
who proposed it in 1911, based on the Geiger–Marsden gold foil experiment, 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 spectrum of atoms. When a gas is heated, it gives off light only at discrete frequencies. For example, the visible light given off by
hydrogen Hydrogen is the chemical element with the symbol H and atomic number 1. Hydrogen is the lightest element. At standard conditions hydrogen is a gas of diatomic molecules having the formula . It is colorless, odorless, tasteless, non-toxic ...
consists of four different colors, as shown in the picture below. The intensity of the light at different frequencies is also different. By contrast, white light consists of a continuous emission across the whole range of visible frequencies. By the end of the nineteenth century, a simple rule known as Balmer's formula showed how the frequencies of the different lines related to each other, though without explaining why this was, or making any prediction about the intensities. The formula also predicted some additional spectral lines in ultraviolet and infrared light that had not been observed at the time. These lines were later observed experimentally, raising confidence in the value of the formula. In 1885 the Swiss mathematician
Johann Balmer Johann Jakob Balmer (1 May 1825 – 12 March 1898) was a Swiss mathematician best known for his work in physics, the Balmer series of hydrogen atom. Biography Balmer was born in Lausen, Switzerland, the son of a chief justice also named Johan ...
discovered that each wavelength (lambda) in the visible spectrum of hydrogen is related to some integer by the equation :\lambda = B\left(\frac\right) \qquad\qquad n = 3,4,5,6 where is a constant Balmer determined is equal to 364.56 nm. In 1888
Johannes Rydberg Johannes (Janne) Robert Rydberg (; 8 November 1854 – 28 December 1919) was a Swedish physicist mainly known for devising the Rydberg formula, in 1888, which is used to describe the wavelengths of photons (of visible light and other electrom ...
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 formula In atomic physics, the Rydberg formula calculates the wavelengths of a spectral line in many chemical elements. The formula was primarily presented as a generalization of the Balmer series for all atomic electron transitions of hydrogen. It wa ...
: : \frac = R \left(\frac - \frac\right), where ''R'' is the
Rydberg constant In spectroscopy, the Rydberg constant, symbol R_\infty for heavy atoms or R_\text for hydrogen, named after the Swedish physicist Johannes Rydberg, is a physical constant relating to the electromagnetic spectra of an atom. The constant first aro ...
, 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 Lyman Theodore Lyman may refer to: * Theodore B. Lyman (1815–1893), American bishop * Theodore Lyman II (1792–1849), American philanthropist, politician, and author * Theodore Lyman III (1833–1897), American natural scientist, military staff offic ...
found some of the predicted ultraviolet wavelengths. Both Balmer and Rydberg's formulas involve integers: in modern terms, they imply that some property of the atom is quantized. Understanding exactly what this property was, and why it was quantized, was a major part of the development of quantum mechanics, as shown in the rest of this article. In 1913
Niels Bohr Niels Henrik David Bohr (; 7 October 1885 – 18 November 1962) was a Danish physicist who made foundational contributions to understanding atomic structure and quantum theory, for which he received the Nobel Prize in Physics in 1922 ...
proposed a new model of the atom that included quantized electron orbits: electrons still orbit the nucleus much as planets orbit around the sun, but they are permitted to inhabit only certain orbits, not to orbit at any arbitrary distance. 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.Alt URL
/ref> 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. Starting from only one simple assumption about the rule that the orbits must obey, the Bohr model was able to relate the observed spectral lines in the emission spectrum of hydrogen to previously known constants. In Bohr's model, the electron was not allowed to emit energy continuously and crash into the nucleus: once it was in the closest permitted orbit, it was stable forever. Bohr's model did not explain why the orbits should be quantized in that way, nor was it able to make accurate predictions for atoms with more than one electron, or to explain why some spectral lines are brighter than others. Some fundamental assumptions of the Bohr model were soon proven wrong—but the key result that the discrete lines in emission spectra are due to some property of the electrons in atoms being quantized is correct. The way that the electrons actually behave is strikingly different from Bohr's atom, and from what we see in the world of our everyday experience; this modern quantum mechanical model of the atom is discussed below. Bohr theorized that the
angular momentum In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syst ...
, , of an electron is quantized: :L = n\frac=n\hbar where is an integer and and are the
Planck constant The Planck constant, or Planck's constant, is a fundamental physical constant of foundational importance in quantum mechanics. The constant gives the relationship between the energy of a photon and its frequency, and by the mass-energy equivale ...
and Planck reduced constant respectively. Starting from this assumption,
Coulomb's law Coulomb's inverse-square law, or simply Coulomb's law, is an experimental law of physics that quantifies the amount of force between two stationary, electrically charged particles. The electric force between charged bodies at rest is convention ...
and the equations of circular motion show that an electron with units of angular momentum orbits a proton at a distance given by :r = \frac, where is the Coulomb constant, is the mass of an electron, and is the charge on an electron. For simplicity this is written as :r = n^2 a_0,\! where , called the
Bohr radius The Bohr radius (''a''0) is a physical constant, approximately equal to the most probable distance between the nucleus and the 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 ...
, is equal to 0.0529 nm. The Bohr radius is the radius of the smallest allowed orbit. The energy of the electronIn this case, the energy of the electron is the sum of its
kinetic Kinetic (Ancient Greek: κίνησις “kinesis”, movement or to move) may refer to: * Kinetic theory of gases, Kinetic theory, describing a gas as particles in random motion * Kinetic energy, the energy of an object that it possesses due to i ...
and
potential Potential generally refers to a currently unrealized ability. The term is used in a wide variety of fields, from physics to the social sciences to indicate things that are in a state where they are able to change in ways ranging from the simple r ...
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 :E = -\frac \frac. Thus Bohr's assumption that angular momentum is quantized means that an electron can inhabit only certain orbits around the nucleus and that it can have only certain energies. A consequence of these constraints is that the electron does 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: :E_ = E_n - E_m = \frac\left(\frac-\frac\right) 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 :\frac = \frac\left(\frac-\frac\right). This equation has the same form as the
Rydberg formula In atomic physics, the Rydberg formula calculates the wavelengths of a spectral line in many chemical elements. The formula was primarily presented as a generalization of the Balmer series for all atomic electron transitions of hydrogen. It wa ...
, and predicts that the constant should be given by :R = \frac . 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 for the emission spectrum of any system consisting of a nucleus and a single electron (that is,
ion An ion () is an atom or molecule with a net electrical charge. The charge of an electron is considered to be negative by convention and this charge is equal and opposite to the charge of a proton, which is considered to be positive by conve ...
s such as He+ or O7+, which contain only one electron) but cannot be extended to an atom with two electrons such as neutral helium.
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

