Atomic theory

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This article is about the historical models of the atom. For a history of the study of how atoms combine to form molecules, see history of molecular theory. For the modern view of the atom which developed from atomic theory, see atomic physics.

The current theoretical model of the atom involves a dense nucleus surrounded by a probabilistic "cloud" of electrons

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 and cut it into ever smaller pieces, one would reach a point where the pieces could not be further cut into anything smaller. Ancient Greek philosophers called these hypothetical ultimate particles of matter atomos, a word which meant "uncut".

In the early 1800s, the scientist John Dalton noticed that chemical substances seemed to combine and break down into other substances by weight in proportions that indicated that matter is indeed made of atoms as the ancient philosophers suspected. Shortly after 1850, certain physicists developed the kinetic theory of gases and of heat, which mathematically modelled the behavior of gases by assuming that gases are made of particles. In the early 20th century, Albert Einstein and Jean Perrin proved that Brownian motion (the erratic motion of pollen grains in water) is caused by the action of water molecules — this third line of evidence silenced remaining doubts among scientists as to whether atoms and molecules are real. Throughout the nineteenth century, some scientists had cautioned that the evidence for atoms was indirect, and therefore atoms might not actually be real, but only seem to be real.

By the early 20th century, scientists had developed fairly detailed and precise models for the structure of matter, which led to more rigorously-defined classifications for the tiny invisible particles that make up ordinary matter. An atom is now defined as the basic particle that composes a chemical element. Around the turn of the 20th century, physicists discovered that the particles that chemists called "atoms" are in fact agglomerations of even smaller particles (subatomic particles), but scientists kept the name out of convention. The term elementary particle is now used to refer to particles that are actually indivisible.

History

Philosophical atomism

The idea that matter is made up of discrete units is a very old idea, appearing in many ancient cultures such as Greece and India. The word "atom" (Greek: ἄτομος; atomos), meaning "uncuttable", was coined by the Pre-Socratic Greek philosophers Leucippus and his pupil Democritus (c.460–c.370 BC).

 Democritus taught that atoms were infinite in number, uncreated, and eternal, and that the qualities of an object result from the kind of atoms that compose it. Democritus's atomism was refined and elaborated by the later Greek philosopher Epicurus (341–270 BC), and by the Roman Epicurean poet Lucretius (c.99–c.55 BC). During the Early Middle Ages, atomism was mostly forgotten in western Europe.During the 12th century, atomism became known again in western Europe through references to it in the newly-rediscovered writings of Aristotle.

In the 14th century, the rediscovery of major works describing atomist teachings, including Lucretius's De rerum natura and Diogenes Laërtius's Lives and Opinions of Eminent Philosophers, led to increased scholarly attention on the subject. Nonetheless, because atomism was associated with the philosophy of Epicureanism, which contradicted orthodox Christian teachings, belief in atoms was not considered acceptable by most European philosophers. The French Catholic priest Pierre Gassendi (1592–1655) revived Epicurean atomism with modifications, arguing that atoms were created by God and, though extremely numerous, are not infinite and the first person who use term "molecule" to describe aggregate of atom. Gassendi's modified theory of atoms was popularized in France by the physician François Bernier (1620–1688) and in England by the natural philosopher Walter Charleton (1619–1707). The chemist Robert Boyle (1627–1691) and the physicist Isaac Newton (1642–1727) both defended atomism and, by the end of the 17th century, it had become accepted by portions of the scientific community.

John Dalton

Near the end of the 18th century, two laws about chemical reactions emerged without referring to the notion of an atomic theory. The first was the law of conservation of mass, closely associated with the work of Antoine Lavoisier, which states that the total mass in a chemical reaction remains constant (that is, the reactants have the same mass as the products). The second was the law of definite proportions. First established by the French chemist Joseph Proust in 1797 this law states that if a compound is broken down into its constituent chemical elements, then the masses of the constituents will always have the same proportions by weight, regardless of the quantity or source of the original substance.

John Dalton studied and expanded upon this previous work and defended a new idea, later known as the law of multiple proportions: if the same two elements can be combined to form a number of different compounds, then the ratios of the masses of the two elements in their various compounds will be represented by small whole numbers. This is a common pattern in chemical reactions that was observed by Dalton and other chemists at the time.

Example 1 — tin oxides: Dalton identified two oxides of tin. One is a grey powder in which for every 100 parts of tin there is 13.5 parts of oxygen. The other oxide is a white powder in which for every 100 parts of tin there is 27 parts of oxygen. 13.5 and 27 form a ratio of 1:2. These oxides are today known as tin(II) oxide (SnO) and tin(IV) oxide (SnO₂) respectively.

Example 2 — iron oxides: Dalton identified two oxides of iron. One is a black powder in which for every 100 parts of iron there is about 28 parts of oxygen. The other is a red powder in which for every 100 parts of iron there is 42 parts of oxygen. 28 and 42 form a ratio of 2:3. These oxides are today known as iron(II) oxide (better known as wüstite) and iron(III) oxide (the major constituent of rust). Their formulas are FeO and Fe₂O₃ respectively.

Example 3 — nitrogen oxides: There are three oxides of nitrogen in which for every 140 g of nitrogen, there is 80 g, 160 g, and 320 g of oxygen respectively, which gives a ratio of 1:2:4. These are nitrous oxide (N₂O), nitric oxide (NO), and nitrogen dioxide (NO₂) respectively.

This recurring pattern suggested that chemicals do not react in any arbitrary quantity, but in multiples of some basic indivisible unit of mass.

In his writings, Dalton used the term "atom" to refer to the basic particle of any chemical substance, not strictly for elements as is the practice today. Dalton did not use the word "molecule"; instead, he used the terms "compound atom" and "elementary atom".

Dalton believed atomic theory could also explain why water absorbed different gases in different proportions—for example, he found that water absorbed carbon dioxide far better than it absorbed nitrogen. Dalton hypothesized this was due to the differences in mass and complexity of the gases' respective particles. Indeed, carbon dioxide molecules (CO₂) are heavier and larger than nitrogen molecules (N₂).

Dalton proposed that each chemical element is composed of atoms of a single, unique type, and though they cannot be altered or destroyed by chemical means, they can combine to form more complex structures (chemical compounds). This marked the first truly scientific theory of the atom, since Dalton reached his conclusions by experimentation and examination of the results in an empirical fashion.

Various atoms and molecules as depicted in John Dalton's A New System of Chemical Philosophy (1808).

In 1803 Dalton orally presented his first list of relative atomic weights for a number of substances. This paper was published in 1805, but he did not discuss there exactly how he obtained these figures. The method was first revealed in 1807 by his acquaintance Thomas Thomson, in the third edition of Thomson's textbook, A System of Chemistry. Finally, Dalton published a full account in his own textbook, A New System of Chemical Philosophy, 1808 and 1810.

