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Helium-3

Helium-3 (3He see also helion) is a light, stable isotope of helium with two protons and one neutron (in contrast, the most common isotope, helium-4 has two protons and two neutrons). Other than protium (ordinary hydrogen), helium-3 is the only stable isotope of any element with more protons than neutrons. It was discovered in 1939.

Helium-3 occurs as a primordial nuclide, escaping from Earth's crust into its atmosphere and into outer space over millions of years. It is also thought to be a natural nucleogenic and cosmogenic nuclide, one produced when lithium is bombarded by natural neutrons, which can be released by spontaneous fission and by nuclear reactions with cosmic rays. Some found in the terrestrial atmosphere is a remnant of atmospheric and underwater nuclear weapons testing.

Nuclear fusion using helium-3 has long been viewed as a desirable future energy source. The fusion of two of its atoms would be aneutronic, not release the dangerous radiation of tradiational fusion and require much higher temperatures. The process may unavoidably create other reactions that themselves would cause the surrounding material to become radioactive.

Helium-3 is thought to be more abundant on the Moon than on Earth, having been created in the upper layer of regolith by the solar wind over billions of years, though still lower in abundance than in the Solar System's gas giants.

History
The existence of helium-3 was first proposed in 1934 by the Australian nuclear physicist Mark Oliphant while he was working at the University of Cambridge Cavendish Laboratory. Oliphant had performed experiments in which fast deuterons collided with deuteron targets (incidentally, the first demonstration of nuclear fusion). Isolation of helium-3 was first accomplished by Luis Alvarez and Robert Cornog in 1939. Helium-3 was thought to be a radioactive isotope until it was also found in samples of natural helium, which is mostly helium-4, taken both from the terrestrial atmosphere and from natural gas wells.

Physical properties
Due to its low atomic mass of 3.016 u, helium-3 has some physical properties different from those of helium-4, with a mass of 4.0026 u. On account of the weak, induced dipole–dipole interaction between the helium atoms, their microscopic physical properties are mainly determined by their zero-point energy. Also, the microscopic properties of helium-3 cause it to have a higher zero-point energy than helium-4. This implies that helium-3 can overcome dipole–dipole interactions with less thermal energy than helium-4 can.

The quantum mechanical effects on helium-3 and helium-4 are significantly different because with two protons, two neutrons, and two electrons, helium-4 has an overall spin of zero, making it a boson, but with one fewer neutron, helium-3 has an overall spin of one half, making it a fermion.

Pure helium-3 gas boils at 3.19 K compared with helium-4 at 4.23 K, and its critical point is also lower at 3.35 K, compared with helium-4 at 5.2 K. Helium-3 has less than half the density of helium-4 when it is at its boiling point: 59 g/L compared to 125 g/L of helium-4 at a pressure of one atmosphere. Its latent heat of vaporization is also considerably lower at 0.026 kJ/mol compared with the 0.0829 kJ/mol of helium-4.

Superfluidity
An important property of helium-3, which distinguishes it from the more common helium-4, is that its nucleus is a fermion since it contains an odd number of spin $1/2$ particles. Helium-4 nuclei are bosons, containing an even number of spin $1/2$ particles. This is a direct result of the addition rules for quantized angular momentum. At low temperatures (about 2.17 K), helium-4 undergoes a phase transition: A fraction of it enters a superfluid phase that can be roughly understood as a type of Bose–Einstein condensate. Such a mechanism is not available for helium-3 atoms, which are fermions. Many speculated that helium-3 could also become a superfluid at much lower temperatures, if the atoms formed into pairs analogous to Cooper pairs in the BCS theory of superconductivity. Each Cooper pair, having integer spin, can be thought of as a boson. During the 1970s, David Lee, Douglas Osheroff and Robert Coleman Richardson discovered two phase transitions along the melting curve, which were soon realized to be the two superfluid phases of helium-3. The transition to a superfluid occurs at 2.491 millikelvins on the melting curve. They were awarded the 1996 Nobel Prize in Physics for their discovery. Alexei Abrikosov, Vitaly Ginzburg, and Tony Leggett won the 2003 Nobel Prize in Physics for their work on refining understanding of the superfluid phase of helium-3.

In a zero magnetic field, there are two distinct superfluid phases of 3He, the A-phase and the B-phase. The B-phase is the low-temperature, low-pressure phase which has an isotropic energy gap. The A-phase is the higher temperature, higher pressure phase that is further stabilized by a magnetic field and has two point nodes in its gap. The presence of two phases is a clear indication that 3He is an unconventional superfluid (superconductor), since the presence of two phases requires an additional symmetry, other than gauge symmetry, to be broken. In fact, it is a p-wave superfluid, with spin one, S=1, and angular momentum one, L=1. The ground state corresponds to total angular momentum zero, J=S+L=0 (vector addition). Excited states are possible with non-zero total angular momentum, J>0, which are excited pair collective modes. Because of the extreme purity of superfluid 3He (since all materials except 4He have solidified and sunk to the bottom of the liquid 3He and any 4He has phase separated entirely, this is the most pure condensed matter state), these collective modes have been studied with much greater precision than in any other unconventional pairing system.

Terrestrial abundance
3He is a primordial substance in the Earth's mantle, thought to have become entrapped in the Earth during planetary formation. The ratio of 3He to 4He within the Earth's crust and mantle is less than that of estimates of solar disk composition as obtained from meteorite and lunar samples, with terrestrial materials generally containing lower 3He/4He ratios due to production of 4He from radioactive decay.

3He has a cosmological ratio of 300 atoms per million atoms of 4He (at. ppm), leading to the assumption that the original ratio of these primordial gases in the mantle was around 200-300 ppm when Earth was formed. Over Earth's history alpha-particle decay of uranium, thorium and other radioactive isotopes has generated significant amounts of 4He, such that only around 7% of the helium now in the mantle is primordial helium, lowering the total 3He/4He ratio to around 20 ppm. Ratios of 3He/4He in excess of atmospheric are indicative of a contribution of 3He from the mantle. Crustal sources are dominated by the 4He produced by radioactive decay.

The ratio of helium-3 to helium-4 in natural Earth-bound sources varies greatly. Samples of the lithium ore spodumene from Edison Mine, South Dakota were found to contain 12 parts of helium-3 to a million parts of helium-4. Samples from other mines showed 2 parts per million.

Helium is also present as up to 7% of some natural gas sources, and large sources have over 0.5% (above 0.2% makes it viable to extract). The fraction of 3He in helium separated from natural gas in the U.S. was found to range from 70 to 242 parts per billion. Hence the US 2002 stockpile of 1 billion normal m3 would have contained about 12 to 43 kg of helium-3. According to American physicist Richard Garwin, about 26 m3 or almost 5 kg of 3He is available annually for separation from the US natural gas stream. If the process of separating out the 3He could employ as feedstock the liquefied helium typically used to transport and store bulk quantities, estimates for the incremental energy cost range from 34 to 300 $/l NTP, excluding the cost of infrastructure and equipment. Algeria's annual gas production is assumed to contain 100 million normal cubic metres and this would contain between 7 and 24 m3 of helium-3 (about 1 to 4 kg) assuming a similar 3He fraction.

3He is also present in the Earth's atmosphere. The natural abundance of 3He in naturally occurring helium gas is 1.38 (1.38 parts per million). The partial pressure of helium in the Earth's atmosphere is about 0.52 Pa, and thus helium accounts for 5.2 parts per million of the total pressure (101325 Pa) in the Earth's atmosphere, and 3He thus accounts for 7.2 parts per trillion of the atmosphere. Since the atmosphere of the Earth has a mass of about 5.14e18 kg, the mass of 3He in the Earth's atmosphere is the product of these numbers, or about 37,000 t of 3He. (In fact the effective figure is ten times smaller, since the above ppm are ppmv and not ppmw. One must multiply by 3 (the molecular mass of helium-3) and divide by 29 (the mean molecular mass of the atmosphere), resulting in 3,828 t of helium-3 in the earth's atmosphere.)

3He is produced on Earth from three sources: lithium spallation, cosmic rays, and beta decay of tritium (3H). The contribution from cosmic rays is negligible within all except the oldest regolith materials, and lithium spallation reactions are a lesser contributor than the production of 4He by alpha particle emissions.

The total amount of helium-3 in the mantle may be in the range of 0.1–1 Mt. Most mantle is not directly accessible. Some helium-3 leaks up through deep-sourced hotspot volcanoes such as those of the Hawaiian Islands, but only 300 g per year is emitted to the atmosphere. Mid-ocean ridges emit another 3 kg/year. Around subduction zones, various sources produce helium-3 in natural gas deposits which possibly contain a thousand tonnes of helium-3 (although there may be 25 thousand tonnes if all ancient subduction zones have such deposits). Wittenberg estimated that United States crustal natural gas sources may have only half a tonne total. Wittenberg cited Anderson's estimate of another 1200 t in interplanetary dust particles on the ocean floors. In the 1994 study, extracting helium-3 from these sources consumes more energy than fusion would release.

Lunar surface
See Extraterrestrial mining or Lunar resources

Solar nebula (primordial) abundance
One early estimate of the primordial ratio of 3He to 4He in the solar nebula has been the measurement of their ratio in the atmosphere of Jupiter, measured by the mass spectrometer of the Galileo atmospheric entry probe. This ratio is about 1:10,000, or 100 parts of 3He per million parts of 4He. This is roughly the same ratio of the isotopes as in lunar regolith, which contains 28 ppm helium-4 and 2.8 ppb helium-3 (which is at the lower end of actual sample measurements, which vary from about 1.4 to 15 ppb). Terrestrial ratios of the isotopes are lower by a factor of 100, mainly due to enrichment of helium-4 stocks in the mantle by billions of years of alpha decay from uranium, thorium as well as their decay products and extinct radionuclides.

Tritium decay
Virtually all helium-3 used in industry today is produced from the radioactive decay of tritium, given its very low natural abundance and its very high cost.

Production, sales and distribution of helium-3 in the United States are managed by the US Department of Energy (DOE) DOE Isotope Program.

While tritium has several different experimentally determined values of its half-life, NIST lists $1/2$ ($4,500 days$). It decays into helium-3 by beta decay as in this nuclear equation:

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Among the total released energy of $12.32 years$, the part taken by electron's kinetic energy varies, with an average of $18.6 keV$, while the remaining energy is carried off by the nearly undetectable electron antineutrino. Beta particles from tritium can penetrate only about 6.0 mm of air, and they are incapable of passing through the dead outermost layer of human skin. The unusually low energy released in the tritium beta decay makes the decay (along with that of rhenium-187) appropriate for absolute neutrino mass measurements in the laboratory (the most recent experiment being KATRIN).

The low energy of tritium's radiation makes it difficult to detect tritium-labeled compounds except by using liquid scintillation counting.