Just as light has both wave-like and particle-like properties, matter also has wave-like properties. Matter behaving as a wave was first demonstrated experimentally for electrons: a beam of electrons can exhibit diffraction, just like a beam of light or a water wave.Electron diffraction was first demonstrated three years after de Broglie published his hypothesis. At the
University of Aberdeen , mottoeng = The fear of the Lord is the beginning of wisdom , established = , type = Public research universityAncient university , endowment = £58.4 million (2021) , budget ...
, George Thomson passed a beam of electrons through a thin metal film and observed diffraction patterns, as would be predicted by the de Broglie hypothesis. At
Bell Labs Nokia Bell Labs, originally named Bell Telephone Laboratories (1925–1984), then AT&T Bell Laboratories (1984–1996) and Bell Labs Innovations (1996–2007), is an American industrial Research and development, research and scientific developm ...
, Davisson and Germer guided an electron beam through a crystalline grid. De Broglie was awarded the
Nobel Prize in Physics ) , image = Nobel Prize.png , alt = A golden medallion with an embossed image of a bearded man facing left in profile. To the left of the man is the text "ALFR•" then "NOBEL", and on the right, the text (smaller) "NAT•" then " ...
in 1929 for his hypothesis; Thomson and Davisson shared the Nobel Prize for Physics in 1937 for their experimental work.
Similar wave-like phenomena were later shown for atoms and even molecules. The wavelength, ''λ'', associated with any object is related to its momentum, ''p'', through the
Planck constant The Planck constant, or Planck's constant, is a fundamental physical constant of foundational importance in quantum mechanics. The constant gives the relationship between the energy of a photon and its frequency, and by the mass-energy equivale ...
, ''h'': : p = \frac. The relationship, called the de Broglie hypothesis, holds for all types of matter: all matter exhibits properties of both particles and waves. 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. Wave–particle duality is an example of the
principle of complementarity In physics, complementarity is a conceptual aspect of quantum mechanics that Niels Bohr regarded as an essential feature of the theory. The complementarity principle holds that objects have certain pairs of complementary properties which cannot al ...
in quantum physics. An elegant example of wave-particle duality, the double-slit experiment, is discussed in the section below.


The double-slit experiment

In the double-slit experiment, as originally performed by Thomas Young in 1803, and then Augustin Fresnel a decade later, a beam of light is directed through two narrow, closely spaced slits, producing an
interference pattern In physics, interference is a phenomenon in which two waves combine by adding their displacement together at every single point in space and time, to form a resultant wave of greater, lower, or the same amplitude. Constructive and destructive ...
of light and dark bands on a screen. If one of the slits is covered up, one might naïvely expect that the intensity of the fringes due to interference would be halved everywhere. In fact, a much simpler pattern is seen, a
diffraction pattern Diffraction is defined as the interference or bending of waves around the corners of an obstacle or through an aperture into the region of geometrical shadow of the obstacle/aperture. The diffracting object or aperture effectively becomes a ...
diametrically opposite the open slit. The same behavior can be demonstrated in water waves, and so the double-slit experiment was seen as a demonstration of the wave nature of light. Variations of the double-slit experiment have been performed using electrons, atoms, and even large molecules, and the same type of interference pattern is seen. Thus it has been demonstrated that all
matter In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume. All everyday objects that can be touched are ultimately composed of atoms, which are made up of interacting subatomic part ...
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 acts as a wave in an experiment to measure its wave-like properties, and like a particle in an experiment to measure its particle-like properties. The point on the detector screen where any individual particle shows up is the result of a random process. However, the distribution pattern of many individual particles mimics the diffraction pattern produced by waves.


Application to the Bohr model

De Broglie expanded the
Bohr model of the atom In atomic physics, the Bohr model or Rutherford–Bohr model, presented by Niels Bohr and Ernest Rutherford in 1913, is a system consisting of a small, dense nucleus surrounded by orbiting electrons—similar to the structure of the Solar Sy ...
by showing that an electron in orbit around a nucleus could be thought of as having wave-like properties. In particular, an
electron The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family, and are generally thought to be elementary particles because they have no ...
is observed only in situations that permit a standing wave around a
nucleus Nucleus ( : nuclei) is a Latin word for the seed inside a fruit. It most often refers to: *Atomic nucleus, the very dense central region of an atom * Cell nucleus, a central organelle of a eukaryotic cell, containing most of the cell's DNA Nucl ...
. An example of a standing wave is a violin string, which is fixed at both ends and can be made to vibrate. The waves created by a stringed instrument appear to oscillate in place, moving from crest to trough in an up-and-down motion. The wavelength of a standing wave is related to the length of the vibrating object and the boundary conditions. For example, because the violin string is fixed at both ends, it can carry standing waves of wavelengths \frac, 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. The electron's wavelength, therefore, determines that only Bohr orbits of certain distances from the nucleus are possible. In turn, at any distance from the nucleus smaller than a certain value, it would be impossible to establish an orbit. The minimum possible distance from the nucleus is called the Bohr radius. De Broglie's treatment of quantum events served as a starting point for Schrödinger when he set out to construct a wave equation to describe quantum-theoretical events.