Dalton estimated the atomic weights according to the mass ratios in which they combined, with the hydrogen atom taken as unity. However, Dalton did not conceive that with some elements atoms exist in molecules—e.g. pure oxygen exists as O₂. He also mistakenly believed that the simplest compound between any two elements is always one atom of each (so he thought water was HO, not H₂O). This, in addition to the crudity of his equipment, flawed his results. For instance, in 1803 he believed that oxygen atoms were 5.5 times heavier than hydrogen atoms, because in water he measured 5.5 grams of oxygen for every 1 gram of hydrogen and believed the formula for water was HO. Adopting better data, in 1806 he concluded that the atomic weight of oxygen must actually be 7 rather than 5.5, and he retained this weight for the rest of his life. Others at this time had already concluded that the oxygen atom must weigh 8 relative to hydrogen equals 1, if one assumes Dalton's formula for the water molecule (HO), or 16 if one assumes the modern water formula (H₂O).

Avogadro

The flaw in Dalton's theory was corrected in principle in 1811 by Amedeo Avogadro. Avogadro had proposed that equal volumes of any two gases, at equal temperature and pressure, contain equal numbers of molecules (in other words, the mass of a gas's particles does not affect the volume that it occupies). Avogadro's law allowed him to deduce the diatomic nature of numerous gases by studying the volumes at which they reacted. For instance: since two liters of hydrogen will react with just one liter of oxygen to produce two liters of water vapor (at constant pressure and temperature), it meant a single oxygen molecule splits in two in order to form two particles of water. Thus, Avogadro was able to offer more accurate estimates of the atomic mass of oxygen and various other elements, and made a clear distinction between molecules and atoms.

Brownian Motion

In 1827, the British botanist Robert Brown observed that dust particles inside pollen grains floating in water constantly jiggled about for no apparent reason. In 1905, Albert Einstein theorized that this Brownian motion was caused by the water molecules continuously knocking the grains about, and developed a hypothetical mathematical model to describe it. This model was validated experimentally in 1908 by French physicist Jean Perrin, thus providing additional validation for particle theory (and by extension atomic theory).

Discovery of subatomic particles

The cathode rays (blue) were emitted from the cathode, sharpened to a beam by the slits, then deflected as they passed between the two electrified plates.

Atoms were thought to be the smallest possible division of matter until 1897 when J. J. Thomson discovered the electron through his work on cathode rays.

A Crookes tube is a sealed glass container in which two electrodes are separated by a vacuum. When a voltage is applied across the electrodes, cathode rays are generated, creating a glowing patch where they strike the glass at the opposite end of the tube. Through experimentation, Thomson discovered that the rays could be deflected by an electric field (in addition to magnetic fields, which was already known). He concluded that these rays, rather than being a form of light, were composed of very light negatively charged particles he called "corpuscles" (they would later be renamed electrons by other scientists). He measured the mass-to-charge ratio and discovered it was 1800 times smaller than that of hydrogen, the smallest atom. These corpuscles were a particle unlike any other previously known.

Thomson suggested that atoms were divisible, and that the corpuscles were their building blocks. To explain the overall neutral charge of the atom, he proposed that the corpuscles were distributed in a uniform sea of positive charge; this was the plum pudding model as the electrons were embedded in the positive charge like raisins in a plum pudding (although in Thomson's model they were not stationary).

Discovery of the nucleus

The Geiger–Marsden experiment
Left: Expected results: alpha particles passing through the plum pudding model of the atom with negligible deflection.
Right: Observed results: a small portion of the particles were deflected by the concentrated positive charge of the nucleus.

Thomson's plum pudding model was disproved in 1909 by one of his former students, Ernest Rutherford, who discovered that most of the mass and positive charge of an atom is concentrated in a very small fraction of its volume, which he assumed to be at the very center.

Ernest Rutherford and his colleagues Hans Geiger and Ernest Marsden came to have doubts about the Thomson model after they encountered difficulties when they tried to build an instrument to measure the charge-to-mass ratio of alpha particles (these are positively-charged particles emitted by certain radioactive substances such as radium). The alpha particles were being scattered by the air in the detection chamber, which made the measurements unreliable. Thomson had encountered a similar problem in his work on cathode rays, which he solved by creating a near-perfect vacuum in his instruments. Rutherford didn't think he'd run into this same problem because alpha particles are much heavier than electrons. According to Thomson's model of the atom, the positive charge in the atom is not concentrated enough to produce an electric field strong enough to deflect an alpha particle, and the electrons are so lightweight they should be pushed aside effortlessly by the much heavier alpha particles. Yet there was scattering, so Rutherford and his colleagues decided to investigate this scattering carefully.

Between 1908 and 1913, Rutheford and his colleagues performed a series of experiments in which they bombarded thin foils of metal with alpha particles. They spotted alpha particles being deflected by angles greater than 90°. To explain this, Rutherford proposed that the positive charge of the atom is not distributed throughout the atom's volume as Thomson believed, but is concentrated in a tiny nucleus at the center. Only such an intense concentration of charge could produce an electric field strong enough to deflect the alpha particles as observed.

First steps toward a quantum physical model of the atom

The planetary model of the atom had two significant shortcomings. The first is that, unlike planets orbiting a sun, electrons are charged particles. An accelerating electric charge is known to emit electromagnetic waves according to the Larmor formula in classical electromagnetism. An orbiting charge should steadily lose energy and spiral toward the nucleus, colliding with it in a small fraction of a second. The second problem was that the planetary model could not explain the highly peaked emission and absorption spectra of atoms that were observed.

The Bohr model of the atom

Quantum theory revolutionized physics at the beginning of the 20th century, when Max Planck and Albert Einstein postulated that light energy is emitted or absorbed in discrete amounts known as quanta (singular, quantum). In 1913, Niels Bohr incorporated this idea into his Bohr model of the atom, in which an electron could only orbit the nucleus in particular circular orbits with fixed angular momentum and energy, its distance from the nucleus (i.e., their radii) being proportional to its energy. Under this model an electron could not spiral into the nucleus because it could not lose energy in a continuous manner; instead, it could only make instantaneous "quantum leaps" between the fixed energy levels. When this occurred, light was emitted or absorbed at a frequency proportional to the change in energy (hence the absorption and emission of light in discrete spectra).

Bohr's model was not perfect. It could only predict the spectral lines of hydrogen; it couldn't predict those of multielectron atoms. Worse still, as spectrographic technology improved, additional spectral lines in hydrogen were observed which Bohr's model couldn't explain. In 1916, Arnold Sommerfeld added elliptical orbits to the Bohr model to explain the extra emission lines, but this made the model very difficult to use, and it still couldn't explain more complex atoms.

Discovery of isotopes

While experimenting with the products of radioactive decay, in 1913 radiochemist Frederick Soddy discovered that there appeared to be more than one element at each position on the periodic table. The term isotope was coined by Margaret Todd as a suitable name for these elements.

