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Encyclopedia > Introduction to quantum mechanics
Quantum mechanics
Delta x , Delta p ge frac{hbar}{2}

Introduction to...
Mathematical formulation of... For a less technical and generally accessible introduction to the topic, see Introduction to quantum mechanics. ... The mathematical formulation of quantum mechanics is the body of mathematical formalisms which permits a rigorous description of quantum mechanics. ...

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Werner Heisenberg and Erwin Schrödinger, founders of Quantum Mechanics.
Werner Heisenberg and Erwin Schrödinger, founders of Quantum Mechanics.

Quantum mechanics (QM, or quantum theory) is a physical science dealing with the behaviour of matter and energy on the scale of atoms and subatomic particles / waves.[1] QM also forms the basis for the contemporary understanding of how very large objects such as stars and galaxies, and cosmological events such as the Big Bang, can be analyzed and explained. Quantum mechanics is the foundation of several related disciplines including nanotechnology, condensed matter physics, quantum chemistry, structural biology, particle physics, and electronics. Image File history File links Quantum_intro_pic-smaller. ... Image File history File links Quantum_intro_pic-smaller. ... Werner Karl Heisenberg (December 5, 1901 – February 1, 1976) was a celebrated German physicist and Nobel laureate, one of the founders of quantum mechanics and acknowledged to be one of the most important physicists of the twentieth century. ... Schrödinger in 1933, when he was awarded the Nobel Prize in Physics Bust of Schrödinger, in the courtyard arcade of the main building, University of Vienna, Austria. ... A magnet levitating above a high-temperature superconductor demonstrates the Meissner effect. ... This article is about matter in physics and chemistry. ... Properties For alternative meanings see atom (disambiguation). ... Helium atom (not to scale) Showing two protons (red), two neutrons (green) and a probability cloud (gray) of two electrons (yellow). ... Surface waves in water This article is about waves in the most general scientific sense. ... This article is about the astronomical object. ... This article is about a celestial body. ... This article is about the physics subject. ... According to the Big Bang theory, the universe originated in an infinitely dense and physically paradoxical singularity. ... Nanotechnology refers broadly to a field of applied science and technology whose unifying theme is the control of matter on the atomic and molecular scale, normally 1 to 100 nanometers, and the fabrication of devices within that size range. ... Condensed matter physics is the field of physics that deals with the macroscopic physical properties of matter. ... Quantum chemistry is a branch of theoretical chemistry, which applies quantum mechanics and quantum field theory to address issues and problems in chemistry. ... Structural biology is a branch of molecular biology concerned with the study of the architecture and shape of biological macromolecules--proteins and nucleic acids in particular—and what causes them to have the structures they have. ... Thousands of particles explode from the collision point of two relativistic (100 GeV per nucleon) gold ions in the STAR detector of the Relativistic Heavy Ion Collider. ... This article is about the engineering discipline. ...


The term "quantum mechanics" was first coined by Max Born in 1924. The acceptance by the general physics community of quantum mechanics is due to its accurate prediction of the physical behaviour of systems, including systems where Newtonian mechanics fails. Even general relativity is limited—in ways quantum mechanics is not—for describing systems at the atomic scale or smaller, at very low or very high energies, or at the lowest temperatures. Through a century of experimentation and applied science, quantum mechanical theory has proven to be very successful and practical. Max Born (December 11, 1882 in Breslau – January 5, 1970 in Göttingen) was a mathematician and physicist. ... Classical mechanics (commonly confused with Newtonian mechanics, which is a subfield thereof) is used for describing the motion of macroscopic objects, from projectiles to parts of machinery, as well as astronomical objects, such as spacecraft, planets, stars, and galaxies. ... For a less technical and generally accessible introduction to the topic, see Introduction to general relativity. ...

Contents

Overview

The foundations of quantum mechanics date from the early 1800s, but the real beginnings of QM date from the work of Max Planck in 1900.[2] Albert Einstein[3] and Niels Bohr[4] soon made important contributions to what is now called the "old quantum theory." However, it was not until 1924 that a more complete picture emerged with Louis de Broglie's matter-wave hypothesis and the true importance of quantum mechanics became clear.[5] Some of the most prominent scientists to subsequently contribute in the mid-1920s to what is now called the "new quantum mechanics" or "new physics" were Max Born,[6] Paul Dirac,[7] Werner Heisenberg,[8] Wolfgang Pauli,[9] and Erwin Schrödinger.[10] Later, the field was further expanded with work by Julian Schwinger, Sin-Itiro Tomonaga and Richard Feynman for the development of Quantum Electrodynamics in 1947 and by Murray Gell-Mann in particular for the development of Quantum Chromodynamics. “Planck” redirects here. ... “Einstein” redirects here. ... Niels Henrik David Bohr (October 7, 1885 – November 18, 1962) was a Danish physicist who made fundamental contributions to understanding atomic structure and quantum mechanics, for which he received the Nobel Prize in Physics in 1922. ... Louis-Victor-Pierre-Raymond, 7th duc de Broglie, generally known as Louis de Broglie (August 15, 1892–March 19, 1987), was a French physicist and Nobel Prize laureate. ... In physics, the de Broglie hypothesis is the statement that all matter (any object) has a wave-like nature (wave-particle duality). ... Max Born (December 11, 1882 in Breslau – January 5, 1970 in Göttingen) was a mathematician and physicist. ... Paul Adrien Maurice Dirac, OM, FRS (IPA: [dɪræk]) (August 8, 1902 – October 20, 1984) was a British theoretical physicist and a founder of the field of quantum physics. ... Werner Karl Heisenberg (December 5, 1901 – February 1, 1976) was a celebrated German physicist and Nobel laureate, one of the founders of quantum mechanics and acknowledged to be one of the most important physicists of the twentieth century. ... This article is about the Austrian-Swiss physicist. ... Schrödinger in 1933, when he was awarded the Nobel Prize in Physics Bust of Schrödinger, in the courtyard arcade of the main building, University of Vienna, Austria. ... Julian Seymour Schwinger (February 12, 1918 -- July 16, 1994) was an American theoretical physicist. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... This article is about the physicist. ... Quantum electrodynamics (QED) is a relativistic quantum field theory of electrodynamics. ... Murray Gell-Mann (born September 15, 1929 in Manhattan, New York City, USA) is an American physicist who received the 1969 Nobel Prize in physics for his work on the theory of elementary particles. ... Quantum chromodynamics (abbreviated as QCD) is the theory of the strong interaction (color force), a fundamental force describing the interactions of the quarks and gluons found in hadrons (such as the proton, neutron or pion). ...