Spin

In 1922,
Otto Stern :''Otto Stern was also the pen name of German women's rights activist Louise Otto-Peters (1819–1895)''. Otto Stern (; 17 February 1888 – 17 August 1969) was a German-American physicist and Nobel laureate in physics. He was the second most n ...
and
Walther Gerlach Walther Gerlach (1 August 1889 – 10 August 1979) was a German physicist who co-discovered, through laboratory experiment, spin quantization in a magnetic field, the Stern–Gerlach effect. The experiment was conceived by Otto Stern in 1921 an ...
shot silver atoms through an
inhomogeneous Homogeneity and heterogeneity are concepts often used in the sciences and statistics relating to the uniformity of a substance or organism. A material or image that is homogeneous is uniform in composition or character (i.e. color, shape, siz ...
magnetic field. Relative to its northern pole, pointing up, down, or somewhere in between, in classical mechanics, a magnet thrown through a magnetic field may be deflected a small or large distance upwards or downwards. The atoms that Stern and Gerlach shot through the magnetic field acted similarly. However, while the magnets could be deflected variable distances, the atoms would always be deflected a constant distance either up or down. This implied that the property of the atom that corresponds to the magnet's orientation must be quantized, taking one of two values (either up or down), as opposed to being chosen freely from any angle.
Ralph Kronig Ralph Kronig (10 March 1904 – 16 November 1995) was a German physicist. He is noted for the discovery of particle spin and for his theory of X-ray absorption spectroscopy. His theories include the Kronig–Penney model, the Coster–Kronig tran ...
originated the theory that particles such as atoms or electrons behave as if they rotate, or "spin", about an axis. Spin would account for the missing
magnetic moment In electromagnetism, the magnetic moment is the magnetic strength and orientation of a magnet or other object that produces a magnetic field. Examples of objects that have magnetic moments include loops of electric current (such as electromagne ...
, and allow two electrons in the same orbital to occupy distinct
quantum state In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution i ...
s if they "spun" in opposite directions, thus satisfying the exclusion principle. The quantum number represented the sense (positive or negative) of spin. The choice of the orientation of the magnetic field used in the Stern–Gerlach experiment is arbitrary. In the animation shown here, the field is vertical and so the atoms are deflected either up or down. If the magnet is rotated a quarter turn, the atoms are deflected either left or right. Using a vertical field shows that the spin along the vertical axis is quantized, and using a horizontal field shows that the spin along the horizontal axis is quantized. If instead of hitting a detector screen, one of the beams of atoms coming out of the Stern–Gerlach apparatus is passed into another (inhomogeneous) magnetic field oriented in the same direction, all of the atoms are deflected the same way in this second field. However, if the second field is oriented at 90° to the first, then half of the atoms are deflected one way and half the other so that the atom's spin about the horizontal and vertical axes are independent of each other. However, if one of these beams (e.g. the atoms that were deflected up then left) is passed into a third magnetic field, oriented the same way as the first, half of the atoms go one way and half the other, even though they all went in the same direction originally. The action of measuring the atoms' spin concerning a horizontal field has changed their spin concerning a vertical field. The Stern–Gerlach experiment demonstrates several important features of quantum mechanics: * A feature of the natural world has been demonstrated to be quantized, and able to take only certain discrete values. * Particles possess an intrinsic
angular momentum In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syst ...
that is closely analogous to the angular momentum of a classically spinning object. * Measurement changes the system being measured in quantum mechanics. Only the spin of an object in one direction can be known, and observing the spin in another direction destroys the original information about the spin. * Quantum mechanics is probabilistic: whether the spin of any individual atom sent into the apparatus is positive or negative is random.


Development of modern quantum mechanics

In 1925,
Werner Heisenberg Werner Karl Heisenberg () (5 December 1901 – 1 February 1976) was a German theoretical physicist and one of the main pioneers of the theory of quantum mechanics. He published his work in 1925 in a breakthrough paper. In the subsequent serie ...
attempted to solve one of the problems that the Bohr model left unanswered, explaining the intensities of the different lines in the hydrogen emission spectrum. Through a series of mathematical analogies, he wrote out the quantum-mechanical analog for the classical computation of intensities. See Werner Heisenberg's paper, "Quantum-Theoretical Re-interpretation of Kinematic and Mechanical Relations" pp. 261–76 Shortly afterward, Heisenberg's colleague Max Born realized that Heisenberg's method of calculating the probabilities for transitions between the different energy levels could best be expressed by using the mathematical concept of
matrices Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
.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 a point when the old quantum physics was discovering a field littered with more and more stumbling blocks. He decided that quantum physics had to be re-thought from the ground up. In doing so he excise ...
.
In the same year, building on de Broglie's hypothesis,
Erwin Schrödinger Erwin Rudolf Josef Alexander Schrödinger (, ; ; 12 August 1887 – 4 January 1961), sometimes written as or , was a Nobel Prize-winning Austrian physicist with Irish citizenship who developed a number of fundamental results in quantum theo ...
developed the equation that describes the behavior of a quantum-mechanical wave. The mathematical model, called the
Schrödinger equation The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of th ...
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."Schrodinger Equation (Physics)", ''Encyclopædia Britannica ''
/ref> The wave itself is described by a mathematical function known as a "
wave function A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements ...
". Schrödinger said that the wave function provides the "means for predicting the probability of measurement results". Schrödinger was able to calculate the energy levels of hydrogen by treating a hydrogen atom's
electron The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family, and are generally thought to be elementary particles because they have no ...
as a classical wave, moving in a well of the electrical potential created by the proton. This calculation accurately reproduced the energy levels of the Bohr model. In May 1926, Schrödinger proved that Heisenberg's
matrix mechanics Matrix mechanics is a formulation of quantum mechanics created by Werner Heisenberg, Max Born, and Pascual Jordan in 1925. It was the first conceptually autonomous and logically consistent formulation of quantum mechanics. Its account of quantum j ...
and his own
wave mechanics Wave mechanics may refer to: * the mechanics of waves * the ''wave equation'' in quantum physics, see Schrödinger equation See also * Quantum mechanics * Wave equation The (two-way) wave equation is a second-order linear partial different ...
made the same predictions about the properties and behavior of the electron; mathematically, the two theories had an underlying common form. Yet the two men disagreed on the interpretation of their mutual theory. For instance, Heisenberg accepted the theoretical prediction of jumps of electrons between orbitals in an atom, but Schrödinger hoped that a theory based on continuous wave-like properties could avoid what he called (as paraphrased by
Wilhelm Wien Wilhelm Carl Werner Otto Fritz Franz Wien (; 13 January 1864 – 30 August 1928) 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 ...
) "this nonsense about quantum jumps". In the end, Heisenberg's approach won out, and quantum jumps were confirmed.