That same year, J. J. Thomson conducted an experiment in which he channeled a stream of neon ions through magnetic and electric fields, striking a photographic plate at the other end. He observed two glowing patches on the plate, which suggested two different deflection trajectories. Thomson concluded this was because some of the neon ions had a different mass. The nature of this differing mass would later be explained by the discovery of neutrons in 1932.

Discovery of nuclear particles

In 1917 Rutherford bombarded nitrogen gas with alpha particles and observed hydrogen nuclei being emitted from the gas (Rutherford recognized these, because he had previously obtained them bombarding hydrogen with alpha particles, and observing hydrogen nuclei in the products). Rutherford concluded that the hydrogen nuclei emerged from the nuclei of the nitrogen atoms themselves (in effect, he had split a nitrogen).

From his own work and the work of his students Bohr and Henry Moseley, Rutherford knew that the positive charge of any atom could always be equated to that of an integer number of hydrogen nuclei. This, coupled with the atomic mass of many elements being roughly equivalent to an integer number of hydrogen atoms - then assumed to be the lightest particles - led him to conclude that hydrogen nuclei were singular particles and a basic constituent of all atomic nuclei. He named such particles protons. Further experimentation by Rutherford found that the nuclear mass of most atoms exceeded that of the protons it possessed; he speculated that this surplus mass was composed of previously-unknown neutrally charged particles, which were tentatively dubbed "neutrons".

In 1928, Walter Bothe observed that beryllium emitted a highly penetrating, electrically neutral radiation when bombarded with alpha particles. It was later discovered that this radiation could knock hydrogen atoms out of paraffin wax. Initially it was thought to be high-energy gamma radiation, since gamma radiation had a similar effect on electrons in metals, but James Chadwick found that the ionization effect was too strong for it to be due to electromagnetic radiation, so long as energy and momentum were conserved in the interaction. In 1932, Chadwick exposed various elements, such as hydrogen and nitrogen, to the mysterious "beryllium radiation", and by measuring the energies of the recoiling charged particles, he deduced that the radiation was actually composed of electrically neutral particles which could not be massless like the gamma ray, but instead were required to have a mass similar to that of a proton. Chadwick now claimed these particles as Rutherford's neutrons. For his discovery of the neutron, Chadwick received the Nobel Prize in 1935.

Quantum physical models of the atom

The five filled atomic orbitals of a neon atom separated and arranged in order of increasing energy from left to right, with the last three orbitals being equal in energy. Each orbital holds up to two electrons, which most probably exist in the zones represented by the colored bubbles. Each electron is equally present in both orbital zones, shown here by color only to highlight the different wave phase.

In 1924, Louis de Broglie proposed that all moving particles—particularly subatomic particles such as electrons—exhibit a degree of wave-like behavior. Erwin Schrödinger, fascinated by this idea, explored whether or not the movement of an electron in an atom could be better explained as a wave rather than as a particle. Schrödinger's equation, published in 1926, describes an electron as a wave function instead of as a point particle. This approach elegantly predicted many of the spectral phenomena that Bohr's model failed to explain. Although this concept was mathematically convenient, it was difficult to visualize, and faced opposition. One of its critics, Max Born, proposed instead that Schrödinger's wave function described not the electron but rather all its possible states, and thus could be used to calculate the probability of finding an electron at any given location around the nucleus. This reconciled the two opposing theories of particle versus wave electrons and the idea of wave–particle duality was introduced. This theory stated that the electron may exhibit the properties of both a wave and a particle. For example, it can be refracted like a wave, and has mass like a particle.

A consequence of describing electrons as waveforms is that it is mathematically impossible to simultaneously derive the position and momentum of an electron. This became known as the Heisenberg uncertainty principle after the theoretical physicist Werner Heisenberg, who first described it and published it in 1927. This invalidated Bohr's model, with its neat, clearly defined circular orbits. The modern model of the atom describes the positions of electrons in an atom in terms of probabilities. An electron can potentially be found at any distance from the nucleus, but, depending on its energy level, exists more frequently in certain regions around the nucleus than others; this pattern is referred to as its atomic orbital. The orbitals come in a variety of shapes-sphere, dumbbell, torus, etc.-with the nucleus in the middle.

History of quantum mechanics

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/History_of_quantum_mechanics 

 

10 influential figures in the history of quantum mechanics. Left to right:

Max Planck, Albert Einstein,
Niels Bohr, Louis de Broglie,
Max Born, Paul Dirac,
Werner Heisenberg, Wolfgang Pauli,
Erwin Schrödinger, Richard Feynman.

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 1838 discovery of cathode rays by Michael Faraday; the 1859–60 winter statement of the black-body radiation problem by Gustav Kirchhoff; the 1877 suggestion by Ludwig Boltzmann that the energy states of a physical system could be discrete; the discovery of the photoelectric effect by Heinrich Hertz in 1887; and the 1900 quantum hypothesis by Max Planck that any energy-radiating atomic system can theoretically be divided into a number of discrete "energy elements" ε (Greek letter epsilon) such that each of these energy elements is proportional to the frequency ν with which each of them individually radiate energy, as defined by the following formula:

${\displaystyle \varepsilon =h\nu \,}$

where h is a numerical value called Planck's constant.

Then, Albert Einstein in 1905, in order to explain the photoelectric effect previously reported by Heinrich Hertz in 1887, postulated consistently with Max Planck's quantum hypothesis that light itself is made of individual quantum particles, which in 1926 came to be called photons by Gilbert N. Lewis. The photoelectric effect was observed upon shining light of particular wavelengths on certain materials, such as metals, which caused electrons to be ejected from those materials only if the light quantum energy was greater than the work function of the metal's surface.

The phrase "quantum mechanics" was coined (in German, Quantenmechanik) by the group of physicists including Max Born, Werner Heisenberg, and Wolfgang Pauli, at the University of Göttingen in the early 1920s, and was first used in Born's 1924 paper "Zur Quantenmechanik". In the years to follow, this theoretical basis slowly began to be applied to chemical structure, reactivity, and bonding.

Overview

Ludwig Boltzmann's diagram of the I₂ molecule proposed in 1898 showing the atomic "sensitive region" (α, β) of overlap.

Ludwig Boltzmann suggested in 1877 that the energy levels of a physical system, such as a molecule, could be discrete (as opposed to continuous). He was a founder of the Austrian Mathematical Society, together with the mathematicians Gustav von Escherich and Emil Müller. Boltzmann's rationale for the presence of discrete energy levels in molecules such as those of iodine gas had its origins in his statistical thermodynamics and statistical mechanics theories and was backed up by mathematical arguments, as would also be the case twenty years later with the first quantum theory put forward by Max Planck.