The interference that produces colored bands on bubbles cannot be explained by a model that depicts light as a particle. It can be explained by a model that depicts it as a wave. The drawing shows sine waves that resemble waves on the surface of water being reflected from two surfaces of a film of varying width, but that depiction of the wave nature of light is only a crude analogy.
The interference that produces colored bands on bubbles cannot be explained by a model that depicts light as a particle. It can be explained by a model that depicts it as a wave. The drawing shows sine waves that resemble waves on the surface of water being reflected from two surfaces of a film of varying width, but that depiction of the wave nature of light is only a crude analogy.

Early researchers differed in their explanations of the fundamental nature of what we now call electromagnetic radiation. Some maintained that light and other frequencies of electromagnetic radiation are composed of particles, while others asserted that electromagnetic radiation is a wave phenomenon. In classical physics these ideas are mutually contradictory. Ever since the early days of QM scientists have acknowledged that neither idea by itself can explain electromagnetic radiation. Image File history File links Download high resolution version (720x704, 28 KB) File links The following pages link to this file: Basics of quantum mechanics ... Image File history File links Download high resolution version (720x704, 28 KB) File links The following pages link to this file: Basics of quantum mechanics ... For other uses, see Light (disambiguation). ... Helium atom (schematic) Showing two protons (red), two neutrons (green) and two electrons (yellow). ... Surface waves in water This article is about waves in the most general scientific sense. ... In trigonometry, an ideal sine wave is a waveform whose graph is identical to the generalized sine function y = Asin[ω(x − α)] + C, where A is the amplitude, ω is the angular frequency (2π/P where P is the wavelength), α is the phase shift, and C... Impact from a water drop causes an upward rebound jet surrounded by circular capillary waves. ... This box:      Electromagnetic (EM) radiation is a self-propagating wave in space with electric and magnetic components. ...


In 1690, Christian Huygens explained the laws of reflection and refraction on the basis of a wave theory.[11] Sir Isaac Newton believed that light consisted of infinitesimally small particles which he designated "corpuscles". In 1827 Thomas Young and Augustin Fresnel made experiments on interference that showed that a corpuscular theory of light was inadequate. Then in 1873 James Clerk Maxwell showed that by making an electrical circuit oscillate it should be possible to produce electromagnetic waves. His theory made it possible to compute the speed of electromagnetic radiation purely on the basis of electrical and magnetic measurements, and the computed value corresponded very closely to the empirically measured speed of light.[12] In 1888, Heinrich Hertz made an electrical device that actually produced what we would now call microwaves — essentially radiation at a lower frequency than visible light.[citation needed] Everything up to that point suggested that Newton had been entirely wrong to regard light as corpuscular. Christiaan Huygens Christiaan Huygens (approximate pronunciation: HOW-khens; SAMPA /h9yGEns/ or /h@YG@ns/) (April 14, 1629–July 8, 1695), was a Dutch mathematician and physicist; born in The Hague as the son of Constantijn Huygens. ... Sir Isaac Newton FRS (4 January 1643 – 31 March 1727) [ OS: 25 December 1642 – 20 March 1727][1] was an English physicist, mathematician, astronomer, natural philosopher, and alchemist. ... Thomas Young, English scientist Thomas Young (June 13, 1773-May 10, 1829) was an English polymath, contributing to the scientific understanding of vision, light, solid mechanics, energy, physiology, and Egyptology. ... Augustin Fresnel Augustin-Jean Fresnel (pronounced fray-NELL) (May 10, 1788 – July 14, French physicist who contributed significantly to the establishment of the wave theory of light and optics. ... For other uses, see Interference (disambiguation). ... This article does not cite its references or sources. ... James Clerk Maxwell (13 June 1831 – 5 November 1879) was a Scottish mathematician and theoretical physicist from Edinburgh, Scotland, UK. His most significant achievement was aggregating a set of equations in electricity, magnetism and inductance — eponymously named Maxwells equations — including an important modification (extension) of the Ampères... For thermodynamic relations, see Maxwell relations. ... The speed of light in a vacuum is an important physical constant denoted by the letter c for constant or the Latin word celeritas meaning swiftness.[1] It is the speed of all electromagnetic radiation, including visible light, in a vacuum. ... Heinrich Rudolf Hertz (February 22, 1857 - January 1, 1894) was the German physicist and mechanician for whom the hertz, an SI unit, is named. ... For other uses, see Frequency (disambiguation). ...