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 A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements ...
, usually represented by the Greek letter \psi ("psi"). (Heisenberg) # How \psi 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 shows up in the double-slit experiment—is related to the square of the absolute value of the amplitude of its wave function. (
Born rule The Born rule (also called Born's rule) is a key postulate of quantum mechanics which gives the probability that a measurement of a quantum system will yield a given result. In its simplest form, it states that the probability density of findi ...
, due to Max Born, which gives a physical meaning to the wave function in the Copenhagen interpretation: the
probability amplitude In quantum mechanics, a probability amplitude is a complex number used for describing the behaviour of systems. The modulus squared of this quantity represents a probability density. Probability amplitudes provide a relationship between the qu ...
) # 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 uncertainty principle (also known as Heisenberg's uncertainty principle) is any of a variety of mathematical inequalities asserting a fundamental limit to the accuracy with which the values for certain pairs of physic ...
) # 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 of Bohr and Heisenberg) Various consequences of these principles are discussed in more detail in the following subsections.


Uncertainty principle

Suppose it is desired to measure the position and speed of an object—for example, a car going through a radar speed trap. It can be assumed that the car has a definite position and speed at a particular moment in time. How accurately these values can be measured depends on the quality of the measuring equipment. If the precision of the measuring equipment is improved, it provides a result closer to the true value. It might be assumed that the speed of the car and its position could be operationally defined and measured simultaneously, as precisely as might be desired. In 1927, Heisenberg proved that this last assumption is not correct. Quantum mechanics shows that certain pairs of physical properties, for example, position and speed, cannot be simultaneously measured, nor defined in operational terms, to arbitrary precision: the more precisely one property is measured, or defined in operational terms, the less precisely can the other. This statement is known as the
uncertainty principle In quantum mechanics, the uncertainty principle (also known as Heisenberg's uncertainty principle) is any of a variety of mathematical inequalities asserting a fundamental limit to the accuracy with which the values for certain pairs of physic ...
. The uncertainty principle is not only a statement about the accuracy of our measuring equipment but, more deeply, is about the conceptual nature of the measured quantities—the assumption that the car had simultaneously defined position and speed does not work in quantum mechanics. On a scale of cars and people, these uncertainties are negligible, 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 of the photon with the electron, but the greater is the disturbance of the electron. This is because from the impact with the photon, the electron absorbs a random amount of energy, rendering the measurement obtained of its momentum increasingly uncertain (momentum is velocity multiplied by mass), for one is necessarily measuring its post-impact disturbed momentum from the collision products and not its original momentum (momentum which should be simultaneously measured with position). With a photon of lower frequency, the disturbance (and hence uncertainty) in the momentum is less, but so is the accuracy of the measurement of the position of the impact."Uncertainty principle", ''Encyclopædia Britannica''
/ref> At the heart of the uncertainty principle is a fact that for any mathematical analysis in the position and velocity domains, achieving a sharper (more precise) curve in the position domain can only be done at the expense of a more gradual (less precise) curve in the speed domain, and vice versa. More sharpness in the position domain requires contributions from more frequencies in the speed domain to create the narrower curve, and vice versa. It is a fundamental tradeoff inherent in any such related or complementary measurements, but is only really noticeable at the smallest (Planck) scale, near the size of
elementary particles In particle physics, an elementary particle or fundamental particle is a subatomic particle that is not composed of other particles. Particles currently thought to be elementary include electrons, the fundamental fermions (quarks, leptons, anti ...
. The uncertainty principle shows mathematically that the product of the uncertainty in the position and momentum 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'' means that a measurement has forced or converted a quantum (probabilistic or potential) state into a definite measured value. This phenomenon is only seen in quantum mechanics rather than classical mechanics. For example, before a photon actually "shows up" on a detection screen it can be described only with a set of probabilities for where it might show up. When it does appear, for instance in the CCD of an electronic camera, the time and space where it interacted with the device are known within very tight limits. However, the photon has disappeared in the process of being captured (measured), and its quantum
wave function A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements ...
has disappeared with it. In its place, some macroscopic physical change in the detection screen has appeared, e.g., an exposed spot in a sheet of photographic film, or a change in electric potential in some cell of a CCD.


Eigenstates and eigenvalues

Because of the
uncertainty principle In quantum mechanics, the uncertainty principle (also known as Heisenberg's uncertainty principle) is any of a variety of mathematical inequalities asserting a fundamental limit to the accuracy with which the values for certain pairs of physic ...
, statements about both the position and momentum of particles can assign only a
probability Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speakin ...
that the position or momentum has some numerical value. Therefore, it is necessary to formulate clearly the difference between the state of something 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. In the Stern–Gerlach experiment discussed above, the spin of the atom about the vertical axis has two eigenstates: up and down. Before measuring it, we can only say that any individual atom has an equal probability of being found to have spin up or spin down. The measurement process causes the wave function to collapse into one of the two states. The eigenstates of spin about the vertical axis are not simultaneously eigenstates of spin about the horizontal axis, so this atom has an equal probability of being found to have either value of spin about the horizontal axis. As described in the section above, measuring the spin about the horizontal axis can allow an atom that was spun up to spin down: measuring its spin about the horizontal axis collapses its wave function into one of the eigenstates of this measurement, which means it is no longer in an eigenstate of spin about the vertical axis, so can take either value.