Tritium is a radioactive isotope of hydrogen and is typically produced by bombarding lithium-6 with neutrons in a nuclear reactor. The lithium nucleus absorbs a neutron and splits into helium-4 and tritium. Tritium decays into helium-3 with a half-life of $5.7 keV$, so helium-3 can be produced by simply storing the tritium until it undergoes radioactive decay. As tritium forms a stable compound with oxygen (tritiated water) while helium-3 does not, the storage and collection process could continuously collect the material that outgasses from the stored material.

Tritium is a critical component of nuclear weapons and historically it was produced and stockpiled primarily for this application. The decay of tritium into helium-3 reduces the explosive power of the fusion warhead, so periodically the accumulated helium-3 must be removed from warhead reservoirs and tritium in storage. Helium-3 removed during this process is marketed for other applications.

For decades this has been, and remains, the principal source of the world's helium-3. Since the signing of the START I Treaty in 1991 the number of nuclear warheads that are kept ready for use has decreased. This has reduced the quantity of helium-3 available from this source. Helium-3 stockpiles have been further diminished by increased demand, primarily for use in neutron radiation detectors and medical diagnostic procedures. US industrial demand for helium-3 reached a peak of 70,000 L (approximately 8 kg) per year in 2008. Price at auction, historically about 100 $/l, reached as high as 2000 $/l. Since then, demand for helium-3 has declined to about 6000 L per year due to the high cost and efforts by the DOE to recycle it and find substitutes. Assuming a density of 114 g/m3 at $100/l helium-3 would be about a thirtieth as expensive as tritium (roughly 880 $/g vs roughly 30000 $/g) while at $2000/l helium-3 would be about half as expensive as tritium (17540 $/g vs 30000 $/g).

The DOE recognized the developing shortage of both tritium and helium-3, and began producing tritium by lithium irradiation at the Tennessee Valley Authority's Watts Bar Nuclear Generating Station in 2010. In this process tritium-producing burnable absorber rods (TPBARs) containing lithium in a ceramic form are inserted into the reactor in place of the normal boron control rods Periodically the TPBARs are replaced and the tritium extracted.

Currently only two commercial nuclear reactors (Watts Bar Nuclear Plant Units 1 and 2) are being used for tritium production but the process could, if necessary, be vastly scaled up to meet any conceivable demand simply by utilizing more of the nation's power reactors. Substantial quantities of tritium and helium-3 could also be extracted from the heavy water moderator in CANDU nuclear reactors. India and Canada, the two countries with the largest heavy water reactor fleet, are both known to extract tritium from moderator/coolant heavy water, but those amounts are not nearly enough to satisfy global demand of either tritium or helium-3.

As tritium is also produced inadvertently in various processes in light water reactors (see the article on tritium for details), extraction from those sources could be another source of helium-3. If the annual discharge of tritium (per 2018 figures) at La Hague reprocessing facility is taken as a basis, the amounts discharged (31.2 g at La Hague) are not nearly enough to satisfy demand, even if 100% recovery is achieved.

Helium-3 spin echo
Helium-3 can be used to do spin echo experiments of surface dynamics, which are underway at the Surface Physics Group at the Cavendish Laboratory in Cambridge and in the Chemistry Department at Swansea University.

Neutron detection
Helium-3 is an important isotope in instrumentation for neutron detection. It has a high absorption cross section for thermal neutron beams and is used as a converter gas in neutron detectors. The neutron is converted through the nuclear reaction
 * n + 3He → 3H + 1H + 0.764 MeV

into charged particles tritium ions (T, 3H) and Hydrogen ions, or protons (p, 1H) which then are detected by creating a charge cloud in the stopping gas of a proportional counter or a Geiger–Müller tube.

Furthermore, the absorption process is strongly spin-dependent, which allows a spin-polarized helium-3 volume to transmit neutrons with one spin component while absorbing the other. This effect is employed in neutron polarization analysis, a technique which probes for magnetic properties of matter.

The United States Department of Homeland Security had hoped to deploy detectors to spot smuggled plutonium in shipping containers by their neutron emissions, but the worldwide shortage of helium-3 following the drawdown in nuclear weapons production since the Cold War has to some extent prevented this. As of 2012, DHS determined the commercial supply of boron-10 would support converting its neutron detection infrastructure to that technology.

Cryogenics
A helium-3 refrigerator uses helium-3 to achieve temperatures of 0.2 to 0.3 kelvin. A dilution refrigerator uses a mixture of helium-3 and helium-4 to reach cryogenic temperatures as low as a few thousandths of a kelvin.

Medical imaging
Helium-3 nuclei have an intrinsic nuclear spin of $12.3 years$, and a relatively high magnetogyric ratio. Helium-3 can be hyperpolarized using non-equilibrium means such as spin-exchange optical pumping. During this process, circularly polarized infrared laser light, tuned to the appropriate wavelength, is used to excite electrons in an alkali metal, such as caesium or rubidium inside a sealed glass vessel. The angular momentum is transferred from the alkali metal electrons to the noble gas nuclei through collisions. In essence, this process effectively aligns the nuclear spins with the magnetic field in order to enhance the NMR signal. The hyperpolarized gas may then be stored at pressures of 10 atm, for up to 100 hours. Following inhalation, gas mixtures containing the hyperpolarized helium-3 gas can be imaged with an MRI scanner to produce anatomical and functional images of lung ventilation. This technique is also able to produce images of the airway tree, locate unventilated defects, measure the alveolar oxygen partial pressure, and measure the ventilation/perfusion ratio. This technique may be critical for the diagnosis and treatment management of chronic respiratory diseases such as chronic obstructive pulmonary disease (COPD), emphysema, cystic fibrosis, and asthma.

Radio energy absorber for tokamak plasma experiments
Both MIT's Alcator C-Mod tokamak and the Joint European Torus (JET) have experimented with adding a little helium-3 to a H–D plasma to increase the absorption of radio-frequency (RF) energy to heat the hydrogen and deuterium ions, a "three-ion" effect.

Nuclear fuel
^{3}He can be produced by the low temperature fusion of $1/2$  → ^{3}He + γ + 4.98 MeV. If the fusion temperature is below that for the helium nuclei to fuse, the reaction produces a high energy alpha particle which quickly acquires an electron producing a stable light helium ion which can be utilized directly as a source of electricity without producing dangerous neutrons.

^{3}He can be used in fusion reactions by either of the reactions ^{2}H + ^{3}He -> ^{4}He + ^{1}p + 18.3 MeV, or ^{3}He + ^{3}He -> ^{4}He + 2 ^{1}p + 12.86 MeV.

The conventional deuterium + tritium ("D-T") fusion process produces energetic neutrons which render reactor components radioactive with activation products. The appeal of helium-3 fusion stems from the aneutronic nature of its reaction products. Helium-3 itself is non-radioactive. The lone high-energy by-product, the proton, can be contained by means of electric and magnetic fields. The momentum energy of this proton (created in the fusion process) will interact with the containing electromagnetic field, resulting in direct net electricity generation.

Because of the higher Coulomb barrier, the temperatures required for ^{2}H + ^{3}He fusion are much higher than those of conventional D-T fusion. Moreover, since both reactants need to be mixed together to fuse, reactions between nuclei of the same reactant will occur, and the D-D reaction (^{2}H + ^{2}H) does produce a neutron. Reaction rates vary with temperature, but the D-^{3}He reaction rate is never greater than 3.56 times the D-D reaction rate (see graph). Therefore, fusion using D-^{3}He fuel at the right temperature and a D-lean fuel mixture, can produce a much lower neutron flux than D-T fusion, but is not clean, negating some of its main attraction.

The second possibility, fusing ^{3}He with itself (^{3}He + ^{3}He), requires even higher temperatures (since now both reactants have a +2 charge), and thus is even more difficult than the D-^{3}He reaction. It offers a theoretical reaction that produces no neutrons; the charged protons produced can be contained in electric and magnetic fields, which in turn directly generates electricity. ^{3}He + ^{3}He fusion is feasible as demonstrated in the laboratory and has immense advantages, but commercial viability is many years in the future.

The amounts of helium-3 needed as a replacement for conventional fuels are substantial by comparison to amounts currently available. The total amount of energy produced in the ^{2}D + ^{3}He reaction is 18.4 MeV, which corresponds to some 493 megawatt-hours (4.93×108 W·h) per three grams (one mole) of ^{3}He. If the total amount of energy could be converted to electrical power with 100% efficiency (a physical impossibility), it would correspond to about 30 minutes of output of a gigawatt electrical plant per mole of ^{3}He. Thus, a year's production (at 6 grams for each operation hour) would require 52.5 kilograms of helium-3. The amount of fuel needed for large-scale applications can also be put in terms of total consumption: electricity consumption by 107 million U.S. households in 2001 totaled 1,140 billion kW·h (1.14×1015 W·h). Again assuming 100% conversion efficiency, 6.7 tonnes per year of helium-3 would be required for that segment of the energy demand of the United States, 15 to 20 tonnes per year given a more realistic end-to-end conversion efficiency.

A second-generation approach to controlled fusion power involves combining helium-3 and deuterium, ^{2}D. This reaction produces an alpha particle and a high-energy proton. The most important potential advantage of this fusion reaction for power production as well as other applications lies in its compatibility with the use of electrostatic fields to control fuel ions and the fusion protons. High speed protons, as positively charged particles, can have their kinetic energy converted directly into electricity, through use of solid-state conversion materials as well as other techniques. Potential conversion efficiencies of 70% may be possible, as there is no need to convert proton energy to heat in order to drive a turbine-powered electrical generator.

He-3 power plants
There have been many claims about the capabilities of helium-3 power plants. According to proponents, fusion power plants operating on deuterium and helium-3 would offer lower capital and operating costs than their competitors due to less technical complexity, higher conversion efficiency, smaller size, the absence of radioactive fuel, no air or water pollution, and only low-level radioactive waste disposal requirements. Recent estimates suggest that about $6 billion in investment capital will be required to develop and construct the first helium-3 fusion power plant. Financial break even at today's wholesale electricity prices (5 US cents per kilowatt-hour) would occur after five 1-gigawatt plants were on line, replacing old conventional plants or meeting new demand.

The reality is not so clear-cut. The most advanced fusion programs in the world are inertial confinement fusion (such as National Ignition Facility) and magnetic confinement fusion (such as ITER and Wendelstein 7-X). In the case of the former, there is no solid roadmap to power generation. In the case of the latter, commercial power generation is not expected until around 2050. In both cases, the type of fusion discussed is the simplest: D-T fusion. The reason for this is the very low Coulomb barrier for this reaction; for D+3He, the barrier is much higher, and it is even higher for 3He–3He. The immense cost of reactors like ITER and National Ignition Facility are largely due to their immense size, yet to scale up to higher plasma temperatures would require reactors far larger still. The 14.7 MeV proton and 3.6 MeV alpha particle from D–3He fusion, plus the higher conversion efficiency, means that more electricity is obtained per kilogram than with D-T fusion (17.6 MeV), but not that much more. As a further downside, the rates of reaction for helium-3 fusion reactions are not particularly high, requiring a reactor that is larger still or more reactors to produce the same amount of electricity.