In 1900, the German physicist Max Planck reluctantly introduced the idea that energy is quantized in order to derive a formula for the observed frequency dependence of the energy emitted by a black body, called Planck's law, that included a Boltzmann distribution (applicable in the classical limit). Planck's law can be stated as follows: ${\displaystyle I(\nu ,T)={\frac {2h\nu ^{3}}{c^{2}}}{\frac {1}{e^{\frac {h\nu }{kT}}-1}},}$ where:

I(ν,T) is the energy per unit time (or the power) radiated per unit area of emitting surface in the normal direction per unit solid angle per unit frequency by a black body at temperature T;

h is the Planck constant;

c is the speed of light in a vacuum;

k is the Boltzmann constant;

ν (nu) is the frequency of the electromagnetic radiation; and

T is the temperature of the body in kelvins.

The earlier Wien approximation may be derived from Planck's law by assuming ${\displaystyle h\nu \gg kT}$.

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

Photoelectric effect

The emission of electrons from a metal plate caused by light quanta (photons) with energy greater than the work function of the metal.

The photoelectric effect reported by Heinrich Hertz in 1887,

and explained by Albert Einstein in 1905.

Low-energy phenomena: Photoelectric effect

Mid-energy phenomena: Compton scattering

High-energy phenomena: Pair production

 

In 1905, Albert Einstein explained the photoelectric effect by postulating that light, or more generally all electromagnetic radiation, can be divided into a finite number of "energy quanta" that are localized points in space. From the introduction section of his March 1905 quantum paper, "On a heuristic viewpoint concerning the emission and transformation of light", Einstein states:

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

This statement has been called the most revolutionary sentence written by a physicist of the twentieth century. These energy quanta later came to be called "photons", a term introduced by Gilbert N. Lewis in 1926. The idea that each photon had to consist of energy in terms of quanta was a remarkable achievement; it effectively solved the problem of black-body radiation attaining infinite energy, which occurred in theory if light were to be explained only in terms of waves. In 1913, Bohr explained the spectral lines of the hydrogen atom, again by using quantization, in his paper of July 1913 On the Constitution of Atoms and Molecules.

These theories, though successful, were strictly phenomenological: during this time, there was no rigorous justification for quantization, aside, perhaps, from Henri Poincaré's discussion of Planck's theory in his 1912 paper Sur la théorie des quanta. They are collectively known as the old quantum theory.

The phrase "quantum physics" was first used in Johnston's Planck's Universe in Light of Modern Physics (1931). 

 

With decreasing temperature, the peak of the blackbody radiation curve shifts to longer wavelengths and also has lower intensities. The blackbody radiation curves (1862) at left are also compared with the early, classical limit model of Rayleigh and Jeans (1900) shown at right. The short wavelength side of the curves was already approximated in 1896 by the Wien distribution law.

 

Niels Bohr's 1913 quantum model of the atom, which incorporated an explanation of Johannes Rydberg's 1888 formula, Max Planck's 1900 quantum hypothesis, i.e. that atomic energy radiators have discrete energy values (ε = hν), J. J. Thomson's 1904 plum pudding model, Albert Einstein's 1905 light quanta postulate, and Ernest Rutherford's 1907 discovery of the atomic nucleus. Note that the electron does not travel along the black line when emitting a photon. It jumps, disappearing from the outer orbit and appearing in the inner one and cannot exist in the space between orbits 2 and 3.

In 1923, the French physicist Louis de Broglie put forward his theory of matter waves by stating that particles can exhibit wave characteristics and vice versa. This theory was for a single particle and derived from special relativity theory. Building on de Broglie's approach, modern quantum mechanics was born in 1925, when the German physicists Werner Heisenberg, Max Born, and Pascual Jordandeveloped matrix mechanics and the Austrian physicist Erwin Schrödinger invented wave mechanics and the non-relativistic Schrödinger equation as an approximation of the generalised case of de Broglie's theory. Schrödinger subsequently showed that the two approaches were equivalent.

Heisenberg formulated his uncertainty principle in 1927, and the Copenhagen interpretation started to take shape at about the same time. Starting around 1927, Paul Dirac began the process of unifying quantum mechanics with special relativity by proposing the Dirac equation for the electron. The Dirac equation achieves the relativistic description of the wavefunction of an electron that Schrödinger failed to obtain. It predicts electron spin and led Dirac to predict the existence of the positron. He also pioneered the use of operator theory, including the influential bra–ket notation, as described in his famous 1930 textbook. During the same period, Hungarian polymath John von Neumann formulated the rigorous mathematical basis for quantum mechanics as the theory of linear operators on Hilbert spaces, as described in his likewise famous 1932 textbook. These, like many other works from the founding period, still stand, and remain widely used.

The field of quantum chemistry was pioneered by physicists Walter Heitler and Fritz London, who published a study of the covalent bond of the hydrogen molecule in 1927. Quantum chemistry was subsequently developed by a large number of workers, including the American theoretical chemist Linus Pauling at Caltech, and John C. Slater into various theories such as Molecular Orbital Theory or Valence Theory.

Beginning in 1927, researchers attempted to apply quantum mechanics to fields instead of single particles, resulting in quantum field theories. Early workers in this area include P.A.M. Dirac, W. Pauli, V. Weisskopf, and P. Jordan. This area of research culminated in the formulation of quantum electrodynamics by R.P. Feynman, F. Dyson, J. Schwinger, and S. Tomonaga during the 1940s. Quantum electrodynamics describes a quantum theory of electrons, positrons, and the electromagnetic field, and served as a model for subsequent quantum field theories.

Feynman diagram of gluon radiation in quantum chromodynamics

The theory of quantum chromodynamics was formulated beginning in the early 1960s. The theory as we know it today was formulated by Politzer, Gross and Wilczek in 1975.

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

Founding experiments

• Thomas Young's double-slit experiment demonstrating the wave nature of light. (c. 1801)
• Henri Becquerel discovers radioactivity. (1896)
• J. J. Thomson's cathode ray tube experiments (discovers the electron and its negative charge). (1897)
• The study of black-body radiation between 1850 and 1900, which could not be explained without quantum concepts.
• The photoelectric effect: Einstein explained this in 1905 (and later received a Nobel prize for it) using the concept of photons, particles of light with quantized energy.
• Robert Millikan's oil-drop experiment, which showed that electric charge occurs as quanta (whole units). (1909)
• Ernest Rutherford's gold foil experiment disproved the plum pudding model of the atom which suggested that the mass and positive charge of the atom are almost uniformly distributed. This led to the planetary model of the atom (1911).
• James Franck and Gustav Hertz's electron collision experiment shows that energy absorption by mercury atoms is quantized. (1914)
• Otto Stern and Walther Gerlach conduct the Stern–Gerlach experiment, which demonstrates the quantized nature of particle spin. (1920)
• Clinton Davisson and Lester Germer demonstrate the wave nature of the electron^[10] in the Electron diffraction experiment. (1927)
• Clyde L. Cowan and Frederick Reines confirm the existence of the neutrino in the neutrino experiment. (1955)
• Clauss Jönsson's double-slit experiment with electrons. (1961)
• The Quantum Hall effect, discovered in 1980 by Klaus von Klitzing. The quantized version of the Hall effect has allowed for the definition of a new practical standard for electrical resistance and for an extremely precise independent determination of the fine structure constant.
• The experimental verification of quantum entanglement by John Clauser and Stuart Freedman. (1972)
• The Mach–Zehnder interferometer experiment conducted by Paul Kwiat, Harold Wienfurter, Thomas Herzog, Anton Zeilinger, and Mark Kasevich, providing experimental verification of the Elitzur–Vaidman bomb tester, proving interaction-free measurement is possible. (1994)

 

Particle in a box

 From Wikipedia, the free encyclopedia

Some trajectories of a particle in a box according to Newton's laws of classical mechanics (A), and according to the Schrödinger equation of quantum mechanics (B–F). In (B–F), the horizontal axis is position, and the vertical axis is the real part (blue) and imaginary part (red) of the wavefunction. The states (B,C,D) are energy eigenstates, but (E,F) are not.