Later experiments indicated that a packet or quantum model was needed to explain some phenomena. When light strikes an electrical conductor it causes electrons to move away from their original positions. The observed phenomenon could only be explained by assuming that the light delivers energy in definite packets. In a photoelectric device such as the light meter in a camera, light hitting the metallic detector causes electrons to move. Greater intensities of light at one frequency can cause more electrons to move, but they will not move faster. In contrast, higher frequencies of light can cause electrons to move faster. Ergo, intensity of light controls current, but frequency of light controls voltage. These observations raised a contradiction when compared with sound waves and ocean waves, where only intensity was needed to predict the energy of the wave. In the case of light, frequency appeared to predict energy. Something was needed to explain this phenomenon and to reconcile experiments that had shown light to have particle nature with experiments that had shown it to have wave nature. In science and engineering, conductors, such as copper or aluminum, are materials with atoms having loosely held valence electrons. ... The photoelectric effect is the emission of electrons from matter upon the absorption of electromagnetic radiation, such as ultraviolet radiation or x-rays. ... For other uses, see Electron (disambiguation). ... In physics, intensity is a measure of the time-averaged energy flux. ... For other uses, see Frequency (disambiguation). ... For other uses, see Frequency (disambiguation). ... This box:      Electric current is the flow (movement) of electric charge. ... International safety symbol Caution, risk of electric shock (ISO 3864), colloquially known as high voltage symbol. ... This article is about compression waves. ... Categories: Physics stubs | Physical oceanography | Waves ...


Despite the success of quantum mechanics, it does have some controversial elements. For example, the behaviour of microscopic objects described in quantum mechanics is very different from our everyday experience, which may provoke some degree of incredulity. Most of classical physics is now recognized to be composed of special cases of quantum physics theory and/or relativity theory. Dirac brought relativity theory to bear on quantum physics so that it could properly deal with events that occur at a substantial fraction of the speed of light. Classical physics, however, also deals with mass attraction (gravity), and no one has yet been able to bring gravity into a unified theory with the relativized quantum theory. Paul Adrien Maurice Dirac, OM, FRS (IPA: [dɪræk]) (August 8, 1902 – October 20, 1984) was a British theoretical physicist and a founder of the field of quantum physics. ...


Spectroscopy and onward

It is fairly easy to see a spectrum produced by white light when it passes through a prism, the bevelled edge of a mirror or a tapered pane of glass, or through drops of rain to form a rainbow. When samples of single elements are caused to emit light they may emit light at several characteristic frequencies. The frequency profile produced is characteristic of that element. Instead of there being a wide band filled with colors from violet to red, there will be isolated bands of single colors separated by darkness. Such a display is called a line spectrum. Some lines go beyond the visible frequencies and can only be detected by special photographic film or other such devices. Scientists hypothesized that an atom could radiate light the way the string on a fine violin radiates sound – not only with a fundamental frequency (in which the entire string moves the same way at once) but with several higher harmonics (formed when the string divides itself into halves and other divisions that vibrate in coordination with each other as when one half of the string is going one way as the other half of the string is going the opposite way). For a long time nobody could find a mathematical way to relate the frequencies of the line spectrum of any element. Vibration and standing waves in a string, The fundamental and the first 6 overtones The fundamental tone, often referred to simply as the fundamental and abbreviated fo, is the lowest frequency in a harmonic series. ...

NASA photo of the bright-line spectrum of hydrogen
NASA photo of the bright-line spectrum of hydrogen
Photo of the bright-line spectrum of nitrogen
Photo of the bright-line spectrum of nitrogen

In 1885, Johann Jakob Balmer (1825-1898) figured out how the frequencies of atomic hydrogen are related to each other. The formula is a simple one: Image File history File linksMetadata NASA_Hydrogen_spectrum. ... Image File history File linksMetadata NASA_Hydrogen_spectrum. ... Image File history File links Nitrogen. ... Image File history File links Nitrogen. ... Johann Jakob Balmer (May 1, 1825 – March 12, 1898) was a Swiss mathematician and an honorary physicist. ...

frac{1}{lambda} = Rleft ( frac{1}{2^2} - frac{1}{n^2} right )

where λ is wavelength, R is the Rydberg constant and n is an integer (n =3, 4,...) This formula can be generalized to apply to atoms that are more complicated than hydrogen, but we will stay with hydrogen for this general exposition. (That is the reason that the denominator in the first fraction is expressed as a square.) The Rydberg constant, named after physicist Janne Rydberg, is a physical constant discovered when measuring the spectrum of hydrogen, and building upon results from Anders Jonas Ã…ngström and Johann Balmer. ...


The next development was the discovery of the Zeeman effect, named after Pieter Zeeman (1865-1943). The physical explanation of the Zeeman effect was worked out by Hendrik Antoon Lorentz (1853-1928). Lorentz hypothesized that the light emitted by hydrogen was produced by vibrating electrons. It was possible to get feedback on what goes on within the atom because moving electrons create a magnetic field and so can be influenced by the imposition of an external magnetic field in a manner analogous to the way that one iron magnet will attract or repel another magnet. The Zeeman effect (IPA ) is the splitting of a spectral line into several components in the presence of a magnetic field. ... Hendrik Antoon Lorentz (July 18, 1853, Arnhem – February 4, 1928, Haarlem) was a Dutch physicist and the winner of the 1902 Nobel Prize in Physics for his work on electromagnetic radiation. ...


The Zeeman effect could be interpreted to mean that light waves are originated by electrons vibrating in their orbits, but classical physics could not explain why electrons should not fall out of their orbits and into the nucleus of their atoms, nor could classical physics explain why their orbits would be such as to produce the series of frequencies derived by Balmer’s formula and displayed in the line spectra. Why did the electrons not produce a continuous spectrum?


Old quantum theory

Quantum mechanics developed from the study of electromagnetic waves through spectroscopy which includes visible light seen in the colors of the rainbow, but also other waves including the more energetic waves like ultraviolet light, x-rays, and gamma rays plus the waves with longer wavelengths including infrared waves, microwaves and radio waves. We are not, however, speaking of sound waves, but only of those waves that travel at the speed of light. Also, when the word "particle" is used below, it always refers to elementary or subatomic particles. Electromagnetic radiation or EM radiation is a combination (cross product) of oscillating electric and magnetic fields perpendicular to each other, moving through space as a wave, effectively transporting energy and momentum. ... Animation of the dispersion of light as it travels through a triangular prism. ...