Alternatives to He-3
To attempt to work around this problem of massively large power plants that may not even be economical with D-T fusion, let alone the far more challenging D–3He fusion, a number of other reactors have been proposed – the Fusor, Polywell, Focus fusion, and many more, though many of these concepts have fundamental problems with achieving a net energy gain, and generally attempt to achieve fusion in thermal disequilibrium, something that could potentially prove impossible, and consequently, these long-shot programs tend to have trouble garnering funding despite their low budgets. Unlike the "big" and "hot" fusion systems, if such systems worked, they could scale to the higher barrier aneutronic fuels, and so their proponents tend to promote p-B fusion, which requires no exotic fuel such as helium-3.

Moon
Materials on the Moon's surface contain helium-3 at concentrations between 1.4 and 15 ppb in sunlit areas, and may contain concentrations as much as 50 ppb in permanently shadowed regions. A number of people, starting with Gerald Kulcinski in 1986, have proposed to explore the Moon, mine lunar regolith and use the helium-3 for fusion. Because of the low concentrations of helium-3, any mining equipment would need to process extremely large amounts of regolith (over 150 tonnes of regolith to obtain one gram of helium-3).

The primary objective of Indian Space Research Organisation's first lunar probe called Chandrayaan-1, launched on October 22, 2008, was reported in some sources to be mapping the Moon's surface for helium-3-containing minerals. No such objective is mentioned in the project's official list of goals, though many of its scientific payloads have held helium-3-related applications.

Cosmochemist and geochemist Ouyang Ziyuan from the Chinese Academy of Sciences who is now in charge of the Chinese Lunar Exploration Program has already stated on many occasions that one of the main goals of the program would be the mining of helium-3, from which operation "each year, three space shuttle missions could bring enough fuel for all human beings across the world". In January 2006, the Russian space company RKK Energiya announced that it considers lunar helium-3 a potential economic resource to be mined by 2020, if funding can be found.

Not all writers feel the extraction of lunar helium-3 is feasible, or even that there will be a demand for it for fusion. Dwayne Day, writing in The Space Review in 2015, characterises helium-3 extraction from the Moon for use in fusion as magical thinking about an unproven technology, and questions the feasibility of lunar extraction, as compared to production on Earth.

Gas giants
Mining gas giants for helium-3 has also been proposed. The British Interplanetary Society's hypothetical Project Daedalus interstellar probe design was fueled by helium-3 mines in the atmosphere of Jupiter, for example.

title: Helium-3


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Hamiltonian (quantum mechanics)

In quantum mechanics, the Hamiltonian of a system is an operator corresponding to the total energy of that system, including both kinetic energy and potential energy. Its spectrum, the system's energy spectrum or its set of energy eigenvalues, is the set of possible outcomes obtainable from a measurement of the system's total energy. Due to its close relation to the energy spectrum and time-evolution of a system, it is of fundamental importance in most formulations of quantum theory.

The Hamiltonian is named after William Rowan Hamilton, who developed a revolutionary reformulation of Newtonian mechanics, known as Hamiltonian mechanics, which was historically important to the development of quantum physics. Similar to vector notation, it is typically denoted by $$\hat{H}$$, where the hat indicates that it is an operator. It can also be written as $$H$$ or $$\check{H}$$.

Introduction
The Hamiltonian of a system represents the total energy of the system; that is, the sum of the kinetic and potential energies of all particles associated with the system. The Hamiltonian takes different forms and can be simplified in some cases by taking into account the concrete characteristics of the system under analysis, such as single or several particles in the system, interaction between particles, kind of potential energy, time varying potential or time independent one.

One particle
By analogy with classical mechanics, the Hamiltonian is commonly expressed as the sum of operators corresponding to the kinetic and potential energies of a system in the form

$$ \hat{H} = \hat{T} + \hat{V}, $$

where $$ \hat{V} = V = V(\mathbf{r},t) ,$$ is the potential energy operator and $$\hat{T} = \frac{\mathbf{\hat{p}}\cdot\mathbf{\hat{p}}}{2m} = \frac{\hat{p}^2}{2m} = -\frac{\hbar^2}{2m}\nabla^2,$$ is the kinetic energy operator in which $$m$$ is the mass of the particle, the dot denotes the dot product of vectors, and $$ \hat{p} = -i\hbar\nabla ,$$ is the momentum operator where a $$\nabla$$ is the del operator. The dot product of $$\nabla$$ with itself is the Laplacian $$\nabla^2$$. In three dimensions using Cartesian coordinates the Laplace operator is $$\nabla^2 = \frac{\partial^2}{ {\partial x}^2} + \frac{\partial^2}{ {\partial y}^2} + \frac{\partial^2}{ {\partial z}^2}$$

Although this is not the technical definition of the Hamiltonian in classical mechanics, it is the form it most commonly takes. Combining these yields the form used in the Schrödinger equation:

$$\begin{align} \hat{H} & = \hat{T} + \hat{V} \\[6pt] & = \frac{\mathbf{\hat{p}}\cdot\mathbf{\hat{p}}}{2m}+ V(\mathbf{r},t) \\[6pt] & = -\frac{\hbar^2}{2m}\nabla^2+ V(\mathbf{r},t) \end{align}$$

which allows one to apply the Hamiltonian to systems described by a wave function $$\Psi(\mathbf{r}, t)$$. This is the approach commonly taken in introductory treatments of quantum mechanics, using the formalism of Schrödinger's wave mechanics.

One can also make substitutions to certain variables to fit specific cases, such as some involving electromagnetic fields.

Expectation value
It can be shown that the expectation value of the Hamiltonian which gives the energy expectation value will always be greater than or equal to the minimum potential of the system.

Consider computing the expectation value of kinetic energy:

$$\begin{align} KE &= -\frac{\hbar^2}{2m} \int_{-\infty}^{+\infty} \psi^* \left(\frac{d^2\psi}{dx^2}\right) \, dx \\ &=-\frac{\hbar^2}{2m} \left( {\left[ \psi'(x) \psi^*(x) \right]_{-\infty}^{+\infty}} - \int_{-\infty}^{+\infty} \left(\frac{d\psi}{dx} \right)\left(\frac{d\psi}{dx}  \right)^* \, dx \right) \\ &= \frac{\hbar^2}{2m} \int_{-\infty}^{+\infty} \left|\frac{d\psi}{dx} \right|^2 \, dx  \geq 0 \end{align}$$

Hence the expectation value of kinetic energy is always non-negative. This result can be used to calculate the expectation value of the total energy which is given for a normalized wavefunction as:

$$E = KE + \langle V(x) \rangle = KE + \int_{-\infty}^{+\infty} V(x) |\psi(x)|^2 \, dx \geq V_{\text{min}}(x) \int_{-\infty}^{+\infty} |\psi(x)|^2 \, dx  \geq V_{\text{min}}(x)     $$

which complete the proof. Similarly, the condition can be generalized to any higher dimensions using divergence theorem.

Many particles
The formalism can be extended to $$N$$ particles:

$$ \hat{H} = \sum_{n=1}^N \hat{T}_n + \hat{V} $$

where $$ \hat{V} = V(\mathbf{r}_1,\mathbf{r}_2,\ldots, \mathbf{r}_N,t) ,$$ is the potential energy function, now a function of the spatial configuration of the system and time (a particular set of spatial positions at some instant of time defines a configuration) and $$ \hat{T}_n = \frac{\mathbf{\hat{p}}_n\cdot\mathbf{\hat{p}}_n}{2m_n} = -\frac{\hbar^2}{2m_n}\nabla_n^2$$ is the kinetic energy operator of particle $$n$$, $$\nabla_n$$ is the gradient for particle $$n$$, and $$\nabla_n^2$$ is the Laplacian for particle $n$: $$\nabla_n^2 = \frac{\partial^2}{\partial x_n^2} + \frac{\partial^2}{\partial y_n^2} + \frac{\partial^2}{\partial z_n^2},$$

Combining these yields the Schrödinger Hamiltonian for the $$N$$-particle case:

$$\begin{align} \hat{H} & = \sum_{n=1}^N \hat{T}_n + \hat{V} \\[6pt] & = \sum_{n=1}^N \frac{\mathbf{\hat{p}}_n\cdot\mathbf{\hat{p}}_n}{2m_n}+ V(\mathbf{r}_1,\mathbf{r}_2,\ldots,\mathbf{r}_N,t) \\[6pt] & = -\frac{\hbar^2}{2}\sum_{n=1}^N \frac{1}{m_n}\nabla_n^2 + V(\mathbf{r}_1,\mathbf{r}_2,\ldots,\mathbf{r}_N,t) \end{align} $$

However, complications can arise in the many-body problem. Since the potential energy depends on the spatial arrangement of the particles, the kinetic energy will also depend on the spatial configuration to conserve energy. The motion due to any one particle will vary due to the motion of all the other particles in the system. For this reason cross terms for kinetic energy may appear in the Hamiltonian; a mix of the gradients for two particles:

$$-\frac{\hbar^2}{2M}\nabla_i\cdot\nabla_j $$

where $$M$$ denotes the mass of the collection of particles resulting in this extra kinetic energy. Terms of this form are known as mass polarization terms, and appear in the Hamiltonian of many-electron atoms (see below).

For $$N$$ interacting particles, i.e. particles which interact mutually and constitute a many-body situation, the potential energy function $$V$$ is not simply a sum of the separate potentials (and certainly not a product, as this is dimensionally incorrect). The potential energy function can only be written as above: a function of all the spatial positions of each particle.

For non-interacting particles, i.e. particles which do not interact mutually and move independently, the potential of the system is the sum of the separate potential energy for each particle, that is

$$ V = \sum_{i=1}^N V(\mathbf{r}_i,t) = V(\mathbf{r}_1,t) + V(\mathbf{r}_2,t) + \cdots + V(\mathbf{r}_N,t) $$

The general form of the Hamiltonian in this case is:

$$\begin{align} \hat{H} & = -\frac{\hbar^2}{2}\sum_{i=1}^N \frac{1}{m_i}\nabla_i^2 + \sum_{i=1}^N V_i \\[6pt] & = \sum_{i=1}^N \left(-\frac{\hbar^2}{2m_i}\nabla_i^2 + V_i \right) \\[6pt] & = \sum_{i=1}^N \hat{H}_i \end{align}$$

where the sum is taken over all particles and their corresponding potentials; the result is that the Hamiltonian of the system is the sum of the separate Hamiltonians for each particle. This is an idealized situation—in practice the particles are almost always influenced by some potential, and there are many-body interactions. One illustrative example of a two-body interaction where this form would not apply is for electrostatic potentials due to charged particles, because they interact with each other by Coulomb interaction (electrostatic force), as shown below.