In quantum mechanics, the particle in a box model (also known as the infinite potential well or the infinite square well) describes a particle free to move in a small space surrounded by impenetrable barriers. The model is mainly used as a hypothetical example to illustrate the differences between classical and quantum systems. In classical systems, for example, a particle trapped inside a large box can move at any speed within the box and it is no more likely to be found at one position than another. However, when the well becomes very narrow (on the scale of a few nanometers), quantum effects become important. The particle may only occupy certain positive energy levels. Likewise, it can never have zero energy, meaning that the particle can never "sit still". Additionally, it is more likely to be found at certain positions than at others, depending on its energy level. The particle may never be detected at certain positions, known as spatial nodes.

The particle in a box model is one of the very few problems in quantum mechanics which can be solved analytically, without approximations. Due to its simplicity, the model allows insight into quantum effects without the need for complicated mathematics. It serves as a simple illustration of how energy quantizations (energy levels), which are found in more complicated quantum systems such as atoms and molecules, come about. It is one of the first quantum mechanics problems taught in undergraduate physics courses, and it is commonly used as an approximation for more complicated quantum systems.

One-dimensional solution

The barriers outside a one-dimensional box have infinitely large potential, while the interior of the box has a constant, zero potential.

The simplest form of the particle in a box model considers a one-dimensional system. Here, the particle may only move backwards and forwards along a straight line with impenetrable barriers at either end. The walls of a one-dimensional box may be visualised as regions of space with an infinitely large potential energy. Conversely, the interior of the box has a constant, zero potential energy. This means that no forces act upon the particle inside the box and it can move freely in that region. However, infinitely large forces repel the particle if it touches the walls of the box, preventing it from escaping. The potential energy in this model is given as

${\displaystyle V(x)={\begin{cases}0,&x_{c}-{\tfrac {L}{2}}<x<x_{c}+{\tfrac {L}{2}},\\\infty ,&{\text{otherwise,}}\end{cases}},}$

where L is the length of the box, x_c is the location of the center of the box and x is the position of the particle within the box. Simple cases include the centered box (x_c = 0 ) and the shifted box (x_c = L/2 ).

Position wave function

In quantum mechanics, the wavefunction gives the most fundamental description of the behavior of a particle; the measurable properties of the particle (such as its position, momentum and energy) may all be derived from the wavefunction. The wavefunction ${\displaystyle \psi (x,t)}$ can be found by solving the Schrödinger equation for the system

${\displaystyle i\hbar {\frac {\partial }{\partial t}}\psi (x,t)=-{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}}{\partial x^{2}}}\psi (x,t)+V(x)\psi (x,t),}$

where ${\displaystyle \hbar }$ is the reduced Planck constant, ${\displaystyle m}$ is the mass of the particle, ${\displaystyle i}$ is the imaginary unit and ${\displaystyle t}$ is time.

Inside the box, no forces act upon the particle, which means that the part of the wavefunction inside the box oscillates through space and time with the same form as a free particle:

${\displaystyle \psi (x,t)=[A\sin(kx)+B\cos(kx)]\mathrm {e} ^{-i\omega t},}$

(1)

where ${\displaystyle A}$ and ${\displaystyle B}$ are arbitrary complex numbers. The frequency of the oscillations through space and time is given by the wavenumber ${\displaystyle k}$ and the angular frequency ${\displaystyle \omega }$ respectively. These are both related to the total energy of the particle by the expression

${\displaystyle E=\hbar \omega ={\frac {\hbar ^{2}k^{2}}{2m}},}$

which is known as the dispersion relation for a free particle. Here one must notice that now, since the particle is not entirely free but under the influence of a potential (the potential V described above), the energy of the particle given above is not the same thing as ${\displaystyle {\frac {p^{2}}{2m}}}$ where p is the momentum of the particle, and thus the wavenumber k above actually describes the energy states of the particle, not the momentum states (i.e. it turns out that the momentum of the particle is not given by ${\displaystyle p=\hbar k}$). In this sense, it is quite dangerous to call the number k a wavenumber, since it is not related to momentum like "wavenumber" usually is. The rationale for calling k the wavenumber is that it enumerates the number of crests that the wavefunction has inside the box, and in this sense it is a wavenumber. This discrepancy can be seen more clearly below, when we find out that the energy spectrum of the particle is discrete (only discrete values of energy are allowed) but the momentum spectrum is continuous (momentum can vary continuously) and in particular, the relation ${\displaystyle E={\frac {p^{2}}{2m}}}$ for the energy and momentum of the particle does not hold. As said above, the reason this relation between energy and momentum does not hold is that the particle is not free, but there is a potential V in the system, and the energy of the particle is ${\displaystyle E=T+V}$, where T is the kinetic and V the potential energy.

Initial wavefunctions for the first four states in a one-dimensional particle in a box

The size (or amplitude) of the wavefunction at a given position is related to the probability of finding a particle there by ${\displaystyle P(x,t)=|\psi (x,t)|^{2}}$. The wavefunction must therefore vanish everywhere beyond the edges of the box. Also, the amplitude of the wavefunction may not "jump" abruptly from one point to the next. These two conditions are only satisfied by wavefunctions with the form

${\displaystyle \psi _{n}(x,t)={\begin{cases}A\sin \left(k_{n}\left(x-x_{c}+{\tfrac {L}{2}}\right)\right)\mathrm {e} ^{-i\omega _{n}t}\quad &x_{c}-{\tfrac {L}{2}}<x<x_{c}+{\tfrac {L}{2}}\\0&{\text{otherwise}}\end{cases}},}$

where 

${\displaystyle k_{n}={\frac {n\pi }{L}}}$,

and

${\displaystyle E_{n}=\hbar \omega _{n}={\frac {n^{2}\pi ^{2}\hbar ^{2}}{2mL^{2}}}}$,

where n is a positive integer (1,2,3,4...). For a shifted box (x_c = L/2), the solution is particularly simple. The simplest solutions, ${\displaystyle k_{n}=0}$ or ${\displaystyle A=0}$ both yield the trivial wavefunction ${\displaystyle \psi (x)=0}$, which describes a particle that does not exist anywhere in the system. Negative values of ${\displaystyle n}$ are neglected, since they give wavefunctions identical to the positive ${\displaystyle n}$ solutions except for a physically unimportant sign change. Here one sees that only a discrete set of energy values and wavenumbers k are allowed for the particle. Usually in quantum mechanics it is also demanded that the derivative of the wavefunction in addition to the wavefunction itself be continuous; here this demand would lead to the only solution being the constant zero function, which is not what we desire, so we give up this demand (as this system with infinite potential can be regarded as a nonphysical abstract limiting case, we can treat it as such and "bend the rules"). Note that giving up this demand means that the wavefunction is not a differentiable function at the boundary of the box, and thus it can be said that the wavefunction does not solve the Schrödinger equation at the boundary points ${\displaystyle x=0}$ and ${\displaystyle x=L}$ (but does solve everywhere else).