The Pauli exclusion principle

In 1924,
Wolfgang Pauli Wolfgang Ernst Pauli (; ; 25 April 1900 – 15 December 1958) was an Austrian theoretical physicist and one of the pioneers of quantum physics. In 1945, after having been nominated by Albert Einstein, Pauli received the Nobel Prize in Physics ...
proposed a new quantum degree of freedom (or quantum number), 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 doublet, or pair of lines differing by a small amount, where only one line was expected. Pauli formulated his ''exclusion principle'', stating, "There cannot exist an atom in such a quantum state that two electrons within thave the same set of quantum numbers." A year later, Uhlenbeck and Goudsmit identified Pauli's new degree of freedom with the property called spin whose effects were observed in the Stern–Gerlach experiment.


Application to the hydrogen atom

Bohr's model of the atom was essentially a planetary one, with the electrons orbiting around the nuclear "sun". However, the uncertainty principle states that an electron cannot simultaneously have an exact location and velocity in the way that a planet does. Instead of classical orbits, electrons are said to inhabit ''
atomic orbital In atomic theory and quantum mechanics, an atomic orbital is a function describing the location and wave-like behavior of an electron in an atom. This function can be used to calculate the probability of finding any electron of an atom in an ...
s''. An orbital is the "cloud" of possible locations in which an electron might be found, a distribution of probabilities rather than a precise location. Each orbital is three dimensional, rather than the two-dimensional orbit, and is often depicted as a three-dimensional region within which there is a 95 percent probability of finding the electron."Orbital (chemistry and physics)", ''Encyclopædia Britannica ''
/ref> Schrödinger was able to calculate the energy levels of hydrogen by treating a hydrogen atom's
electron The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family, and are generally thought to be elementary particles because they have no ...
as a wave, represented by the "
wave function A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements ...
" , in an
electric potential The electric potential (also called the ''electric field potential'', potential drop, the electrostatic potential) is defined as the amount of work energy needed to move a unit of electric charge from a reference point to the specific point in ...
well A well is an excavation or structure created in the ground by digging, driving, or drilling to access liquid resources, usually water. The oldest and most common kind of well is a water well, to access groundwater in underground aquifers. The ...
, , 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 match 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 In electromagnetism, the magnetic moment is the magnetic strength and orientation of a magnet or other object that produces a magnetic field. Examples of objects that have magnetic moments include loops of electric current (such as electromagne ...
of the orbital around the -axis. # The "spin" of the electron. The collective name for these properties is the
quantum state In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution i ...
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 In quantum physics and chemistry, quantum numbers describe values of conserved quantities in the dynamics of a quantum system. Quantum numbers correspond to eigenvalues of operators that commute with the Hamiltonian—quantities that can be k ...
. The quantum state of the electron is described by its wave function. The Pauli exclusion principle demands that no two electrons within an atom may have the same values of all four numbers. The first property describing the orbital is the
principal quantum number In quantum mechanics, the principal quantum number (symbolized ''n'') is one of four quantum numbers assigned to each electron in an atom to describe that electron's state. Its values are natural numbers (from 1) making it a discrete variable. A ...
, , which is the same as in Bohr's model. denotes the energy level of each orbital. The possible values for are integers: :n = 1, 2, 3\ldots The next quantum number, the
azimuthal quantum number The azimuthal quantum number is a quantum number for an atomic orbital that determines its orbital angular momentum and describes the shape of the orbital. The azimuthal quantum number is the second of a set of quantum numbers that describe ...
, denoted , describes the shape of the orbital. The shape is a consequence of the
angular momentum In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syst ...
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 (where is the principal quantum number of the electron): :l = 0, 1, \ldots, n-1. The shape of each orbital is usually referred to by a letter, rather than by its azimuthal quantum number. The first shape (=0) is denoted by the letter (a
mnemonic A mnemonic ( ) device, or memory device, is any learning technique that aids information retention or retrieval (remembering) in the human memory for better understanding. Mnemonics make use of elaborative encoding, retrieval cues, and imag ...
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 orbital In atomic theory and quantum mechanics, an atomic orbital is a function describing the location and wave-like behavior of an electron in an atom. This function can be used to calculate the probability of finding any electron of an atom in an ...
), and are denoted by the letters , , , etc. The third quantum number, the
magnetic quantum number In atomic physics, the magnetic quantum number () is one of the four quantum numbers (the other three being the principal, azimuthal, and spin) which describe the unique quantum state of an electron. The magnetic quantum number distinguishes the ...
, describes the
magnetic moment In electromagnetism, the magnetic moment is the magnetic strength and orientation of a magnet or other object that produces a magnetic field. Examples of objects that have magnetic moments include loops of electric current (such as electromagne ...
of the electron, and is denoted by (or simply ''m''). The possible values for are integers from to (where is the azimuthal quantum number of the electron): :m_l = -l, -(l-1), \ldots, 0, \ldots, (l-1), l. The magnetic quantum number measures the component of the angular momentum in a particular direction. The choice of direction is arbitrary; conventionally the z-direction is chosen. The fourth quantum number, the spin quantum number (pertaining to the "orientation" of the electron's spin) is denoted , with values + or −. The chemist Linus Pauling wrote, by way of example: It is the underlying structure and symmetry of atomic orbitals, and the way that electrons fill them, that leads to the organization of the periodic table. The way the atomic orbitals on different atoms combine to form
molecular orbital In chemistry, a molecular orbital is a mathematical function describing the location and 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 findin ...
s determines the structure and strength of chemical bonds between atoms.