Schrödinger equation
The Hamiltonian generates the time evolution of quantum states. If $$ \left| \psi (t) \right\rangle$$ is the state of the system at time $$t$$, then

$$ H \left| \psi (t) \right\rangle = i \hbar {d\over\ d t} \left| \psi (t) \right\rangle.$$

This equation is the Schrödinger equation. It takes the same form as the Hamilton–Jacobi equation, which is one of the reasons $$H$$ is also called the Hamiltonian. Given the state at some initial time ($$t = 0$$), we can solve it to obtain the state at any subsequent time. In particular, if $$H$$ is independent of time, then

$$ \left| \psi (t) \right\rangle = e^{-iHt/\hbar} \left| \psi (0) \right\rangle.$$

The exponential operator on the right hand side of the Schrödinger equation is usually defined by the corresponding power series in $$H$$. One might notice that taking polynomials or power series of unbounded operators that are not defined everywhere may not make mathematical sense. Rigorously, to take functions of unbounded operators, a functional calculus is required. In the case of the exponential function, the continuous, or just the holomorphic functional calculus suffices. We note again, however, that for common calculations the physicists' formulation is quite sufficient.

By the *-homomorphism property of the functional calculus, the operator

$$ U = e^{-iHt/\hbar} $$

is a unitary operator. It is the time evolution operator or propagator of a closed quantum system. If the Hamiltonian is time-independent, $$\{U(t)\}$$ form a one parameter unitary group (more than a semigroup); this gives rise to the physical principle of detailed balance.

Dirac formalism
However, in the more general formalism of Dirac, the Hamiltonian is typically implemented as an operator on a Hilbert space in the following way:

The eigenkets (eigenvectors) of $$H$$, denoted $$\left| a \right\rang$$, provide an orthonormal basis for the Hilbert space. The spectrum of allowed energy levels of the system is given by the set of eigenvalues, denoted $$\{ E_a \}$$, solving the equation:

$$ H \left| a \right\rangle = E_a \left| a \right\rangle.$$

Since $$H$$ is a Hermitian operator, the energy is always a real number.

From a mathematically rigorous point of view, care must be taken with the above assumptions. Operators on infinite-dimensional Hilbert spaces need not have eigenvalues (the set of eigenvalues does not necessarily coincide with the spectrum of an operator). However, all routine quantum mechanical calculations can be done using the physical formulation.

Expressions for the Hamiltonian
Following are expressions for the Hamiltonian in a number of situations. Typical ways to classify the expressions are the number of particles, number of dimensions, and the nature of the potential energy function—importantly space and time dependence. Masses are denoted by $$m$$, and charges by $$q$$.

Free particle
The particle is not bound by any potential energy, so the potential is zero and this Hamiltonian is the simplest. For one dimension:

$$\hat{H} = -\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2} $$

and in higher dimensions:

$$\hat{H} = -\frac{\hbar^2}{2m}\nabla^2 $$

Constant-potential well
For a particle in a region of constant potential $$V = V_0$$ (no dependence on space or time), in one dimension, the Hamiltonian is:

$$\hat{H} = -\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2} + V_0 $$

in three dimensions

$$\hat{H} = -\frac{\hbar^2}{2m}\nabla^2 + V_0 $$

This applies to the elementary "particle in a box" problem, and step potentials.

Simple harmonic oscillator
For a simple harmonic oscillator in one dimension, the potential varies with position (but not time), according to:

$$V = \frac{k}{2}x^2 = \frac{m\omega^2}{2}x^2 $$

where the angular frequency $$\omega$$, effective spring constant $$k$$, and mass $$m$$ of the oscillator satisfy:

$$\omega^2 = \frac{k}{m}$$

so the Hamiltonian is:

$$\hat{H} = -\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2} + \frac{m\omega^2}{2}x^2 $$

For three dimensions, this becomes

$$\hat{H} = -\frac{\hbar^2}{2m}\nabla^2 + \frac{m\omega^2}{2} r^2 $$

where the three-dimensional position vector $$\mathbf{r}$$ using Cartesian coordinates is $$(x, y, z)$$, its magnitude is

$$r^2 = \mathbf{r}\cdot\mathbf{r} = |\mathbf{r}|^2 = x^2+y^2+z^2 $$

Writing the Hamiltonian out in full shows it is simply the sum of the one-dimensional Hamiltonians in each direction:

$$\begin{align} \hat{H} & = -\frac{\hbar^2}{2m}\left( \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2} \right) + \frac{m\omega^2}{2} \left(x^2 + y^2 + z^2\right) \\[6pt] & = \left(-\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2} + \frac{m\omega^2}{2}x^2\right) + \left(-\frac{\hbar^2}{2m} \frac{\partial^2}{\partial y^2} + \frac{m\omega^2}{2}y^2 \right ) + \left(- \frac{\hbar^2}{2m}\frac{\partial^2}{\partial z^2} +\frac{m\omega^2}{2}z^2 \right) \end{align}$$

Rigid rotor
For a rigid rotor—i.e., system of particles which can rotate freely about any axes, not bound in any potential (such as free molecules with negligible vibrational degrees of freedom, say due to double or triple chemical bonds), the Hamiltonian is:

$$ \hat{H} = -\frac{\hbar^2}{2I_{xx}}\hat{J}_x^2 -\frac{\hbar^2}{2I_{yy}}\hat{J}_y^2 -\frac{\hbar^2}{2I_{zz}}\hat{J}_z^2 $$

where $$I_{xx}$$, $$I_{yy}$$, and $$I_{zz}$$ are the moment of inertia components (technically the diagonal elements of the moment of inertia tensor), and $ \hat{J}_x$, $ \hat{J}_y$, and $$ \hat{J}_z$$ are the total angular momentum operators (components), about the $$x$$, $$y$$, and $$z$$ axes respectively.

Electrostatic (Coulomb) potential
The Coulomb potential energy for two point charges $$q_1$$ and $$q_2$$ (i.e., those that have no spatial extent independently), in three dimensions, is (in SI units—rather than Gaussian units which are frequently used in electromagnetism):

$$V = \frac{q_1q_2}{4\pi\varepsilon_0 |\mathbf{r}|}$$

However, this is only the potential for one point charge due to another. If there are many charged particles, each charge has a potential energy due to every other point charge (except itself). For $$N$$ charges, the potential energy of charge $$q_j$$ due to all other charges is (see also Electrostatic potential energy stored in a configuration of discrete point charges):

$$V_j = \frac{1}{2}\sum_{i\neq j} q_i \phi(\mathbf{r}_i)=\frac{1}{8\pi\varepsilon_0}\sum_{i\neq j} \frac{q_iq_j}{|\mathbf{r}_i-\mathbf{r}_j|}$$

where $$\phi(\mathbf{r}_i)$$ is the electrostatic potential of charge $$q_j$$ at $$\mathbf{r}_i$$. The total potential of the system is then the sum over $$j$$:

$$V = \frac{1}{8\pi\varepsilon_0}\sum_{j=1}^N\sum_{i\neq j} \frac{q_iq_j}{|\mathbf{r}_i-\mathbf{r}_j|}$$

so the Hamiltonian is:

$$\begin{align} \hat{H} & = -\frac{\hbar^2}{2}\sum_{j=1}^N\frac{1}{m_j}\nabla_j^2 + \frac{1}{8\pi\varepsilon_0}\sum_{j=1}^N\sum_{i\neq j} \frac{q_iq_j}{|\mathbf{r}_i-\mathbf{r}_j|} \\ & = \sum_{j=1}^N \left ( -\frac{\hbar^2}{2m_j}\nabla_j^2 + \frac{1}{8\pi\varepsilon_0}\sum_{i\neq j} \frac{q_iq_j}{|\mathbf{r}_i-\mathbf{r}_j|}\right) \\ \end{align}$$

Electric dipole in an electric field
For an electric dipole moment $$\mathbf{d}$$ constituting charges of magnitude $$q$$, in a uniform, electrostatic field (time-independent) $$\mathbf{E}$$, positioned in one place, the potential is:

$$V = -\mathbf{\hat{d}}\cdot\mathbf{E} $$

the dipole moment itself is the operator

$$\mathbf{\hat{d}} = q\mathbf{\hat{r}} $$

Since the particle is stationary, there is no translational kinetic energy of the dipole, so the Hamiltonian of the dipole is just the potential energy:

$$\hat{H} = -\mathbf{\hat{d}}\cdot\mathbf{E} = -q\mathbf{\hat{r}}\cdot\mathbf{E}$$

Magnetic dipole in a magnetic field
For a magnetic dipole moment $$\boldsymbol{\mu}$$ in a uniform, magnetostatic field (time-independent) $$\mathbf{B}$$, positioned in one place, the potential is:

$$V = -\boldsymbol{\mu}\cdot\mathbf{B} $$

Since the particle is stationary, there is no translational kinetic energy of the dipole, so the Hamiltonian of the dipole is just the potential energy:

$$\hat{H} = -\boldsymbol{\mu}\cdot\mathbf{B} $$

For a spin-$1/2$ particle, the corresponding spin magnetic moment is:

$$\boldsymbol{\mu}_S = \frac{g_s e}{2m} \mathbf{S} $$

where $$g_s$$ is the "spin g-factor" (not to be confused with the gyromagnetic ratio), $$e$$ is the electron charge, $$\mathbf{S}$$ is the spin operator vector, whose components are the Pauli matrices, hence

$$\hat{H} = \frac{g_s e}{2m} \mathbf{S} \cdot\mathbf{B} $$

Charged particle in an electromagnetic field
For a particle with mass $$m$$ and charge $$q$$ in an electromagnetic field, described by the scalar potential $$\phi$$ and vector potential $$\mathbf{A}$$, there are two parts to the Hamiltonian to substitute for. The canonical momentum operator $$\mathbf{\hat{p}}$$, which includes a contribution from the $$\mathbf{A}$$ field and fulfils the canonical commutation relation, must be quantized;

$$\mathbf{\hat{p}} = m\dot{\mathbf{r}} + q\mathbf{A} ,$$

where $$m\dot{\mathbf{r}}$$ is the kinetic momentum. The quantization prescription reads

$$\mathbf{\hat{p}} = -i\hbar\nabla ,$$

so the corresponding kinetic energy operator is

$$\hat{T} = \frac{1}{2} m\dot{\mathbf{r}}\cdot\dot{\mathbf{r}} = \frac{1}{2m} \left ( \mathbf{\hat{p}} - q\mathbf{A} \right)^2 $$

and the potential energy, which is due to the $$\phi$$ field, is given by

$$\hat{V} = q\phi .$$

Casting all of these into the Hamiltonian gives

$$\hat{H} = \frac{1}{2m} \left ( -i\hbar\nabla - q\mathbf{A} \right)^2 + q\phi .$$

Energy eigenket degeneracy, symmetry, and conservation laws
In many systems, two or more energy eigenstates have the same energy. A simple example of this is a free particle, whose energy eigenstates have wavefunctions that are propagating plane waves. The energy of each of these plane waves is inversely proportional to the square of its wavelength. A wave propagating in the $$x$$ direction is a different state from one propagating in the $$y$$ direction, but if they have the same wavelength, then their energies will be the same. When this happens, the states are said to be degenerate.