Planck's constant

Classical physics predicted that a black-body radiator would produce infinite energy, but that result was not observed in the laboratory. If black-body radiation was dispersed into a spectrum, then the amount of energy radiated at various frequencies rose from zero at one end, peaked at a frequency related to the temperature of the radiating object, and then fell back to zero. In 1900, Max Planck developed an empirical equation that could account for the observed energy curves, but he could not harmonize it with classical theory. He concluded that the classical laws of physics do not apply on the atomic scale as had been assumed. As the temperature decreases, the peak of the black-body radiation curve moves to lower intensities and longer wavelengths. ... “Planck” redirects here. ...


In this theoretical account, Planck allowed all possible frequencies, all possible wavelengths. However, he restricted the energy that is delivered. "In classical physics,... the energy of a given oscillator depends merely on its amplitude, and this amplitude is subject to no restriction."[13] But, according to Planck's theory, the energy emitted by an oscillator is strictly proportional to its frequency. The higher the frequency, the greater the energy. To reach this theoretical conclusion, he postulated that a radiating body consisted of an enormous number of elementary oscillators, some vibrating at one frequency and some at another, with all frequencies from zero to infinity being represented. The energy E of any one oscillator was not permitted to take on any arbitrary value, but was proportional to some integral multiple of the frequency f of the oscillator. That is,

E = nhf,,!

where n =1, 2, 3,... The proportionality constant h is called Planck's constant.[14] A commemoration plaque for Max Planck on his discovery of Plancks constant, in front of Humboldt University, Berlin. ...


One of the most direct applications is finding the energy of photons. If you know h, and you know the frequency of the photon, then you can calculate the energy of the photons. For instance, if a beam of light illuminated a target, and the light frequency was 540 × 1012 hertz, then the energy of each photon would be h × 540 × 1012 joules. The value of h itself is exceedingly small, about 6.6260693 × 10-34 joule seconds. This means that the photons in the beam of light have an energy of about 3.58 × 10-19 joules or approximately 2.23 eV. In physics, the photon (from Greek φως, phōs, meaning light) is the quantum of the electromagnetic field; for instance, light. ... In physics, the photon (from Greek φως, phōs, meaning light) is the quantum of the electromagnetic field; for instance, light. ... In modern physics the photon is the elementary particle responsible for electromagnetic phenomena. ... A joule-second is a unit of power over time called action or a unit of angular momentum. ...


When you describe the energy of a wave in this manner, it seems that the wave is carrying its energy in a certain number of little packets per second. This discovery then seemed to remake the wave into a particle. These packets of energy carried along with the wave were called quanta by Planck. Quantum mechanics began with the discovery that energy is delivered in packets whose size is related to the frequencies of all electromagnetic waves (and to the color of visible light since in that case frequency determines color). Be aware, however, that these descriptions in terms of wave and particle import macro-world concepts into the quantum world, where they have only provisional relevance or appropriateness. In physics quanta is the plural of quantum. ...


In early research on light, there were two competing ways to describe light, either as a wave propagated through empty space, or as small particles traveling in straight lines. Because Planck showed that the energy of the wave is made up of packets, the particle analogy became favored to help understand how light delivers energy in multiples of certain set values designated as quanta of energy. Nevertheless, the wave analogy is also indispensable for helping to understand other light phenomena. In 1905, Albert Einstein used Planck's constant to postulate that the energy in a beam of light occurs in concentrations that he called photons.[15] According to that account, 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. Although the description that stemmed from Planck's research sounds like Newton's corpuscular account, Einstein's photon was still said to have a frequency, and the energy of the photon was accounted proportional to that frequency. The particle account had been compromised once again. “Einstein” redirects here. ... In modern physics the photon is the elementary particle responsible for electromagnetic phenomena. ...


Both the idea of a wave and the idea of a particle are models derived from our everyday experience. We cannot see individual photons. We can only investigate their properties indirectly. We look at some phenomena, such as the rainbow of colors that we see when a thin film of oil rests on the surface of a puddle of water, and we can explain that phenomenon to ourselves by comparing light with waves.[16] We look at other phenomena, such as the way a photoelectric meter in our camera works, and we explain it by analogy to particles colliding with the detection screen in the meter. In both cases we take concepts from our everyday experience and apply them to a world we have never seen. A mental model is an explanation in someones thought process for how something works in the real world. ... The photoelectric effect is the emission of electrons from matter upon the absorption of electromagnetic radiation, such as visible light or ultraviolet radiation. ...


Neither form of explanation is entirely satisfactory. In general any model can only approximate that which it models. A model is useful only within the range of conditions where it is able to predict the real thing with accuracy. Newtonian physics is still a good predictor of many of the phenomena in our everyday life. To remind us that both "wave" and "particle" are concepts imported from our macro world to explain the world of atomic-scale phenomena, some physicists such as George Gamow have used the term "wavicle" to refer to whatever it is that is really there. In the following discussion, "wave" and "particle" may both be used depending on which aspect of quantum mechanical phenomena is under discussion. Model may refer to more than one thing : For models in society, art, fashion, and cosmetics, see; role model model (person) supermodel figure drawing modeling section In science and technology, a model (abstract) is understood as an abstract or theoretical representation of a phenomenon,see; geologic modeling model (economics) model... Classical mechanics is a model of the physics of forces acting upon bodies. ... George Gamow (pronounced GAM-off) (March 4, 1904 – August 19, 1968) , born Georgiy Antonovich Gamov (Георгий Антонович Гамов) was a Ukrainian born physicist and cosmologist. ... In the quantum world, particles and waves are simply twin facets of all entities. ...