Dirac wave equation

In 1928,
Paul Dirac Paul Adrien Maurice Dirac (; 8 August 1902 – 20 October 1984) was an English theoretical physicist who is regarded as one of the most significant physicists of the 20th century. He was the Lucasian Professor of Mathematics at the Univer ...
extended the
Pauli equation In quantum mechanics, the Pauli equation or Schrödinger–Pauli equation is the formulation of the Schrödinger equation for spin-½ particles, which takes into account the interaction of the particle's spin with an external electromagnetic f ...
, which described spinning electrons, to account for
special relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates: # The laws ...
. 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 light The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit ...
. By using the simplest
electromagnetic interaction In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions o ...
, 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 physics. He was able to solve for the spectral lines of the hydrogen atom and to reproduce from physical first principles
Sommerfeld Arnold Johannes Wilhelm Sommerfeld, (; 5 December 1868 – 26 April 1951) was a German theoretical physicist who pioneered developments in atomic and quantum physics, and also educated and mentored many students for the new era of theoretica ...
's successful formula for the fine structure 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 The positron or antielectron is the antiparticle or the antimatter counterpart of the electron. It has an electric charge of +1 '' e'', a spin of 1/2 (the same as the electron), and the same mass as an electron. When a positron collides w ...
and a dynamical vacuum. This led to the many-particle quantum field theory.


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 each of 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 probability that is the square of the absolute value of the sum of the complex-valued 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 we have two color-coded states of photons: one state labeled ''blue'' and another state labeled ''red''. Let the superposition of the red and the blue state appear (in imagination) as a ''purple'' state. We consider a case in which two photons are produced as the result of one single 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. In this case, the photons are interconnected via their shared origin in a single atomic event. This setup results in superimposed states of the photons. So the two photons come out ''purple.'' If the experimenter now performs some experiment that determines 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 after its twin had been measured, it would necessarily show up 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 classically, 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 paradox EPR may refer to: Science and technology * EPR (nuclear reactor), European Pressurised-Water Reactor * EPR paradox (Einstein–Podolsky–Rosen paradox), in physics * Earth potential rise, in electrical engineering * East Pacific Rise, a mid-oc ...
. 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 interpretation The Copenhagen interpretation is a collection of views about the meaning of quantum mechanics, principally attributed to Niels Bohr and Werner Heisenberg. It is one of the oldest of numerous proposed interpretations of quantum mechanics, as feat ...
, 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 that objectively exist in nature. (Einstein, Podolsky, & Rosen 1935 is currently Einstein's most cited publication in physics journals.) In the same year,
Erwin Schrödinger Erwin Rudolf Josef Alexander Schrödinger (, ; ; 12 August 1887 – 4 January 1961), sometimes written as or , was a Nobel Prize-winning Austrian physicist with Irish citizenship who developed a number of fundamental results in quantum theo ...
used the word "entanglement" and declared: "I would not call that ''one'' but rather ''the'' characteristic trait of quantum mechanics." Ever since Irish physicist John Stewart Bell theoretically and experimentally disproved the "hidden variables" theory of Einstein, Podolsky, and Rosen, most physicists have accepted entanglement as a real phenomenon. However, there is some minority 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 Dirac Paul Adrien Maurice Dirac (; 8 August 1902 – 20 October 1984) was an English theoretical physicist who is regarded as one of the most significant physicists of the 20th century. He was the Lucasian Professor of Mathematics at the Univer ...
, when he attempted to quantize the energy of the electromagnetic field; just like in quantum mechanics the energy of an electron in the hydrogen atom was quantized. Quantization is a procedure for constructing a quantum theory starting from a classical theory. '' Merriam-Webster'' defines a ''field'' in physics as "a region or space in which a given effect (such as magnetism) exists". Other effects that manifest themselves as fields are gravitation and static electricity."Field"
''Encyclopædia Britannica''
In 2008, physicist Richard Hammond wrote:
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".Richard Hammond, ''The Unknown Universe'', New Page Books, 2008. In 1931, Dirac proposed the existence of particles that later became known as
antimatter In modern physics, antimatter is defined as matter composed of the antiparticles (or "partners") of the corresponding particles in "ordinary" matter. Antimatter occurs in natural processes like cosmic ray collisions and some types of radioac ...
. Dirac shared the
Nobel Prize in Physics ) , image = Nobel Prize.png , alt = A golden medallion with an embossed image of a bearded man facing left in profile. To the left of the man is the text "ALFR•" then "NOBEL", and on the right, the text (smaller) "NAT•" then " ...
for 1933 with Schrödinger "for the discovery of new productive forms of
atomic theory Atomic theory is the scientific theory that matter is composed of particles called atoms. Atomic theory traces its origins to an ancient philosophical tradition known as atomism. According to this idea, if one were to take a lump of matter ...
". On its face, quantum field theory allows infinite numbers of particles and leaves it up to the theory itself to predict how many and with which probabilities or numbers they should exist. When developed further, the theory often contradicts observation, so that its creation and annihilation operators can be empirically tied down. Furthermore, empirical conservation laws such as that of mass–energy suggest certain constraints on the mathematical form of the theory, which are mathematically speaking finicky. The latter fact makes quantum field theories difficult to handle, but has also led to further restrictions on admissible forms of the theory; the complications are mentioned below under the rubric of
renormalization Renormalization is a collection of techniques in quantum field theory, the statistical mechanics of fields, and the theory of self-similar geometric structures, that are used to treat infinities arising in calculated quantities by altering va ...
.