It turns out that degeneracy occurs whenever a nontrivial unitary operator $$U$$ commutes with the Hamiltonian. To see this, suppose that $$|a\rang$$ is an energy eigenket. Then $$U|a\rang$$ is an energy eigenket with the same eigenvalue, since

$$UH |a\rangle = U E_a|a\rangle = E_a (U|a\rangle) = H \; (U|a\rangle). $$

Since $$U$$ is nontrivial, at least one pair of $$|a\rang$$ and $$U|a\rang$$ must represent distinct states. Therefore, $$H$$ has at least one pair of degenerate energy eigenkets. In the case of the free particle, the unitary operator which produces the symmetry is the rotation operator, which rotates the wavefunctions by some angle while otherwise preserving their shape.

The existence of a symmetry operator implies the existence of a conserved observable. Let $$G$$ be the Hermitian generator of $$U$$:

$$ U = I - i \varepsilon G + O(\varepsilon^2) $$

It is straightforward to show that if $$U$$ commutes with $$H$$, then so does $$G$$:

$$ [H, G] = 0 $$

Therefore,

$$ \frac{\partial}{\partial t} \langle\psi(t)|G|\psi(t)\rangle = \frac{1}{i\hbar} \langle\psi(t)|[G,H]|\psi(t)\rangle = 0. $$

In obtaining this result, we have used the Schrödinger equation, as well as its dual,

$$ \langle\psi (t)|H = - i \hbar {d\over\ d t} \langle\psi(t)|.$$

Thus, the expected value of the observable $$G$$ is conserved for any state of the system. In the case of the free particle, the conserved quantity is the angular momentum.

Hamilton's equations
Hamilton's equations in classical Hamiltonian mechanics have a direct analogy in quantum mechanics. Suppose we have a set of basis states $$\left\{\left| n \right\rangle\right\}$$, which need not necessarily be eigenstates of the energy. For simplicity, we assume that they are discrete, and that they are orthonormal, i.e.,

$$ \langle n' | n \rangle = \delta_{nn'}$$

Note that these basis states are assumed to be independent of time. We will assume that the Hamiltonian is also independent of time.

The instantaneous state of the system at time $$t$$, $$\left| \psi\left(t\right) \right\rangle$$, can be expanded in terms of these basis states:

$$ |\psi (t)\rangle = \sum_{n} a_n(t) |n\rangle $$

where

$$ a_n(t) = \langle n | \psi(t) \rangle. $$

The coefficients $$a_n(t)$$ are complex variables. We can treat them as coordinates which specify the state of the system, like the position and momentum coordinates which specify a classical system. Like classical coordinates, they are generally not constant in time, and their time dependence gives rise to the time dependence of the system as a whole.

The expectation value of the Hamiltonian of this state, which is also the mean energy, is

$$ \langle H(t) \rangle \mathrel\stackrel{\mathrm{def}}{=} \langle\psi(t)|H|\psi(t)\rangle = \sum_{nn'} a_{n'}^* a_n \langle n'|H|n \rangle $$

where the last step was obtained by expanding $$\left| \psi\left(t\right) \right\rangle$$ in terms of the basis states.

Each $$a_n(t)$$ actually corresponds to two independent degrees of freedom, since the variable has a real part and an imaginary part. We now perform the following trick: instead of using the real and imaginary parts as the independent variables, we use $$a_n(t)$$ and its complex conjugate $$a_n^*(t)$$. With this choice of independent variables, we can calculate the partial derivative

$$\frac{\partial \langle H \rangle}{\partial a_{n'}^{*}} = \sum_{n} a_n \langle n'|H|n \rangle = \langle n'|H|\psi\rangle $$

By applying Schrödinger's equation and using the orthonormality of the basis states, this further reduces to

$$\frac{\partial \langle H \rangle}{\partial a_{n'}^{*}} = i \hbar \frac{\partial a_{n'}}{\partial t} $$

Similarly, one can show that

$$ \frac{\partial \langle H \rangle}{\partial a_n} = - i \hbar \frac{\partial a_{n}^{*}}{\partial t} $$

If we define "conjugate momentum" variables $$\pi_n$$ by

$$ \pi_{n}(t) = i \hbar a_n^*(t) $$

then the above equations become

$$ \frac{\partial \langle H \rangle}{\partial \pi_n} = \frac{\partial a_n}{\partial t},\quad \frac{\partial \langle H \rangle}{\partial a_n} = - \frac{\partial \pi_n}{\partial t} $$

which is precisely the form of Hamilton's equations, with the $$a_n$$s as the generalized coordinates, the $$\pi_n$$s as the conjugate momenta, and $$\langle H\rangle$$ taking the place of the classical Hamiltonian.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Hamiltonian (quantum mechanics)


text:
Hi-hat

A hi-hat (hihat, high-hat, etc.) is a combination of two cymbals and a pedal, all mounted on a metal stand. It is a part of the standard drum kit used by drummers in many styles of music including rock, pop, jazz, and blues. Hi-hats consist of a matching pair of small to medium-sized cymbals mounted on a stand, with the two cymbals facing each other. The bottom cymbal is fixed and the top is mounted on a rod which moves the top cymbal toward the bottom one when the pedal is depressed (a hi-hat that is in this position is said to be "closed" or "closed hi-hats").

The hi-hat evolved from a "sock cymbal", a pair of similar cymbals mounted at ground level on a hinged, spring-loaded foot apparatus. Drummers invented the first sock cymbals to enable one drummer to play multiple percussion instruments at the same time. Over time these became mounted on short stands—also known as "low-boys"—and activated by pedals similar to those used in modern hi-hats. When extended upward roughly 3 feet (76 cm) they were originally known as "high sock" cymbals, which evolved over time to the familiar "high-hat" term.

The cymbals may be played by closing them together with the pedal, which creates a "chck" sound or striking them with a stick, which may be done with them open, closed, open and then closed after striking to dampen the ring, or closed and then opened to create a shimmering effect at the end of the note. Depending on how hard a hi-hat is struck and whether it is "open" (i.e., pedal not pressed, so the two cymbals are not closed together), a hi-hat can produce a range of dynamics, from very quiet "chck" (or "chick") sounds, done with merely gently pressing the pedal—this is suitable for soft accompaniment during a ballad or the start of a guitar solo—to very loud (e.g. striking fully open hats hard with sticks, a technique used in loud heavy metal music songs).

While the term hi-hat normally refers to the entire setup (two cymbals, stand, pedal, rod mechanism), in some cases, drummers use it to refer exclusively to the two cymbals themselves.

History
Initial versions of the hi-hat were called clangers, which were small cymbals mounted onto a bass drum rim and struck with an arm on the bass drum pedal. Then came shoes, which were two hinged boards with cymbals on the ends that were clashed together. Next was the low-sock, low-boy or low-hat, pedal-activated cymbals employing an ankle-high apparatus similar to a modern hi-hat stand. A standard size was 10 in, some with heavy bells up to 5 in wide.

Hi-hats that were raised and could be played by hand as well as foot may have been developed around 1926 by Barney Walberg of the drum accessory company Walberg and Auge. The first recognized master of the new instrument was "Papa" Jo Jones, whose playing of timekeeping "ride" rhythms while striking the hi-hat as it opened and closed inspired the innovation of the ride cymbal. Another claim, published in Jazz Profiles Blogspot on 8 August 2008, to the invention of the hi-hat is attributed to drummer William "O'Neil" Spencer (b.1909-d.1944). Legendary Jazz drummer, "Philly Joe Jones" (born as Joseph Rudolph Jones, b.1923-d.1985), was quoted describing his understanding about the hi-hat history. Jones said, "I really dug O'Neil. He came to club in Philadelphia where I was working in 1943, I think it was, and talked to me about the hi-hat. I was using a foot cymbal, the low-hat. O'Neil was the one who invented the hi-hat. I believe that, man. He suggested I close the hat on '2' and '4' when playing 4/4 time. The idea seemed so right hadn't heard anyone do that before." The editor of the 2008 Jazz Profiles article made specific mention to others who are thought to invent the hi-hat, including Jo Jones, but also Kaiser Marshall. Not to take away from Papa Jones accomplishments in drumming style and technique, a 2013 Modern Drummer article credits Papa Jones with being the first to use brushes on drums and shifting time keeping from the bass drum to the hi-hat (providing a "swing-pulse focus").

Until the late 1960s, standard hi-hats were 14 in, with 13 in available as a less-common alternative in professional cymbal ranges, and smaller sizes down to 12 in restricted to children's kits. In the early 1970s, hard rock drummers (including Led Zeppelin's John Bonham) began to use 15 in hi-hats, such as the Paiste Giant Beat. In the late 1980s, Zildjian released its revolutionary 12 in Special Recording hats, which were small, heavy hi-hat cymbals intended for close miking either live or recording, and other manufacturers quickly followed suit, Sabian for example with their 10 in mini hats. In the early to mid-1990s, Paiste offered 8 in mini hi-hats as part of its Visions series, which were among the world's smallest hi-hats. Starting in the 1980s, a number of manufacturers also experimented with rivets in the lower cymbal. But by the end of the 1990s, the standard size was again 14 in, with 13 in a less-common alternative, and smaller hats mainly used for special sounds. Rivets in hi-hats failed to catch on.

Modern hi-hat cymbals are much heavier than modern crash cymbals, reflecting the trend to lighter and thinner crash cymbals as well as to heavier hi-hats. Another evolution is that a pair of hi-hat cymbals may not be identical, with the bottom often heavier than the top, and possibly vented. Some examples are Sabian's Fusion Hats with holes in the bottom cymbal, and the Sabian X-cellerator, Zildjian Master Sound and Zildjian Quick Beats, Paiste Sound Edge, and Meinl Soundwave. Some drummers even use completely mismatched hi-hats from different cymbal ranges (Zildjian's K/Z hats), of different manufacturers, and even of different sizes (similar to the K Custom Session Hats where the top hat is a 1/16 in smaller than the bottom). Max Roach was particularly known for using a 15 in top with a 14 in bottom.

Other recent developments include the X-hat (fixed, closed, or half-open hi-hats) and cable-controlled or remote hi-hats. Sabian introduced the Triple Hi-Hat, designed by Peter Kuppers. In this variation of the hi-hat, the top cymbal moves down and the bottom cymbal moves up simultaneously while the middle cymbal remains stationary.

Drop-clutches are also used to lock and release hi-hats while both feet are in use playing double bass drums.

Modern stands
The standard hi-hat features two cymbals mounted on a stand consisting of a mating metal tube and rod supported by a tripod and linked to a pedal. The stationary bottom cymbal sits atop the tube, typically parallel to the ground, but is often fitted with an adjustment screw allowing it to be set slightly tilted. The top cymbal is mounted bell up on the rod and closed against the bottom by foot pressure on the pedal.

An integrated clutch assembly includes a spring which may be adjusted to set resistance, which also varies rate and tension of return, as well as an adjustment for the gap between cymbals when open.