Reduced Planck's constant

Relation between a cycle and a wave; half of a circle describes half of the cycle of a wave
Relation between a cycle and a wave; half of a circle describes half of the cycle of a wave

Planck's constant originally represented the energy that a light wave carries as a function of its frequency. A step in the development of this concept appeared in Bohr's work. Bohr was using a "planetary" or particle model of the electron, and could not understand why a 2π factor was essential to his experimentally derived formulae. Later, de Broglie postulated that electrons have frequencies, just as do photons, and that the frequency of an electron must conform to the conditions for a standing wave that can exist in a certain orbit. That is to say, the beginning of one cycle of a wave at some point on the circumference of a circle (since that is what an orbit is) must coincide with the end of some cycle. There can be no gap, no length along the circumference that is not participating in the vibration, and there can be no overlap of cycles. So the circumference of the orbit, C, must equal the wavelength, λ , of the electron multiplied by some positive integer (n = 1, 2, 3...). Knowing the circumference one can calculate wavelengths that fit that orbit, and knowing the radius, r, of the orbit one can calculate its circumference. To put all that in mathematical form, Image File history File links Gallery_SineWave_Generation. ... Image File history File links Gallery_SineWave_Generation. ... Vibration and standing waves in a string, The fundamental and the first 6 overtones A standing wave, also known as a stationary wave, is a wave that remains in a constant position. ...

C = 2 pi r = n lambda ,!

and so

lambda = 2 pi r/n ,!

and the appearance of the 2π factor is seen to occur simply because we need it to calculate possible wavelengths (and therefore possible frequencies) when we already know the radius of an orbit.[17]


Again in 1925 when Werner Heisenberg developed his full quantum theory, calculations involving wave analysis called Fourier series were fundamental, and so the "reduced" version of Planck's constant (h/2π) became invaluable because it includes a conversion factor to facilitate calculations involving wave analysis. Finally, when this reduced Planck's constant appeared naturally in Dirac's equation it was then given an alternate designation, "Dirac's constant." Therefore, it is appropriate to begin with an explanation of what this constant is, even though we haven't yet touched on the theories that made its use convenient. The Fourier series is a mathematical tool used for analyzing periodic functions by decomposing such a function into a weighted sum of much simpler sinusoidal component functions sometimes referred to as normal Fourier modes, or simply modes for short. ...


As noted above, the energy of any wave is given by its frequency multiplied by Planck's constant. A wave is made up of crests and troughs. In a wave, a cycle is defined by the return from a certain position to the same position such as from the top of one crest to the next crest. A cycle actually is mathematically related to a circle, and both have 360 degrees. A degree is a unit of measure for the amount of turn needed to produce an arc of a certain length at a given distance. A sine curve is generated by a point on the circumference of a circle as that circle rotates. (See a demonstration at: Rotation Applet) There are 2 π radians per cycle in a wave, which is mathematically related to the way a circle has 360° (which are equal to two π radians). (A radian is simply the angle you would get if you measured a distance along the circumference of the circle equal to the radius of the circle, and then drew lines to the center of the circle and looked at the angle thus formed.) Since one cycle is 2 π radians, when h is divided by 2π the two "2 π" factors will cancel out leaving just the radian to contend with. So, dividing h by 2π describes a constant that, when multiplied by the frequency of a wave, gives the energy in joules per radian. The reduced Planck's constant is written in mathematical formulas as ħ, and is read as "h-bar". In trigonometry, an ideal sine wave is a waveform whose graph is identical to the generalized sine function y = Asin[ω(x − α)] + C, where A is the amplitude, ω is the angular frequency (2π/P where P is the wavelength), α is the phase shift, and C... Some common angles, measured in radians. ...

hbar = frac{h}{2 pi} .

The reduced Planck's constant allows computation of the energy of a wave in units per radian instead of in units per cycle. These two constants h and ħ are merely conversion factors between energy units and frequency units. The reduced Planck's constant is used more often than h (Planck's constant) alone in quantum mechanical mathematical formulas for many reasons, one of which is that angular velocity or angular frequency is ordinarily measured in radians per second so using ħ that works in radians too will save a computation to put radians into degrees or vice-versa. Also, when equations relevant to those problems are written in terms of ħ, the frequently occurring 2π factors in numerator and denominator can cancel out, saving a computation. However, in other cases, as in the orbits of the Bohr atom, h/2π was obtained naturally for the angular momentum of the orbits. Another expression for the relation between energy and wave length is given in electron volts for energy and angstroms for wavelength: Ephoton (eV) = 12,400/λ(Å) — it appears not to involve h at all, but that is only because a different system of units has been used and now, numerically, the appropriate conversion factor is 12,400.[18]


Bohr atom

The Bohr model of the atom, showing electron quantum jumping to ground state n=1
The Bohr model of the atom, showing electron quantum jumping to ground state n=1

In 1897 the particle called the electron was discovered. By means of the gold foil experiment physicists discovered that matter is, volume for volume, largely space. Once that was clear, it was hypothesized that negative charge entities called electrons surround positively charged nuclei. So at first all scientists believed that the atom must be like a miniature solar system. But that simple analogy predicted that electrons would, within about one hundredth of a microsecond,[19] crash into the nucleus of the atom. The great question of the early 20th century was, "Why do electrons normally maintain a stable orbit around the nucleus?" Image File history File links Bohr-planetary-atom-model. ... Image File history File links Bohr-planetary-atom-model. ... For other uses, see Electron (disambiguation). ... Top: Expected results: alpha particles passing through the plum pudding model of the atom undisturbed. ... The nucleus of an atom is the very small dense region, of positive charge, in its centre consisting of nucleons (protons and neutrons). ...


In 1913, Niels Bohr removed this substantial problem by applying the idea of discrete (non-continuous) quanta to the orbits of electrons. This account became known as the Bohr model of the atom. Bohr basically theorized that electrons can only inhabit certain orbits around the atom. These orbits could be derived by looking at the spectral lines produced by atoms. Niels Henrik David Bohr (October 7, 1885 – November 18, 1962) was a Danish physicist who made fundamental contributions to understanding atomic structure and quantum mechanics, for which he received the Nobel Prize in Physics in 1922. ... The Bohr model of the atom The Bohr Model is a physical model that depicts the atom as a small positively charged nucleus with electrons in orbit at different levels, similar in structure to the solar system. ... Animation of the dispersion of light as it travels through a triangular prism. ...