Quantum electrodynamics

Quantum electrodynamics (QED) is the name of the quantum theory of the
electromagnetic force In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions o ...
. Understanding QED begins with understanding
electromagnetism In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions of ...
. Electromagnetism can be called "electrodynamics" because it is a dynamic interaction between electrical and
magnetic force In physics (specifically in electromagnetism) the Lorentz force (or electromagnetic force) is the combination of electric and magnetic force on a point charge due to electromagnetic fields. A particle of charge moving with a velocity in an e ...
s. Electromagnetism begins with the
electric charge Electric charge is the physical property of matter that causes charged matter to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative'' (commonly carried by protons and electrons respe ...
. Electric charges are the sources of and create, electric fields. An electric field is a field that 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 changing magnetic field, in turn, causes electric current (often moving electrons). The physical description of interacting
charged particle In physics, a charged particle is a particle with an electric charge. It may be an ion, such as a molecule or atom with a surplus or deficit of electrons relative to protons. It can also be an electron or a proton, or another elementary pa ...
s, electrical currents, electrical fields, and magnetic fields is called electromagnetism. In 1928
Paul Dirac Paul Adrien Maurice Dirac (; 8 August 1902 – 20 October 1984) was an English theoretical physicist who is regarded as one of the most significant physicists of the 20th century. He was the Lucasian Professor of Mathematics at the Univer ...
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 In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
. Years later,
renormalization Renormalization is a collection of techniques in quantum field theory, the statistical mechanics of fields, and the theory of self-similar geometric structures, that are used to treat infinities arising in calculated quantities by altering va ...
largely solved this problem. Initially viewed as a provisional, suspect 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 diagrams depicted all possible interactions on a given event. The diagrams showed in particular that the electromagnetic force is the exchange of photons between interacting particles. The
Lamb shift In physics, the Lamb shift, named after Willis Lamb, is a difference in energy between two energy levels 2''S''1/2 and 2''P''1/2 (in term symbol notation) of the hydrogen atom which was not predicted by the Dirac equation, according to which th ...
is an example of a quantum electrodynamics prediction that has been experimentally verified. It is an effect whereby the quantum nature of the electromagnetic field makes the energy levels in an atom or ion deviate slightly from what they would otherwise be. As a result, spectral lines may shift or split. Similarly, within a freely propagating electromagnetic wave, the current can also be just an abstract
displacement current In electromagnetism, displacement current density is the quantity appearing in Maxwell's equations that is defined in terms of the rate of change of , the electric displacement field. Displacement current density has the same units as electric ...
, instead of involving charge carriers. In QED, its full description makes essential use of short-lived
virtual particles A virtual particle is a theoretical transient particle that exhibits some of the characteristics of an ordinary particle, while having its existence limited by the uncertainty principle. The concept of virtual particles arises in the perturbat ...
. There, QED again validates an earlier, rather mysterious concept.


Standard Model

In the 1960s
physicist A physicist is a scientist who specializes in the field of physics, which encompasses the interactions of matter and energy at all length and time scales in the physical universe. Physicists generally are interested in the root or ultimate cau ...
s realized that QED broke down at extremely high energies. From this inconsistency the Standard Model of particle physics was discovered, which remedied the higher energy breakdown in theory. It is another extended quantum field theory that unifies the electromagnetic and
weak interaction In nuclear physics and particle physics, the weak interaction, which is also often called the weak force or weak nuclear force, is one of the four known fundamental interactions, with the others being electromagnetism, the strong interaction ...
s into one theory. This is called the
electroweak theory In particle physics, the electroweak interaction or electroweak force is the unified description of two of the four known fundamental interactions of nature: electromagnetism and the weak interaction. Although these two forces appear very differe ...
. Additionally, the Standard Model contains a high energy unification of the electroweak theory with the
strong force The strong interaction or strong force is a fundamental interaction that confines quarks into proton, neutron, and other hadron particles. The strong interaction also binds neutrons and protons to create atomic nuclei, where it is called the ...
, described by quantum chromodynamics. It also postulates a connection with
gravity In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the stro ...
as yet another gauge theory, but the connection is as of 2015 still poorly understood. The theory's successful prediction of the
Higgs particle The Higgs boson, sometimes called the Higgs particle, is an elementary particle in the Standard Model of particle physics produced by the quantum excitation of the Higgs field, one of the fields in particle physics theory. In the Stand ...
to explain inertial mass was confirmed by the Large Hadron Collider, and thus the Standard model is now considered the basic and more or less complete description of
particle physics Particle physics or high energy physics is the study of fundamental particles and forces that constitute matter and radiation. The fundamental particles in the universe are classified in the Standard Model as fermions (matter particles) an ...
as we know it.


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. These interpretations are widely varying and sometimes somewhat abstract. For instance, the
Copenhagen interpretation The Copenhagen interpretation is a collection of views about the meaning of quantum mechanics, principally attributed to Niels Bohr and Werner Heisenberg. It is one of the oldest of numerous proposed interpretations of quantum mechanics, as feat ...
states that before a measurement, statements about a particle's properties are completely meaningless, while in the
Many-worlds interpretation The many-worlds interpretation (MWI) is an interpretation of quantum mechanics that asserts that the universal wavefunction is objectively real, and that there is no wave function collapse. This implies that all possible outcomes of quantum ...
describes the existence of a multiverse made up of every possible universe.


Applications

Applications of quantum mechanics include the
laser A laser is a device that emits light through a process of optical amplification based on the stimulated emission of electromagnetic radiation. The word "laser" is an acronym for "light amplification by stimulated emission of radiation". The fi ...
, the
transistor upright=1.4, gate (G), body (B), source (S) and drain (D) terminals. The gate is separated from the body by an insulating layer (pink). A transistor is a semiconductor device used to Electronic amplifier, amplify or electronic switch, switch ...
, the
electron microscope An electron microscope is a microscope that uses a beam of accelerated electrons as a source of illumination. As the wavelength of an electron can be up to 100,000 times shorter than that of visible light photons, electron microscopes have a hi ...
, and magnetic resonance imaging. A special class of quantum mechanical applications is related to
macroscopic quantum phenomena Macroscopic quantum phenomena are processes showing quantum behavior at the macroscopic scale, rather than at the atomic scale where quantum effects are prevalent. The best-known examples of macroscopic quantum phenomena are superfluidity and su ...
such as superfluid helium and superconductors. The study of semiconductors led to the invention of the diode and the
transistor upright=1.4, gate (G), body (B), source (S) and drain (D) terminals. The gate is separated from the body by an insulating layer (pink). A transistor is a semiconductor device used to Electronic amplifier, amplify or electronic switch, switch ...
, which are indispensable for modern
electronics The field of electronics is a branch of physics and electrical engineering that deals with the emission, behaviour and effects of electrons using electronic devices. Electronics uses active devices to control electron flow by amplification ...
. In even the simple
light switch In electrical wiring, a light switch is a switch most commonly used to operate electric lights, permanently connected equipment, or electrical outlets. Portable lamps such as table lamps may have a light switch mounted on the socket, base, or i ...
,
quantum tunneling In physics, a quantum (plural quanta) is the minimum amount of any physical entity (physical property) involved in an interaction. The fundamental notion that a physical property can be "quantized" is referred to as "the hypothesis of quantizati ...
is absolutely vital, as otherwise the electrons in the electric current could not penetrate the potential barrier made up of a layer of oxide. Flash memory chips found in USB drives also use quantum tunneling, to erase their memory cells.