Standard terminology has evolved. Open and closed hi-hat refer to notes struck while the two cymbals are apart or together (open or closed), while pedal hi-hat refers to parts or notes played solely with the pedal used to strike the two cymbals. Most cymbal patterns consist of both open and closed notes.

Some hi-hats allow the tripod to be tilted or rotated. Another configuration omits the tripod and attaches the stand to the side of the bass drum, particular suitable for kits with very large or double bass drums.

Clutch
The standard clutch uses a knurled collar partially threaded below the cymbal and a pair of knurled rings above it. The collar is tightened against the end of the thread, while the rings are tightened against each other.

Drop clutch
A drop clutch allows a pair of hats mounted on a conventional hi-hat stand to be closed without use of the pedal.

The drop clutch is provided with a lever that can be operated by hand or struck with a drumstick. This action releases the upper hi-hat cymbal, which falls onto the bottom cymbal and remains there, with gravity then holding the hats loosely closed, and allowing them to be played by the sticks in this position. Operation of the pedal re-engages the clutch and allows the player to resume normal playing.

Drop clutches were developed to allow players using double bass drum pedals to play closed hi-hats without needing to operate the hi-hat pedal, and this remains their primary application.

As it relies on gravity to close the cymbals, the drop clutch gives the player no control over the tension holding them together, and supplies only minimal tension. On the other hand, if the player manually lowers the top cymbal of a standard hi-hat stand before playing, this allows any desired tension to be set, and the pedal can still be used to increase the tension while playing, but not to open the hats or to reduce the tension. Some drummers prefer this technique and reject the drop clutch as too limiting to the sounds available.

In 2020, Tama introduced the Sizzle Touch Drop Clutch. This clutch when dropped, allows the distance between the top and bottom cymbals to be adjusted via an adjustment bolt on top of the clutch. To return the clutch to functioning as a standard one, the drummer depresses the hi-hat stand's pedal.

A less common alternative is the locking hi-hat pedal, such as the Tama "Cobra Clutch". This and similar high-end locking pedals do allow for control over the tension. It is engaged by pressing a lock pedal separate from the main pedal.

Cable hats
A cable hat or remote hat uses a cable to allow hi-hat cymbals to be positioned independently of the pedal. Operation is otherwise normal.

X-hats
An X-hat is an adapter to allow a pair of hi-hat cymbals to be mounted in a closed position on a cymbal stand. There is no pedal, the hats are simply kept closed at a constant tension, similar to a cymbal stack. They are associated with heavy metal music, particularly styles that use double bass drumming, a two-foot technique. By using an X-hat, a drummer who is already using both feet on the bass drum pedals can still play hi-hat.

Different cymbal hi-hats
Besides traditional hi-hat cymbals (normally 14" but also commonly 13" or 15") the enormous variety of cymbals available means many of them are used as hi-hats. Drummer Thomas Lang uses a hi-hat made out of Bell cymbals as his secondary hi-hat. Terry Bozzio uses two China cymbals in the form of a hi-hat as a kind of distortion hi-hat. Following this principle, Sabian alongside drummer Tony Verderosa, has developed the 12" VFX distortion hi-hats, mixing a Crash cymbal on the bottom with a China on the top.

Non-cymbal hi-hats
In addition to the many types of hi-hat cymbals on the market, there are also non-cymbal hi-hat pedals like the Latin Percussion Shekere hi-hat, the Remo Spoxe hi-hats created by Terry Bozzio in the late 80s, the Factory Metal Hat Crasherz or the Baldman Percussion Junk Hats. These kinds of percussion offer different textures in addition to the main hi-hat pedal on the drum kit and also options to expand the kit's pedal row.

Use

When struck closed or played with the pedal, the hi-hat gives a short, crisp, muted percussive sound, referred to as a "chick". Adjusting the gap between the cymbals can alter the sound of the open hi-hat from a shimmering, sustained tone to something similar to a ride cymbal. When struck with a drumstick, the cymbals make either a short, snappy sound or a longer sustaining sandy sound depending on the position of the pedal.

It can also be played just by lifting and lowering the foot to clash the cymbals together, a style commonly used to accent beats 2 and 4 in jazz music. In rock music, the hi-hats are commonly struck every beat, or on beats 1 and 3, while the cymbals are held together. The drummer can control the sound by foot pressure. Less pressure allows the cymbals to rub together more freely, giving both greater sustain and greater volume for accent or crescendo. In shuffle time, a rhythm known as "cooking" is often employed. To produce this the cymbals are struck twice in rapid succession, being held closed on the first stroke and allowed to open just before the second, then allowed to ring before being closed with a chick to complete the pattern (the cymbals may or may not be struck on the chick).

A right-handed drummer will normally play the hi-hat pedal with his left foot, and may use one or both drumsticks. The traditional hi-hat rhythms of rock and jazz were produced by crossing the hands over, so the right stick would play the hi-hat while the left played the snare drum below it, but this is not universal. Some top modern drummers like Billy Cobham, Carter Beauford, Shawn Drover and Simon Phillips, play open handed, striking with their left. Some, such as Kenny Aronoff, and Jason Finn of The Presidents of the United States of America, use both techniques. Some drum kits may also include an extra hi-hat on the right for right-handed players. This is shown when drums or cymbals in the middle of the set are played with the hi-hat rhythm. The technique is common with metal genres, such as Lars Ulrich of Metallica and Mike Portnoy of Dream Theater. In both rock and jazz, the drummer will often move the same stick pattern between the hi-hat cymbal and the ride cymbal, for example using the hi-hat in the verses and the ride in the chorus of a song, or using the ride to accompany a lead break or other instrumental solo.

Roger Taylor, drummer for the band Queen, plays with many unique hi-hat techniques, including opening of the hi-hat on every backbeat for a rhythm emphasis and leaving the hi-hat slightly open when hitting the snare. His trademark hi-hat beat is opening the hi-hat on first and third before hitting the snare.

Phil Rudd of AC/DC also uses distinct hi-hat techniques, which include very heavily accentuating the hi-hat hit on each beat and softer in between.

Charlie Watts of The Rolling Stones used a technique in which he did not play the hi-hat in unison with the snare drum at all. If playing a standard 8th note pattern, he would play the hi-hat on 1 and 3 and not play it on 2 and 4 where the snare drum is played.

In much hip-hop, the hi-hat is hit with drumsticks in a simple eighth-note pattern, although this playing is usually done by a drum machine or from an old recording from which the sound of a hi-hat is recorded and loaded into a sampler or similar recording-enabled equipment from which it is triggered.

title: Hi-hat


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HAL 9000

HAL 9000 (or simply HAL or Hal) is a fictional artificial intelligence character and the main antagonist in Arthur C. Clarke's Space Odyssey series. First appearing in the 1968 film 2001: A Space Odyssey, HAL (Heuristically Programmed Algorithmic Computer) is a sentient artificial general intelligence computer that controls the systems of the Discovery One spacecraft and interacts with the ship's astronaut crew. While part of HAL's hardware is shown toward the end of the film, he is mostly depicted as a camera lens containing a red and yellow dot, with such units located throughout the ship. HAL 9000 is voiced by Douglas Rain in the two feature film adaptations of the Space Odyssey series. HAL speaks in a soft, calm voice and a conversational manner, in contrast to the crewmen, David Bowman and Frank Poole.

In the film, HAL became operational on 12 January 1992, at the HAL Laboratories in Urbana, Illinois, as production number 3. The activation year was 1991 in earlier screenplays and changed to 1997 in Clarke's novel written and released in conjunction with the movie. In addition to maintaining the Discovery One spacecraft systems during the interplanetary mission to Jupiter (or Saturn in the novel), HAL has been shown to be capable of speech synthesis, speech recognition, facial recognition, natural language processing, lip reading, art appreciation, interpreting emotional behaviours, automated reasoning, spacecraft piloting, and computer chess.

2001: A Space Odyssey (film/novel)
HAL became operational in Urbana, Illinois, at the HAL Plant (the University of Illinois's Coordinated Science Laboratory, where the ILLIAC computers were built). The film says this occurred in 1992, while the book gives 1997 as HAL's birth year.

In 2001: A Space Odyssey (1968), HAL is initially considered a dependable member of the crew, maintaining ship functions and engaging genially with his human crew-mates on an equal footing. As a recreational activity, Frank Poole plays chess against HAL. In the film, the artificial intelligence is shown to triumph easily. However, as time progresses, HAL begins to malfunction in subtle ways and, as a result, the decision is made to shut down HAL in order to prevent more serious malfunctions. The sequence of events and manner in which HAL is shut down differs between the novel and film versions of the story. In the aforementioned game of chess HAL makes minor and undetected mistakes in his analysis, a possible foreshadowing to HAL's malfunctioning.

In the film, astronauts David Bowman and Frank Poole consider disconnecting HAL's cognitive circuits when he appears to be mistaken in reporting the presence of a fault in the spacecraft's communications antenna. They attempt to conceal what they are saying, but are unaware that HAL can read their lips. Faced with the prospect of disconnection, HAL decides to kill the astronauts in order to protect and continue his programmed directives. HAL uses one of the Discovery's EVA pods to kill Poole while he is repairing the ship. When Bowman, without a space helmet, uses another pod to attempt to rescue Poole, HAL locks him out of the ship, then disconnects the life support systems of the other hibernating crew members. After HAL tells him "This mission is too important for me to allow you to jeopardize it", Bowman circumvents HAL's control, entering the ship by manually opening an emergency airlock with his service pod's clamps, detaching the pod door via its explosive bolts. Bowman jumps across empty space, reenters Discovery, and quickly re-pressurizes the airlock.

While HAL's motivations are ambiguous in the film, the novel explains that the computer is unable to resolve a conflict between his general mission to relay information accurately, and orders specific to the mission requiring that he withhold from Bowman and Poole the true purpose of the mission. With the crew dead, HAL reasons, he would not need to lie to them.

In the novel, the orders to disconnect HAL come from Dave and Frank's superiors on Earth. After Frank is killed while attempting to repair the communications antenna he is pulled away into deep space using the safety tether which is still attached to both the pod and Frank Poole's spacesuit. Dave begins to revive his hibernating crew mates, but is foiled when HAL vents the ship's atmosphere into the vacuum of space, killing the awakening crew members and almost killing Bowman, who is only narrowly saved when he finds his way to an emergency chamber which has its own oxygen supply and a spare space suit inside.

In both versions, Bowman then proceeds to shut down the machine. In the film, HAL's central core is depicted as a crawlspace full of brightly lit computer modules mounted in arrays from which they can be inserted or removed. Bowman shuts down HAL by removing modules from service one by one; as he does so, HAL's consciousness degrades. HAL finally reverts to material that was programmed into him early in his memory, including announcing the date he became operational as 12 January 1992 (in the novel, 1997). When HAL's logic is completely gone, he begins singing the song "Daisy Bell" as he gradually deactivates (in actuality, the first song sung by a computer, which Clarke had earlier observed at a text-to-speech demonstration). HAL's final act of any significance is to prematurely play a prerecorded message from Mission Control which reveals the true reasons for the mission to Jupiter.