Bohr explained the orbits that electrons can take by relating the angular momentum of electrons in each "permitted" orbit to the value of h, Planck's constant. He held that an electron in the lowest orbital has an angular momentum equal to h/2π. Each orbit after the initial orbit must provide for an electron's angular momentum being an integer multiple of that lowest value. He depicted electrons in atoms as being analogous to planets in a solar orbit. However, he took Planck's constant to be a fundamental quantity that introduces special requirements at this subatomic level and that explains the spacing of those "planetary" orbits.


Bohr considered one revolution in orbit to be equivalent to one cycle in an oscillator (as in Planck's initial measurements to define the constant h) which is in turn similar to one cycle in a wave. The number of revolutions per second is (or defines) what we call the frequency of that electron or that orbital. Specifying that the frequency of each orbit must be an integer multiple of Planck's constant h would only permit certain orbits, and would also fix their size.


Bohr generalized Balmer's formula for hydrogen by replacing denominator in the term 1/4 with an explicit squared variable: In atomic physics, the Balmer series is the designation of one of a set of six different named series describing the spectral line emissions of the hydrogen atom. ...

frac{1}{lambda} = R_mathrm{H}left(frac{1}{m^2} - frac{1}{n^2}right), m=1,2,3,4,5,..., and n > m

where λ is the wavelength of the light and RH is the Rydberg constant for hydrogen. This generalization predicted many more line spectra than had been previously detected, and experimental confirmation of this prediction followed. The Rydberg constant, named after physicist Janne Rydberg, is a physical constant discovered when measuring the spectrum of hydrogen, and building upon results from Anders Jonas Ã…ngström and Johann Balmer. ...


It follows almost immediately that if λ is quantized as the formula above indicates, then the momentum of any photon must be quantized. The frequency of light, ν, at a given wavelength λ is given by the relationship

nu=frac{c}{lambda} and :lambda=frac{c}{nu} and multiplying by h/h = 1,
lambda=frac{hc}{hnu}, and we know that
E = hν so lambda=frac{hc}{E} which we can rewrite as:
lambda=frac{h}{E/c}, and E/c = p (momentum) so
lambda=frac{h}{p} or p=frac{h}{lambda}

Beginning with line spectra, physicists were able to deduce empirically the rules according to which the orbits of electrons are determined and to discover something vital about the momentums involved--that they are quantized.[20]


Bohr next realized how the angular momentum of an electron in its orbit, L, is quantized, i.e., he determined that there is some constant value K such that when it is multiplied by Planck’s constant, h, it will yield the angular momentum that pertains to the lowest orbital. When it is multiplied by successive integers it will then give the values of other possible orbitals. He later determined that K = 1/2π . (See the detailed argument at [1].)


Bohr's theory represented electrons as orbiting the nucleus of an atom in a way that was amazingly different from what we see in the world of our everyday experience. He showed that when an electron changed orbits it did not move in a continuous trajectory from one orbit around the nucleus to another. Instead, it suddenly disappeared from its original orbit and reappeared in another orbit. Each distance at which an electron can orbit is a function of a quantized amount of energy. The closer to the nucleus an electron orbits, the less energy it takes to remain in that orbital. Electrons that absorb a photon gain a quantum of energy, so they jump to an orbit that is farther from the nucleus, while electrons that emit a photon lose a quantum of energy and so jump to an inner orbital. Electrons cannot gain or lose a fractional quantum of energy, and so, it is argued, they cannot have a position that is at a fractional distance between allowed orbitals. Allowed orbitals were designated as whole numbers using the letter n with the innermost orbital being designated n = 1, the next out being n = 2, and so on. Any orbital with the same value of n is called an electron shell. The term orbital has several meanings: In physics and chemistry it is used to describe an atomic electron configuration, see also molecular orbital and atomic orbital. ...


Bohr's model of the atom was essentially two-dimensional because it depicts electrons as particles in circular orbits. In this context, two-dimensional means something that can be described on the surface of a plane. One-dimensional means something that can be described by a line. Because circles can be described by their radius, which is a line segment, sometimes Bohr's model of the atom is described as one-dimensional.[21]


Wave-particle duality

Probability distribution of the Bohr atom
Probability distribution of the Bohr atom

Niels Bohr determined that it is impossible to describe light adequately by the sole use of either the wave analogy or of the particle analogy. Therefore he enunciated the principle of complementarity, which is a theory of pairs, such as the pairing of wave and particle or the pairing of position and momentum. Louis de Broglie worked out the mathematical consequences of these findings. In quantum mechanics, it was found that electromagnetic waves could react in certain experiments as though they were particles and in other experiments as though they were waves. It was also discovered that subatomic particles could sometimes be described as particles and sometimes as waves. This discovery led to the theory of wave-particle duality by Louis-Victor de Broglie in 1924, which states that subatomic entities have properties of both waves and particles at the same time. Image File history File links Bohr_atomic_wave. ... Image File history File links Bohr_atomic_wave. ... Niels Henrik David Bohr (October 7, 1885 – November 18, 1962) was a Danish physicist who made fundamental contributions to understanding atomic structure and quantum mechanics, for which he received the Nobel Prize in Physics in 1922. ... Complementarity is a concept in a number of fields: Economics In economics is a concept similar to that of externality. ... In physics, wave-particle duality holds that light and matter exhibit properties of both waves and of particles. ... Louis-Victor-Pierre-Raymond, 7th duc de Broglie, generally known as Louis de Broglie (August 15, 1892–March 19, 1987), was a French physicist and Nobel Prize laureate. ...