Finally, the unknown constant ${\displaystyle A}$ may be found by normalizing the wavefunction so that the total probability density of finding the particle in the system is 1. It follows that

${\displaystyle \left|A\right|={\sqrt {\frac {2}{L}}}.}$

Thus, A may be any complex number with absolute value √2/L; these different values of A yield the same physical state, so A = √2/L can be selected to simplify.

It is expected that the eigenvalues, i.e., the energy ${\displaystyle E_{n}}$ of the box should be the same regardless of its position in space, but ${\displaystyle \psi _{n}(x,t)}$ changes. Notice that ${\displaystyle x_{c}-{\tfrac {L}{2}}}$ represents a phase shift in the wave function, This phase shift has no effect when solving the Schrödinger equation, and therefore does not affect the eigenvalue.

If we set the origin of coordinates to the left edge of the box, we can rewrite the spacial part of the wave function succinctly as:

${\displaystyle \psi _{n}(x)={\begin{cases}{\sqrt {\frac {2}{L}}}\sin(k_{n}x)\quad {}{\text{for }}n{\text{ even}}\\{\sqrt {\frac {2}{L}}}\cos(k_{n}x)\quad {}{\text{for }}n{\text{ odd}}\end{cases}}}$.

Momentum wave function

The momentum wavefunction is proportional to the Fourier transform of the position wavefunction. With ${\displaystyle k=p/\hbar }$ (note that the parameter k describing the momentum wavefunction below is not exactly the special k_n above, linked to the energy eigenvalues), the momentum wavefunction is given by

${\displaystyle \phi _{n}(p,t)={\frac {1}{\sqrt {2\pi \hbar }}}\int _{-\infty }^{\infty }\psi _{n}(x,t)e^{-ikx}\,dx={\sqrt {\frac {L}{\pi \hbar }}}\left({\frac {n\pi }{n\pi +kL}}\right)\,{\textrm {sinc}}\left({\tfrac {1}{2}}(n\pi -kL)\right)e^{-ikx_{c}}e^{i(n-1){\tfrac {\pi }{2}}}e^{-i\omega _{n}t},}$

where sinc is the cardinal sine sinc function, sinc(x)=sin(x)/x. For the centered box (x_c= 0), the solution is real and particularly simple, since the phase factor on the right reduces to unity. (With care, it can be written as an even function of p.)

It can be seen that the momentum spectrum in this wave packet is continuous, and one may conclude that for the energy state described by the wavenumber k_n, the momentum can, when measured, also attain other values beyond ${\displaystyle p=\pm \hbar k_{n}}$.

Hence, it also appears that, since the energy is ${\displaystyle E_{n}={\frac {\hbar ^{2}k_{n}^{2}}{2m}}}$ for the nth eigenstate, the relation ${\displaystyle E={\frac {p^{2}}{2m}}}$ does not strictly hold for the measured momentum p; the energy eigenstate ${\displaystyle \psi _{n}}$ is not a momentum eigenstate, and, in fact, not even a superposition of two momentum eigenstates, as one might be tempted to imagine from equation (1) above: peculiarly, it has no well-defined momentum before measurement!

Position and momentum probability distributions

In classical physics, the particle can be detected anywhere in the box with equal probability. In quantum mechanics, however, the probability density for finding a particle at a given position is derived from the wavefunction as ${\displaystyle P(x)=|\psi (x)|^{2}.}$ For the particle in a box, the probability density for finding the particle at a given position depends upon its state, and is given by

${\displaystyle P_{n}(x,t)={\begin{cases}{\frac {2}{L}}\sin ^{2}(k_{n}(x-x_{c}+{\tfrac {L}{2}})),&x_{c}-L/2<x<x_{c}+L/2,\\0,&{\text{otherwise,}}\end{cases}}}$

Thus, for any value of n greater than one, there are regions within the box for which ${\displaystyle P(x)=0}$, indicating that spatial nodes exist at which the particle cannot be found.

In quantum mechanics, the average, or expectation value of the position of a particle is given by

${\displaystyle \langle x\rangle =\int _{-\infty }^{\infty }xP_{n}(x)\,\mathrm {d} x.}$

For the steady state particle in a box, it can be shown that the average position is always ${\displaystyle \langle x\rangle =x_{c}}$, regardless of the state of the particle. For a superposition of states, the expectation value of the position will change based on the cross term which is proportional to ${\displaystyle \cos(\omega t)}$.

The variance in the position is a measure of the uncertainty in position of the particle:

${\displaystyle \mathrm {Var} (x)=\int _{-\infty }^{\infty }(x-\langle x\rangle )^{2}P_{n}(x)\,dx={\frac {L^{2}}{12}}\left(1-{\frac {6}{n^{2}\pi ^{2}}}\right)}$

The probability density for finding a particle with a given momentum is derived from the wavefunction as ${\displaystyle P(x)=|\phi (x)|^{2}}$. As with position, the probability density for finding the particle at a given momentum depends upon its state, and is given by

${\displaystyle P_{n}(p)={\frac {L}{\pi \hbar }}\left({\frac {n\pi }{n\pi +kL}}\right)^{2}\,{\textrm {sinc}}^{2}\left({\tfrac {1}{2}}(n\pi -kL)\right)}$

where, again, ${\displaystyle k=p/\hbar }$. The expectation value for the momentum is then calculated to be zero, and the variance in the momentum is calculated to be:

${\displaystyle \mathrm {Var} (p)=\left({\frac {\hbar n\pi }{L}}\right)^{2}}$

The uncertainties in position and momentum (${\displaystyle \Delta x}$ and ${\displaystyle \Delta p}$) are defined as being equal to the square root of their respective variances, so that:

${\displaystyle \Delta x\Delta p={\frac {\hbar }{2}}{\sqrt {{\frac {n^{2}\pi ^{2}}{3}}-2}}}$

This product increases with increasing n, having a minimum value for n=1. The value of this product for n=1 is about equal to 0.568 ${\displaystyle \hbar }$ which obeys the Heisenberg uncertainty principle, which states that the product will be greater than or equal to ${\displaystyle \hbar /2}$

Another measure of uncertainty in position is the information entropy of the probability distribution H_x:

${\displaystyle H_{x}=\int _{-\infty }^{\infty }P_{n}(x)\log(P_{n}(x)x_{0})\,dx=\log \left({\frac {2L}{e\,x_{0}}}\right)}$

where x₀ is an arbitrary reference length.