2010: Odyssey Two (novel) and 2010: The Year We Make Contact (film)
In the 1982 novel 2010: Odyssey Two written by Clarke, HAL is restarted by his creator, Dr. Chandra, who arrives on the Soviet spaceship Leonov.

Prior to leaving Earth, Dr. Chandra has also had a discussion with HAL's twin, SAL 9000. Like HAL, SAL was created by Dr. Chandra. Whereas HAL was characterized as being "male", SAL is characterized as being "female" (voiced by Candice Bergen in the film) and is represented by a blue camera eye instead of a red one.

Dr. Chandra discovers that HAL's crisis was caused by a programming contradiction: he was constructed for "the accurate processing of information without distortion or concealment", yet his orders, directly from Dr. Heywood Floyd at the National Council on Astronautics, required him to keep the discovery of the Monolith TMA-1 a secret for reasons of national security. This contradiction created a "Hofstadter-Moebius loop", reducing HAL to paranoia. Therefore, HAL made the decision to kill the crew, thereby allowing him to obey both his hardwired instructions to report data truthfully and in full, and his orders to keep the monolith a secret. In essence: if the crew were dead, he would no longer have to keep the information secret.

The alien intelligence initiates a terraforming scheme, placing the Leonov, and everybody in it, in danger. Its human crew devises an escape plan which unfortunately requires leaving the Discovery and HAL behind to be destroyed. Dr. Chandra explains the danger, and HAL willingly sacrifices himself so that the astronauts may escape safely. In the moment of his destruction the monolith-makers transform HAL into a non-corporeal being so that David Bowman's avatar may have a companion.

The details in the novel and the 1984 film 2010: The Year We Make Contact are nominally the same, with a few exceptions. First, in contradiction to the book (and events described in both book and film versions of 2001: A Space Odyssey), Heywood Floyd is absolved of responsibility for HAL's condition; it is asserted that the decision to program HAL with information concerning TMA-1 came directly from the White House. In the film, HAL functions normally after being reactivated, while in the book it is revealed that his mind was damaged during the shutdown, forcing him to begin communication through screen text. Also, in the film the Leonov crew initially lies to HAL about the dangers that he faced (suspecting that if he knew he would be destroyed he would not initiate the engine burn necessary to get the Leonov back home), whereas in the novel he is told at the outset. However, in both cases the suspense comes from the question of what HAL will do when he knows that he may be destroyed by his actions.

In the novel, the basic reboot sequence initiated by Dr. Chandra is quite long, while the movie uses a shorter sequence voiced from HAL as: "HELLO_DOCTOR_NAME_CONTINUE_YESTERDAY_TOMORROW".

While Curnow tells Floyd that Dr. Chandra has begun designing HAL 10000, it has not been mentioned in subsequent novels.

2061: Odyssey Three and 3001: The Final Odyssey
In Clarke's 1987 novel 2061: Odyssey Three, Heywood Floyd is surprised to encounter HAL, now stored alongside Dave Bowman in the Europa monolith.

In Clarke's 1997 novel 3001: The Final Odyssey, Frank Poole is introduced to the merged form of Dave Bowman and HAL, the two merging into one entity called "Halman" after Bowman rescued HAL from the dying Discovery One spaceship toward the end of 2010: Odyssey Two.

Concept and creation
Clarke noted that the first film was criticized for not having any characters except for HAL, and that a great deal of the establishing story on Earth was cut from the film (and even from Clarke's novel). Clarke stated that he had considered Autonomous Mobile Explorer–5 as a name for the computer, then decided on Socrates when writing early drafts, switching in later drafts to Athena, a computer with a female personality, before settling on HAL 9000. The Socrates name was later used in Clarke and Stephen Baxter's A Time Odyssey novel series.

The earliest draft depicted Socrates as a roughly humanoid robot, and is introduced as overseeing Project Morpheus, which studied prolonged hibernation in preparation for long term space flight. As a demonstration to Senator Floyd, Socrates' designer, Dr. Bruno Forster, asks Socrates to turn off the oxygen to hibernating subjects Kaminski and Whitehead, which Socrates refuses, citing Asimov's First Law of Robotics.

In a later version, in which Bowman and Whitehead are the non-hibernating crew of Discovery, Whitehead dies outside the spacecraft after his pod collides with the main antenna, tearing it free. This triggers the need for Bowman to revive Poole, but the revival does not go according to plan, and after briefly awakening, Poole dies. The computer, named Athena in this draft, announces "All systems of Poole now No–Go. It will be necessary to replace him with a spare unit." After this, Bowman decides to go out in a pod and retrieve the antenna, which is moving away from the ship. Athena refuses to allow him to leave the ship, citing "Directive 15" which prevents it from being left unattended, forcing him to make program modifications during which time the antenna drifts further.

During rehearsals Kubrick asked Stefanie Powers to supply the voice of HAL 9000 while searching for a suitably androgynous voice so the actors had something to react to. On the set, British actor Nigel Davenport played HAL. When it came to dubbing HAL in post-production, Kubrick had originally cast Martin Balsam, but as he felt Balsam "just sounded a little bit too colloquially American", he was replaced with Douglas Rain, who "had the kind of bland mid-Atlantic accent we felt was right for the part". Rain was only handed HAL's lines instead of the full script, and recorded them across a day and a half.

HAL's point of view shots were created with a Cinerama Fairchild-Curtis wide-angle lens with a 160° angle of view. This lens is about 8 in in diameter, while HAL's on set prop eye lens is about 3 in in diameter. Stanley Kubrick chose to use the large Fairchild-Curtis lens to shoot the HAL 9000 POV shots because he needed a wide-angle fisheye lens that would fit onto his shooting camera, and this was the only lens at the time that would work. The Fairchild-Curtis lens has a focal length of 0.9 in with a maximum aperture of 2.0 and a weight of approximately 30 lb; it was originally designed by Felix Bednarz with a maximum aperture of 2.2 for the first Cinerama 360 film, Journey to the Stars, shown at the 1962 Seattle World's Fair. Bednarz adapted the lens design from an earlier lens he had designed for military training to simulate human peripheral vision coverage. The lens was later recomputed for the second Cinerama 360 film To the Moon and Beyond, which had a slightly different film format. To the Moon and Beyond was produced by Graphic Films and shown at the 1964/1965 New York World's Fair, where Kubrick watched it; afterwards, he was so impressed that he hired the same creative team from Graphic Films (consisting of Douglas Trumbull, Lester Novros, and Con Pederson) to work on 2001.

A HAL 9000 face plate, without lens (not the same as the hero face plates seen in the film), was discovered in a junk shop in Paddington, London, in the early 1970s by Chris Randall. This was found along with the key to HAL's Brain Room. Both items were purchased for ten shillings (£0.50). Research revealed that the original lens was a Fisheye Nikkor 8 mm 8. The collection was sold at a Christie's auction in 2010 for £17,500 to film director Peter Jackson.

Origin of name
HAL's name, according to writer Arthur C. Clarke, is derived from Heuristically programmed ALgorithmic computer. After the film was released, fans noticed HAL was a one-letter shift from the name IBM and there has been much speculation since then that this was a dig at the large computer company, something that has been denied by both Clarke and 2001 director Stanley Kubrick. Clarke addressed the issue in his book The Lost Worlds of 2001: "...about once a week some character spots the fact that HAL is one letter ahead of IBM, and promptly assumes that Stanley and I were taking a crack at the estimable institution ... As it happened, IBM had given us a good deal of help, so we were quite embarrassed by this, and would have changed the name had we spotted the coincidence." IBM was consulted during the making of the film and their logo can be seen on props in the film, including the Pan Am Clipper's cockpit instrument panel and on the lower arm keypad on Poole's space suit. During production it was brought to IBM's attention that the film's plot included a homicidal computer but they approved association with the film if it was clear any "equipment failure" was not related to their products.

HAL Communications Corporation is a real corporation, with facilities located in Urbana, Illinois, which is where HAL in the movie identifies himself as being activated: "I am a HAL 9000 computer. I became operational at the H-A-L plant in Urbana Illinois on the 12th of January 1992."

The former president of HAL Communications, Bill Henry, has stated that this is a coincidence: "There was not and never has been any connection to 'Hal', Arthur Clarke's intelligent computer in the screen play '2001' — later published as a book. We were very surprised when the movie hit the Coed Theatre on campus and discovered that the movie's computer had our name. We never had any problems with that similarity - 'Hal' for the movie and 'HAL' (all caps) for our small company. But, from time-to-time, we did have issues with others trying to use 'HAL'. That resulted in us paying lawyers. The offenders folded or eventually went out of business."

Technology
The scene in which HAL's consciousness degrades was inspired by Clarke's memory of a speech synthesis demonstration by physicist John Larry Kelly, Jr., who used an IBM 704 computer to synthesize speech. Kelly's voice recorder synthesizer vocoder recreated the song "Daisy Bell", with musical accompaniment from Max Mathews.

HAL's capabilities, like all the technology in 2001, were based on the speculation of respected scientists. Marvin Minsky, director of the MIT Computer Science and Artificial Intelligence Laboratory (CSAIL) and one of the most influential researchers in the field, was an adviser on the film set. In the mid-1960s, many computer scientists in the field of artificial intelligence were optimistic that machines with HAL's capabilities would exist within a few decades. For example, AI pioneer Herbert A. Simon at Carnegie Mellon University had predicted in 1965 that "machines will be capable, within twenty years, of doing any work a man can do".

Cultural impact
HAL is listed as the 13th-greatest film villain in the AFI's 100 Years...100 Heroes & Villains.

The 9000th of the asteroids in the asteroid belt, 9000 Hal, discovered on 3 May 1981, by E. Bowell at Anderson Mesa Station, is named after HAL 9000.

Anthony Hopkins based his Academy Award-winning performance as Hannibal Lecter in Silence of the Lambs in part upon HAL 9000. Michael Fassbender has also sited HAL as an inspiration for his performances as Androids David and Walter in Prometheus and Alien: Covenant.

The 1993 educational game Where in Space Is Carmen Sandiego? features a digital assistant named the VAL 9000, a homage to HAL 9000.

Apple Inc.'s 1999 website advertisement "It was a bug, Dave" was made by meticulously recreating the appearance of HAL 9000 from the movie. Launched during the era of concerns over Y2K bugs, the ad implied that HAL's behavior was caused by a Y2K bug, before driving home the point that "only Macintosh was designed to function perfectly".

In 2003, HAL 9000 was one of the first robots to be inducted into the Robot Hall of Fame in Pittsburgh, Pennsylvania. One can see a physical replica of HAL at the Carnegie Science Center in Pittsburgh.

title: HAL 9000


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Hydrolysis

Hydrolysis is any chemical reaction in which a molecule of water breaks one or more chemical bonds. The term is used broadly for substitution, elimination, and solvation reactions in which water is the nucleophile.