See also

*
Einstein's thought experiments A hallmark of Albert Einstein's career was his use of visualized thought experiments (german: Gedankenexperiment) as a fundamental tool for understanding physical issues and for elucidating his concepts to others. Einstein's thought experiments too ...
*
Macroscopic quantum phenomena Macroscopic quantum phenomena are processes showing quantum behavior at the macroscopic scale, rather than at the atomic scale where quantum effects are prevalent. The best-known examples of macroscopic quantum phenomena are superfluidity and su ...
*
Philosophy of physics In philosophy, philosophy of physics deals with conceptual and interpretational issues in modern physics, many of which overlap with research done by certain kinds of theoretical physicists. Philosophy of physics can be broadly divided into thr ...
* Quantum computing *
Virtual particle A virtual particle is a theoretical transient particle that exhibits some of the characteristics of an ordinary particle, while having its existence limited by the uncertainty principle. The concept of virtual particles arises in the perturba ...
*
List of textbooks on classical and quantum mechanics This is a list of notable textbooks on classical mechanics and quantum mechanics arranged according to level and surnames of the authors in alphabetical order. Undergraduate Classical mechanics * * * * * * * * Quantum mechanics * Three ...


Notes


References


Bibliography

* * * * * * * * * * * * * * * * * * * * ''Scientific American Reader'', 1953. * * ; cited in: * * Van Vleck, J. H.,1928, "The Correspondence Principle in the Statistical Interpretation of Quantum Mechanics", ''Proc. Natl. Acad. Sci.'' 14: 179. * * *


Further reading

The following titles, all by working physicists, attempt to communicate quantum theory to laypeople, using a minimum of technical apparatus. *
Jim Al-Khalili Jameel Sadik "Jim" Al-Khalili ( ar, جميل صادق الخليلي; born 20 September 1962) is an Iraqi-British theoretical physicist, author and broadcaster. He is professor of theoretical physics and chair in the public engagement in scien ...
(2003). ''Quantum: A Guide for the Perplexed''. Weidenfeld & Nicolson. . * Chester, Marvin (1987). ''Primer of Quantum Mechanics''. John Wiley. . * Brian Cox and Jeff Forshaw (2011) ''
The Quantum Universe ''The Quantum Universe: Everything That Can Happen Does Happen'' is a 2011 book by the theoretical physicists Brian Cox (physicist), Brian Cox and Jeff Forshaw. Overview The book aims to provide an explanation of quantum mechanics and its impact ...
''. Allen Lane. . *
Richard Feynman Richard Phillips Feynman (; May 11, 1918 – February 15, 1988) was an American theoretical physicist, known for his work in the path integral formulation of quantum mechanics, the theory of quantum electrodynamics, the physics of the superfl ...
(1985). '' QED: The Strange Theory of Light and Matter''. Princeton University Press. . * 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
algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary ...
,
trigonometry Trigonometry () is a branch of mathematics that studies relationships between side lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies ...
, and bra–ket notation can be passed over on a first reading. * Tony Hey 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. . * J. P. McEvoy and Oscar Zarate (2004). ''Introducing Quantum Theory''. Totem Books. ' *
N. David Mermin Nathaniel David Mermin (; born 30 March 1935) is a solid-state physicist at Cornell University best known for the eponymous Mermin–Wagner theorem, his application of the term " boojum" to superfluidity, his textbook with Neil Ashcroft on sol ...
(1990). "Spooky actions at a distance: mysteries of the QT" in his ''Boojums all the way through''. Cambridge Univ. Press: 110–76. The author is a rare physicist who tries to communicate to philosophers and humanists. . *
Roland Omnès Roland Omnès (born 18 February 1931), is the author of several books which aim to give non-scientists the information required to understand quantum mechanics from an everyday standpoint. Biography Omnès is currently Professor Emeritus of Th ...
(1999). ''Understanding Quantum Mechanics''. Princeton Univ. Press. . *
Victor Stenger Victor John Stenger (; January 29, 1935 – August 25, 2014) was an American particle physicist, philosopher, author, and religious skeptic. Following a career as a research scientist in the field of particle physics, Stenger was associat ...
(2000). ''Timeless Reality: Symmetry, Simplicity, and Multiple Universes''. Buffalo NY: Prometheus Books. Chpts. 5–8. . *
Martinus Veltman Martinus Justinus Godefriedus "Tini" Veltman (; 27 June 1931 – 4 January 2021) was a Dutch theoretical physicist. He shared the 1999 Nobel Prize in physics with his former PhD student Gerardus 't Hooft for their work on particle theory. Biogr ...
(2003). ''Facts and Mysteries in Elementary Particle Physics''. World Scientific Publishing Company. .


External links

*
Microscopic World – Introduction to Quantum Mechanics".
by Takada, Kenjiro, Emeritus professor at
Kyushu University , abbreviated to , is a Japanese national university located in Fukuoka, on the island of Kyushu. It was the 4th Imperial University in Japan, ranked as 4th in 2020 Times Higher Education Japan University Rankings, one of the top 10 Design ...

The Quantum Exchange
(tutorials and open-source learning software).
Atoms and the Periodic Table

Single and double slit interference

Time-Evolution of a Wavepacket in a Square Well
An animated demonstration of a wave packet dispersion over time. * {{DEFAULTSORT:Quantum mechanics, Introduction to Articles containing video clips