Another measure of uncertainty in momentum is the information entropy of the probability distribution H_p:

${\displaystyle H_{p}(n)=\int _{-\infty }^{\infty }P_{n}(p)\log(P_{n}(p)p_{0})\,dp}$

${\displaystyle \lim _{n\to \infty }H_{p}(n)=\log \left({\frac {4\pi \hbar \,e^{2(1-\gamma )}}{L\,p_{0}}}\right)}$

where γ is Euler's constant. The quantum mechanical entropic uncertainty principle states that for ${\displaystyle x_{0}\,p_{0}=\hbar }$

${\displaystyle H_{x}+H_{p}(n)\geq \log(e\,\pi )\approx 2.14473...}$ (nats)

For ${\displaystyle x_{0}\,p_{0}=\hbar }$, the sum of the position and momentum entropies yields:

${\displaystyle H_{x}+H_{p}(\infty )=\log(8\pi \,e^{1-2\gamma })\approx 3.06974...}$ (nats)

which satisfies the quantum entropic uncertainty principle.

Energy levels

The energy of a particle in a box (black circles) and a free particle (grey line) both depend upon wavenumber in the same way. However, the particle in a box may only have certain, discrete energy levels.

The energies which correspond with each of the permitted wavenumbers may be written as

${\displaystyle E_{n}={\frac {n^{2}\hbar ^{2}\pi ^{2}}{2mL^{2}}}={\frac {n^{2}h^{2}}{8mL^{2}}}}$.

The energy levels increase with ${\displaystyle n^{2}}$, meaning that high energy levels are separated from each other by a greater amount than low energy levels are. The lowest possible energy for the particle (its zero-point energy) is found in state 1, which is given by

${\displaystyle E_{1}={\frac {\hbar ^{2}\pi ^{2}}{2mL^{2}}}.}$

The particle, therefore, always has a positive energy. This contrasts with classical systems, where the particle can have zero energy by resting motionlessly. This can be explained in terms of the uncertainty principle, which states that the product of the uncertainties in the position and momentum of a particle is limited by

${\displaystyle \Delta x\Delta p\geq {\frac {\hbar }{2}}}$

It can be shown that the uncertainty in the position of the particle is proportional to the width of the box. Thus, the uncertainty in momentum is roughly inversely proportional to the width of the box. The kinetic energy of a particle is given by ${\displaystyle E=p^{2}/(2m)}$, and hence the minimum kinetic energy of the particle in a box is inversely proportional to the mass and the square of the well width, in qualitative agreement with the calculation above.

Higher-dimensional boxes

(Hyper)rectangular walls

The wavefunction of a 2D well with n_x=4 and n_y=4

If a particle is trapped in a two-dimensional box, it may freely move in the ${\displaystyle x}$ and ${\displaystyle y}$-directions, between barriers separated by lengths ${\displaystyle L_{x}}$ and ${\displaystyle L_{y}}$ respectively. For a centered box, the position wave function may be written including the length of the box as ${\displaystyle \psi _{n}(x,t,L)}$. Using a similar approach to that of the one-dimensional box, it can be shown that the wavefunctions and energies for a centered box are given respectively by

${\displaystyle \psi _{n_{x},n_{y}}=\psi _{n_{x}}(x,t,L_{x})\psi _{n_{y}}(y,t,L_{y})}$,

${\displaystyle E_{n_{x},n_{y}}={\frac {\hbar ^{2}k_{n_{x},n_{y}}^{2}}{2m}}}$,

where the two-dimensional wavevector is given by

${\displaystyle \mathbf {k_{n_{x},n_{y}}} =k_{n_{x}}\mathbf {\hat {x}} +k_{n_{y}}\mathbf {\hat {y}} ={\frac {n_{x}\pi }{L_{x}}}\mathbf {\hat {x}} +{\frac {n_{y}\pi }{L_{y}}}\mathbf {\hat {y}} }$.

For a three dimensional box, the solutions are

${\displaystyle \psi _{n_{x},n_{y},n_{z}}=\psi _{n_{x}}(x,t,L_{x})\psi _{n_{y}}(y,t,L_{y})\psi _{n_{z}}(z,t,L_{z})}$,

${\displaystyle E_{n_{x},n_{y},n_{z}}={\frac {\hbar ^{2}k_{n_{x},n_{y},n_{z}}^{2}}{2m}}}$,

where the three-dimensional wavevector is given by:

${\displaystyle \mathbf {k_{n_{x},n_{y},n_{z}}} =k_{n_{x}}\mathbf {\hat {x}} +k_{n_{y}}\mathbf {\hat {y}} +k_{n_{z}}\mathbf {\hat {z}} ={\frac {n_{x}\pi }{L_{x}}}\mathbf {\hat {x}} +{\frac {n_{y}\pi }{L_{y}}}\mathbf {\hat {y}} +{\frac {n_{z}\pi }{L_{z}}}\mathbf {\hat {z}} }$.

In general for an n-dimensional box, the solutions are

${\displaystyle \psi =\prod _{i}\psi _{n_{i}}(x_{i},t,L_{i})}$

The n-dimensional momentum wave functions may likewise be represented by ${\displaystyle \phi _{n}(x,t,L_{x})}$ and the momentum wave function for an n-dimensional centered box is then:

${\displaystyle \phi =\prod _{i}\phi _{n_{i}}(k_{i},t,L_{i})}$

An interesting feature of the above solutions is that when two or more of the lengths are the same (e.g. ${\displaystyle L_{x}=L_{y}}$), there are multiple wavefunctions corresponding to the same total energy. For example, the wavefunction with ${\displaystyle n_{x}=2,n_{y}=1}$ has the same energy as the wavefunction with ${\displaystyle n_{x}=1,n_{y}=2}$. This situation is called degeneracy and for the case where exactly two degenerate wavefunctions have the same energy that energy level is said to be doubly degenerate. Degeneracy results from symmetry in the system. For the above case two of the lengths are equal so the system is symmetric with respect to a 90° rotation.

More complicated wall shapes

The wavefunction for a quantum-mechanical particle in a box whose walls have arbitrary shape is given by the Helmholtz equation subject to the boundary condition that the wavefunction vanishes at the walls. These systems are studied in the field of quantum chaos for wall shapes whose corresponding dynamical billiard tables are non-integrable.