The Bohr atom model was enlarged upon with the discovery by de Broglie that the electron has wave-like properties. In accord with de Broglie's conclusions, electrons can only appear under conditions that permit a standing wave. A standing wave can be made if a string is fixed on both ends and made to vibrate (as it would in a stringed instrument). That illustration shows that the only standing waves that can occur are those with zero amplitude at the two fixed ends. The waves created by a stringed instrument appear to oscillate in place, simply changing crest for trough in an up-and-down motion. A standing wave can only be formed when the wave's length fits the available vibrating entity. In other words, no partial fragments of wave crests or troughs are allowed. In a round vibrating medium, the wave must be a continuous formation of crests and troughs all around the circle. Each electron must be its own standing wave in its own discrete orbital.


Development of modern quantum mechanics

Full quantum mechanical theory

Werner Heisenberg developed the full quantum mechanical theory in 1925 at the young age of 23. Following his mentor, Niels Bohr, Werner Heisenberg began to work out a theory for the quantum behavior of electron orbitals. Because electrons could not be observed in their orbits, Heisenberg went about creating a mathematical description of quantum mechanics built on what could be observed, that is, the light emitted from atoms in their characteristic atomic spectra. Heisenberg studied the electron orbital on the model of a charged ball on a spring, an oscillator, whose motion is anharmonic (not quite regular). For a picture of the behavior of a charged ball on a spring see: Vibrating Charges. Heisenberg first explained this kind of observed motion in terms of the laws of classical mechanics known to apply in the macro world, and then applied quantum restrictions, discrete (non-continuous) properties, to the picture. Doing so causes gaps to appear between the orbitals so that the mathematical description he formulated would then represent only the electron orbitals predicted on the basis of the atomic spectra. Werner Karl Heisenberg (December 5, 1901 – February 1, 1976) was a celebrated German physicist and Nobel laureate, one of the founders of quantum mechanics and acknowledged to be one of the most important physicists of the twentieth century. ...


In 1925 Heisenberg published a paper (in Z. Phys. vol. 33, p. 879-893) entitled "Quantum-mechanical re-interpretation of kinematic and mechanical relations." So ended the old quantum theory and began the age of quantum mechanics. Heisenberg's paper gave few details that might aid readers in determining how he actually contrived to get his results for the one-dimensional models he used to form the hypothesis that proved so useful. In his paper, Heisenberg proposed to "discard all hope of observing hitherto unobservable quantities, such as the position and period of the electron," and restrict himself strictly to actually observable quantities. He needed mathematical rules for predicting the relations actually observed in nature, and the rules he produced worked differently depending on the sequence in which they were applied. "It quickly became clear that the non-commutativity (in general) of kinematical quantities in quantum theory was the really essential new technical idea in the paper." (Aitchison, p. 5) But it was unclear why this non-commutativity was essential. Could it have a physical interpretation? At least the matter was made more palatable when Max Born discovered that the Heisenberg computational scheme could be put in a more familiar form present in elementary mathematics.


The special type of multiplication that turned out to be required in his formula was most elegantly described by using special arrays of numbers called matrices. In ordinary situations it does not matter in which order the operations involved in multiplication are performed, but matrix multiplication does not commute. Essentially that means that it matters which order given operations are performed in. Multiplying matrix A by matrix B is not the same as multiplying matrix B by matrix A. In symbols, AxB is in general not equal to BxA. (The important thing in quantum theory is that it turned out to matter whether one measures velocity first and then measures position, or vice-versa.) The matrix convention turned out to be a convenient way of organizing information and making clear the exact sequence in which calculations must be made. In mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries), which may be numbers or, more generally, any abstract quantities that can be added and multiplied. ... Example showing the commutativity of addition (3 + 2 = 2 + 3) For other uses, see Commute (disambiguation). ...


In matrix mathematics sets of numbers are given in rows and columns, and there are conventions for the way multiplication of matrices is performed. If everybody arranged their matrices entirely as they pleased, then understanding every new matrix calculation would involve learning the personal plan of the person who made the matrix, so certain conventions have evolved. Reverse the order of the multiplication of the matrices and the numerical results will go wrong. In other words, Matrix multiplication is noncommutative. Because these complex operations are, by analogy, called "multiplication," it is tempting to imagine that it ought to be irrelevant whether one multiplies matrix A by matrix B, or one multiplies matrix B by matrix A. Because of the complications involved in the rules of matrix multiplication, in almost every case the ordinary mathematical expectation of commutation does not hold. In Heisenberg's matrix mechanics, the sets of numbers are infinite, representing all possible positions of the electron, and those matrices cannot be multiplied in reverse order and still produce the correct results. The essential point is that Heisenberg first learned what ways he had to operate on measured quantities to be able to account for the line spectra that had been observed. The operations to be performed seemed complicated and arbitrary, but when specialists realized that what he was doing could be represented in a coherent and structurally straightforward way by the use of matrix mathematics it became much easier to get through the calculations successfully and also lessened the appearance of arbitrariness of the process. In mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries), which may be numbers or, more generally, any abstract quantities that can be added and multiplied. ... In mathematics, especially abstract algebra, a binary operation * on a set S is commutative if x * y = y * x for all x and y in S. Otherwise * is noncommutative. ...


Heisenberg approached quantum mechanics from the historical perspective that treated an electron as an oscillating charged particle. Bohr's use of this analogy had already allowed him to explain why the radii of the orbits of electrons could only take on certain values. It followed from this interpretation of the experimental results available and the quantum theory that Heisenberg subsequently created that an electron could not be at any intermediate position between two "permitted" orbits. Therefore electrons were described as "jumping" from orbit to orbit. The idea that an electron might now be in one place and an instant later be in some other place without having traveled between the two points was one of the earliest indications of the "spookiness" of quantum phenomena. Although the scale is smaller, the "jump" from orbit to orbit is as strange and unexpected as would be a case in which someone stepped out of a doorway in London onto the streets of Los Angeles.