Applications

Because of its mathematical simplicity, the particle in a box model is used to find approximate solutions for more complex physical systems in which a particle is trapped in a narrow region of low electric potential between two high potential barriers. These quantum well systems are particularly important in optoelectronics, and are used in devices such as the quantum well laser, the quantum well infrared photodetector and the quantum-confined Stark effect modulator. It is also used to model a lattice in the Kronig-Penney model and for a finite metal with the free electron approximation.

Conjugated polyenes

β-carotene is a conjugated polyene

Conjugated polyene systems can be modeled using particle in a box. The conjugated system of electrons can be modeled as a one dimensional box with length equal to the total bond distance from one terminus of the polyene to the other. In this case each pair of electrons in each π bond corresponds to one energy level. The energy difference between two energy levels, n_f and n_i is:

${\displaystyle \Delta E={\frac {(n_{f}^{2}-n_{i}^{2})h^{2}}{8mL^{2}}}}$

The difference between the ground state energy, n, and the first excited state, n+1, corresponds to the energy required to excite the system. This energy has a specific wavelength, and therefore color of light, related by:

${\displaystyle \lambda ={\frac {hc}{\Delta E}}}$

A common example of this phenomenon is in β-carotene. β-carotene (C₄₀H₅₆) is a conjugated polyene with an orange color and a molecular length of approximately 3.8 nm (though its chain length is only approximately 2.4 nm). Due to β-carotene's high level of conjugation, electrons are dispersed throughout the length of the molecule, allowing one to model it as a one-dimensional particle in a box. β-carotene has 11 carbon-carbon double bonds in conjugation; each of those double bonds contains two π-electrons, therefore β-carotene has 22 π-electrons. With two electrons per energy level, β-carotene can be treated as a particle in a box at energy level n=11. Therefore, the minimum energy needed to excite an electron to the next energy level can be calculated, n=12, as follows (recalling that the mass of an electron is 9.109 × 10⁻³¹ kg):

${\displaystyle \Delta E={\frac {(n_{f}^{2}-n_{i}^{2})h^{2}}{8mL^{2}}}}$

${\displaystyle ={\frac {(12^{2}-11^{2})h^{2}}{8mL^{2}}}}$

${\displaystyle =2.3658\times 10^{-19}{\text{ J}}}$

Using the previous relation of wavelength to energy, recalling both Planck's constant h and the speed of light c:

${\displaystyle \lambda ={\tfrac {hc}{\Delta E}}}$

${\displaystyle =0.00000084{\text{ m}}=840{\text{ nm}}}$

This indicates that β-carotene primarily absorbs light in the infrared spectrum, therefore it would appear white to a human eye. However the observed wavelength is 450 nm, indicating that the particle in a box is not a perfect model for this system.

Quantum well laser

The particle in a box model can be applied to quantum well lasers, which are laser diodes consisting of one semiconductor “well” material sandwiched between two other semiconductor layers of different material . Because the layers of this sandwich are very thin (the middle layer is typically about 100 Å thick), quantum confinement effects can be observed. The idea that quantum effects could be harnessed to create better laser diodes originated in the 1970s. The quantum well laser was patented in 1976 by R. Dingle and C. H. Henry.

Specifically, the quantum well’s behavior can be represented by the particle in a finite well model. Two boundary conditions must be selected. The first is that the wave function must be continuous. Often, the second boundary condition is chosen to be the derivative of the wave function must be continuous across the boundary, but in the case of the quantum well the masses are different on either side of the boundary. Instead, the second boundary condition is chosen to conserve particle flux as${\displaystyle (1/m)d\phi /dz}$, which is consistent with experiment. The solution to the finite well particle in a box must be solved numerically, resulting in wave functions that are sine functions inside the quantum well and exponentially decaying functions in the barriers. This quantization of the energy levels of the electrons allows a quantum well laser to emit light more efficiently than conventional semiconductor lasers.

Due to their small size, quantum dots do not showcase the bulk properties of the specified semi-conductor but rather show quantised energy states. This effect is known as the quantum confinement and has led to numerous applications of quantum dots such as the quantum well laser.

Researchers at Princeton University have recently built a quantum well laser which is no bigger than a grain of rice. The laser is powered by a single electron which passes through two quantum dots; a double quantum dot. The electron moves from a state of higher energy, to a state of lower energy whilst emitting photons in the microwave region. These photons bounce off mirrors to create a beam of light; the laser.

The quantum well laser is heavily based on the interaction between light and electrons. This relationship is a key component in quantum mechanical theories which include the De Broglie Wavelength and Particle in a box. The double quantum dot allows scientists to gain full control over the movement of an electron which consequently results in the production of a laser beam.

Quantum dots

Quantum dots are extremely small semiconductors (on the scale of nanometers). They display quantum confinement in that the electrons cannot escape the “dot”, thus allowing particle-in-a-box approximations to be applied. Their behavior can be described by three-dimensional particle-in-a-box energy quantization equations.

The energy gap of a quantum dot is the energy gap between its valence and conduction bands. This energy gap ${\displaystyle \Delta E(r)}$ is equal to the band gap of the bulk material ${\displaystyle E_{\text{gap}}}$ plus the energy equation derived from particle-in-a-box, which gives the energy for electrons and holes. This can be seen in the following equation, where ${\displaystyle m_{e}^{*}}$ and ${\displaystyle m_{h}^{*}}$ are the effective masses of the electron and hole, ${\displaystyle r}$ is radius of the dot, and ${\displaystyle h}$ is Planck's constant:

${\displaystyle \Delta E(r)=E_{\text{gap}}+\left({\frac {h^{2}}{8r^{2}}}\right)\left({\frac {1}{m_{e}^{*}}}+{\frac {1}{m_{h}^{*}}}\right)}$

Hence, the energy gap of the quantum dot is inversely proportional to the square of the “length of the box,” i.e. the radius of the quantum dot.

Manipulation of the band gap allows for the absorption and emission of specific wavelengths of light, as energy is inversely proportional to wavelength. The smaller the quantum dot, the larger the band gap and thus the shorter the wavelength absorbed.

Different semiconducting materials are used to synthesize quantum dots of different sizes and therefore emit different wavelengths of light. Materials that normally emit light in the visible region are often used and their sizes are fine-tuned so that certain colors are emitted. Typical substances used to synthesize quantum dots are cadmium (Cd) and selenium (Se). For example, when the electrons of two nanometer CdSe quantum dots relax after excitation, blue light is emitted. Similarly, red light is emitted in four nanometer CdSe quantum dots.

Quantum dots have a variety of functions including but not limited to fluorescent dyes, transistors, LEDs, solar cells, and medical imaging via optical probes.

One function of quantum dots is their use in lymph node mapping, which is feasible due to their unique ability to emit light in the near infrared (NIR) region. Lymph node mapping allows surgeons to track if and where cancerous cells exist.

Quantum dots are useful for these functions due to their emission of brighter light, excitation by a wide variety of wavelengths, and higher resistance to light than other substances.