Amplitudes of position and momentum that have a period of 2 π like a cycle in a wave are called Fourier series variables. Heisenberg described the particle-like properties of the electron in a wave as having position and momentum in his matrix mechanics. When these amplitudes of position and momentum are measured and multiplied together, they give intensity. However, he found that when the position and momentum were multiplied together in that respective order, and then the momentum and position were multiplied together in that respective order, there was a difference or deviation in intensity between them of h/2π. Heisenberg would not understand the reason for this deviation until two more years had passed, but for the time being he satisfied himself with the idea that the math worked and provided an exact description of the quantum behavior of the electron. This article is about momentum in physics. ... The Fourier series is a mathematical tool used for analyzing periodic functions by decomposing such a function into a weighted sum of much simpler sinusoidal component functions sometimes referred to as normal Fourier modes, or simply modes for short. ... Matrix mechanics is a formulation of quantum mechanics created by Werner Heisenberg, Max Born, and Pascual Jordan in 1925. ...


Matrix mechanics was the first complete definition of quantum mechanics, its laws, and properties that described fully the behavior of the electron. It was later extended to apply to all subatomic particles. Schrödinger subsequently produced a quantum wave theory that was computationally easier and avoided some of the odd-sounding ideas like "quantum leaps" of an electron from one orbit to another, and finally Dirac made the idea of non-commutativity central to his own theory that proved the formulations of Heisenberg and of Schrödinger to be special cases of his own.


Schrödinger wave equation

Main article: Schrödinger equation
Model of the Schrödinger atom, showing the nucleus with two protons (blue) and two neutrons (red), orbited by two electrons (waves)
Model of the Schrödinger atom, showing the nucleus with two protons (blue) and two neutrons (red), orbited by two electrons (waves)

Because particles could be described as waves, later in 1925 Erwin Schrödinger analyzed what an electron would look like as a wave around the nucleus of the atom. Using this model, he formulated his equation for particle waves. Rather than explaining the atom by analogy to satellites in planetary orbits, he treated everything as waves whereby each electron has its own unique wavefunction. A wavefunction is described in Schrödinger's equation by three properties (later Wolfgang Pauli added a fourth). The three properties were (1) an "orbital" designation, indicating whether the particle wave is one that is closer to the nucleus with less energy or one that is farther from the nucleus with more energy, (2) the shape of the orbital, i.e., an indication that orbitals were not just spherical but other shapes, and (3) the magnetic moment of the orbital, which is a manifestation of force exerted by the charge of the electron as it rotates around the nucleus. For a non-technical introduction to the topic, please see Introduction to quantum mechanics. ... Image File history File links Helium_atom_with_charge-smaller. ... Image File history File links Helium_atom_with_charge-smaller. ... Schrödinger in 1933, when he was awarded the Nobel Prize in Physics Bust of Schrödinger, in the courtyard arcade of the main building, University of Vienna, Austria. ... For other uses, see Electron (disambiguation). ... This article is about the Austrian-Swiss physicist. ... A bar magnet. ...


These three properties were called collectively the wavefunction of the electron and are said to describe the quantum state of the electron. "Quantum state" means the collective properties of the electron describing what we can say about its condition at a given time. For the electron, the quantum state is described by its wavefunction, which is designated in physics by the Greek letter ψ (psi, pronounced "sigh"). The three properties of Schrödinger's equation that describe the wavefunction of the electron and, therefore, also describe the quantum state of the electron as described in the previous paragraph are each called quantum numbers. The first property that describes the orbital was numbered according to Bohr's model where n is the letter used to describe the energy of each orbital. This is called the principal quantum number. The next quantum number that describes the shape of the orbital is called the azimuthal quantum number and it is represented by the letter l (lower case L). The shape is caused by the angular momentum of the orbital. The rate of change of the angular momentum of any system is equal to the resultant external torque acting on that system. In other words, angular momentum represents the resistance of a spinning object to speed up or slow down under the influence of external force. The azimuthal quantum number "l" represents the orbital angular momentum of an electron around its nucleus. However, the shape of each orbital has its own letter as well. So for the letter "l" there are other letters to describe the shapes of "l". The first shape is spherical and is described by the letter s. The next shape is like a dumbbell and is described by the letter p. The other shapes of orbitals become more complicated (see Atomic Orbitals) and are described by the letters d, f, and g. For the shape of a carbon atom, see Carbon atom. The third quantum number of Schrödinger's equation describes the magnetic moment of the electron and is designated by the letter m and sometimes as the letter m with a subscript l because the magnetic moment depends upon the second quantum number l. Probability densities for the electron at different quantum numbers (l) In quantum mechanics, the quantum state of a system is a set of numbers that fully describe a quantum system. ... A quantum number is a number used to parametrise certain properties of particles or other systems in quantum mechanics. ... In atomic physics, the principal quantum number symbolized as n is the first quantum number of an atomic orbital. ... The Azimuthal quantum number (or orbital angular momentum quantum number) symbolized as l (lower-case L) is a quantum number for an atomic orbital which determines its orbital angular momentum. ... This gyroscope remains upright while spinning due to its angular momentum. ...


In May 1926 Schrödinger published a proof that Heisenberg's matrix mechanics and his own wave mechanics gave equivalent results: mathematically they were the same theory. Yet both men disagreed on the interpretation of this theory. Heisenberg saw no problem in the existence of discontinuous quantum jumps, while Schrödinger hoped that a theory based on continuous wave-like properties could avoid this "nonsense about quantum jumps" (in the words of Wilhelm Wien[22] ). Wilhelm Carl Werner Otto Fritz Franz Wien (January 13, 1864 – August 30, 1928) was a German physicist who, in 1893, used theories about heat and electromagnetism to compose Wi