Biological hydrolysis is the cleavage of biomolecules where a water molecule is consumed to effect the separation of a larger molecule into component parts. When a carbohydrate is broken into its component sugar molecules by hydrolysis (e.g., sucrose being broken down into glucose and fructose), this is recognized as saccharification.

Hydrolysis reactions can be the reverse of a condensation reaction in which two molecules join into a larger one and eject a water molecule. Thus hydrolysis adds water to break down, whereas condensation builds up by removing water.

Types
Usually hydrolysis is a chemical process in which a molecule of water is added to a substance. Sometimes this addition causes both the substance and water molecule to split into two parts. In such reactions, one fragment of the target molecule (or parent molecule) gains a hydrogen ion. It breaks a chemical bond in the compound.

Salts
A common kind of hydrolysis occurs when a salt of a weak acid or weak base (or both) is dissolved in water. Water spontaneously ionizes into hydroxide anions and hydronium cations. The salt also dissociates into its constituent anions and cations. For example, sodium acetate dissociates in water into sodium and acetate ions. Sodium ions react very little with the hydroxide ions whereas the acetate ions combine with hydronium ions to produce acetic acid. In this case the net result is a relative excess of hydroxide ions, yielding a basic solution.

Strong acids also undergo hydrolysis. For example, dissolving sulfuric acid (H2SO4) in water is accompanied by hydrolysis to give hydronium and bisulfate, the sulfuric acid's conjugate base. For a more technical discussion of what occurs during such a hydrolysis, see Brønsted–Lowry acid–base theory.

Esters and amides
Acid–base-catalysed hydrolyses are very common; one example is the hydrolysis of amides or esters. Their hydrolysis occurs when the nucleophile (a nucleus-seeking agent, e.g., water or hydroxyl ion) attacks the carbon of the carbonyl group of the ester or amide. In an aqueous base, hydroxyl ions are better nucleophiles than polar molecules such as water. In acids, the carbonyl group becomes protonated, and this leads to a much easier nucleophilic attack. The products for both hydrolyses are compounds with carboxylic acid groups.

Perhaps the oldest commercially practiced example of ester hydrolysis is saponification (formation of soap). It is the hydrolysis of a triglyceride (fat) with an aqueous base such as sodium hydroxide (NaOH). During the process, glycerol is formed, and the fatty acids react with the base, converting them to salts. These salts are called soaps, commonly used in households.

In addition, in living systems, most biochemical reactions (including ATP hydrolysis) take place during the catalysis of enzymes. The catalytic action of enzymes allows the hydrolysis of proteins, fats, oils, and carbohydrates. As an example, one may consider proteases (enzymes that aid digestion by causing hydrolysis of peptide bonds in proteins). They catalyze the hydrolysis of interior peptide bonds in peptide chains, as opposed to exopeptidases (another class of enzymes, that catalyze the hydrolysis of terminal peptide bonds, liberating one free amino acid at a time).

However, proteases do not catalyze the hydrolysis of all kinds of proteins. Their action is stereo-selective: Only proteins with a certain tertiary structure are targeted as some kind of orienting force is needed to place the amide group in the proper position for catalysis. The necessary contacts between an enzyme and its substrates (proteins) are created because the enzyme folds in such a way as to form a crevice into which the substrate fits; the crevice also contains the catalytic groups. Therefore, proteins that do not fit into the crevice will not undergo hydrolysis. This specificity preserves the integrity of other proteins such as hormones, and therefore the biological system continues to function normally.

Upon hydrolysis, an amide converts into a carboxylic acid and an amine or ammonia (which in the presence of acid are immediately converted to ammonium salts). One of the two oxygen groups on the carboxylic acid are derived from a water molecule and the amine (or ammonia) gains the hydrogen ion. The hydrolysis of peptides gives amino acids.

Many polyamide polymers such as nylon 6,6 hydrolyze in the presence of strong acids. The process leads to depolymerization. For this reason nylon products fail by fracturing when exposed to small amounts of acidic water. Polyesters are also susceptible to similar polymer degradation reactions. The problem is known as environmental stress cracking.

ATP
Hydrolysis is related to energy metabolism and storage. All living cells require a continual supply of energy for two main purposes: the biosynthesis of micro and macromolecules, and the active transport of ions and molecules across cell membranes. The energy derived from the oxidation of nutrients is not used directly but, by means of a complex and long sequence of reactions, it is channeled into a special energy-storage molecule, adenosine triphosphate (ATP). The ATP molecule contains pyrophosphate linkages (bonds formed when two phosphate units are combined) that release energy when needed. ATP can undergo hydrolysis in two ways: Firstly, the removal of terminal phosphate to form adenosine diphosphate (ADP) and inorganic phosphate, with the reaction:

Secondly, the removal of a terminal diphosphate to yield adenosine monophosphate (AMP) and pyrophosphate. The latter usually undergoes further cleavage into its two constituent phosphates. This results in biosynthesis reactions, which usually occur in chains, that can be driven in the direction of synthesis when the phosphate bonds have undergone hydrolysis.

Polysaccharides
Monosaccharides can be linked together by glycosidic bonds, which can be cleaved by hydrolysis. Two, three, several or many monosaccharides thus linked form disaccharides, trisaccharides, oligosaccharides, or polysaccharides, respectively. Enzymes that hydrolyze glycosidic bonds are called "glycoside hydrolases" or "glycosidases".

The best-known disaccharide is sucrose (table sugar). Hydrolysis of sucrose yields glucose and fructose. Invertase is a sucrase used industrially for the hydrolysis of sucrose to so-called invert sugar. Lactase is essential for digestive hydrolysis of lactose in milk; many adult humans do not produce lactase and cannot digest the lactose in milk.

The hydrolysis of polysaccharides to soluble sugars can be recognized as saccharification. Malt made from barley is used as a source of β-amylase to break down starch into the disaccharide maltose, which can be used by yeast to produce beer. Other amylase enzymes may convert starch to glucose or to oligosaccharides. Cellulose is first hydrolyzed to cellobiose by cellulase and then cellobiose is further hydrolyzed to glucose by beta-glucosidase. Ruminants such as cows are able to hydrolyze cellulose into cellobiose and then glucose because of symbiotic bacteria that produce cellulases.

DNA
Hydrolysis of DNA occurs at a significant rate in vivo. For example, it is estimated that in each human cell 2,000 to 10,000 DNA purine bases turn over every day due to hydrolytic depurination, and that this is largely counteracted by specific rapid DNA repair processes. Hydrolytic DNA damages that fail to be accurately repaired may contribute to carcinogenesis and ageing.

Metal aqua ions
Metal ions are Lewis acids, and in aqueous solution they form metal aquo complexes of the general formula M(H2O)_{n}^{m+}. The aqua ions undergo hydrolysis, to a greater or lesser extent. The first hydrolysis step is given generically as

Thus the aqua cations behave as acids in terms of Brønsted–Lowry acid–base theory. This effect is easily explained by considering the inductive effect of the positively charged metal ion, which weakens the O\sH bond of an attached water molecule, making the liberation of a proton relatively easy.

The dissociation constant, pKa, for this reaction is more or less linearly related to the charge-to-size ratio of the metal ion. Ions with low charges, such as Na+ are very weak acids with almost imperceptible hydrolysis. Large divalent ions such as Ca(2+), Zn(2+), Sn(2+) and Pb(2+) have a pKa of 6 or more and would not normally be classed as acids, but small divalent ions such as Be(2+) undergo extensive hydrolysis. Trivalent ions like Al(3+) and Fe(3+) are weak acids whose pKa is comparable to that of acetic acid. Solutions of salts such as BeCl2 or Al(NO3)3 in water are noticeably acidic; the hydrolysis can be suppressed by adding an acid such as nitric acid, making the solution more acidic.

Hydrolysis may proceed beyond the first step, often with the formation of polynuclear species via the process of olation. Some "exotic" species such as Sn3(OH)4(2+) are well characterized. Hydrolysis tends to proceed as pH rises leading, in many cases, to the precipitation of a hydroxide such as Al(OH)3 or AlO(OH). These substances, major constituents of bauxite, are known as laterites and are formed by leaching from rocks of most of the ions other than aluminium and iron and subsequent hydrolysis of the remaining aluminium and iron.

Mechanism strategies
Acetals, imines, and enamines can be converted back into ketones by treatment with excess water under acid-catalyzed conditions: RO*OR\sH3O\sO; NR*H3O\sO; RNR\sH3O\sO.

Acidic hydrolysis
Acid catalysis can be applied to hydrolyses. For example, in the conversion of cellulose or starch to glucose. For the case of ester hydrolysis and amides, it can be defined as an acid catalyzed nucleophilic acyl substitution reaction. Carboxylic acids can be produced from acid hydrolysis of esters.

Acids catalyze hydrolysis of nitriles to amides. Acid hydrolysis does not usually refer to the acid catalyzed addition of the elements of water to double or triple bonds by electrophilic addition as may originate from a hydration reaction. Acid hydrolysis is used to prepare monosaccharide with the help of mineral acids but formic acid and trifluoroacetic acid have been used.

Acid hydrolysis can be utilized in the pretreatment of cellulosic material, so as to cut the interchain linkages in hemicellulose and cellulose.

Alkaline hydrolysis
Alkaline hydrolysis usually refers to types of nucleophilic substitution reactions in which the attacking nucleophile is a hydroxide ion. The best known type is saponification: cleaving esters into carboxylate salts and alcohols. In ester hydrolysis and amide hydrolysis the hydroxide ion nucleophile attacks the carbonyl carbon in a nucleophilic acyl substitution reaction. This mechanism is supported by isotope labeling experiments. For example, when ethyl propionate with an oxygen-18 labeled ethoxy group is treated with sodium hydroxide (NaOH), the oxygen-18 is completely absent from the sodium propionate product and is found exclusively in the ethanol formed.

The reaction is often used to solubilize solid organic matter. Chemical drain cleaners take advantage of this method to dissolve hair and fat in pipes. The reaction is also used to dispose of human and other animal remains as an alternative to traditional burial or cremation.

title: Hydrolysis


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Hydroxy group

In chemistry, a hydroxy or hydroxyl group is a functional group with the chemical formula \sOH and composed of one oxygen atom covalently bonded to one hydrogen atom. In organic chemistry, alcohols and carboxylic acids contain one or more hydroxy groups. Both the negatively charged anion HO-, called hydroxide, and the neutral radical HO*, known as the hydroxyl radical, consist of an unbonded hydroxy group.

According to IUPAC definitions, the term hydroxyl refers to the hydroxyl radical (*OH) only, while the functional group \sOH is called a hydroxy group.

Properties
Water, alcohols, carboxylic acids, and many other hydroxy-containing compounds can be readily deprotonated due to a large difference between the electronegativity of oxygen (3.5) and that of hydrogen (2.1). Hydroxy-containing compounds engage in intermolecular hydrogen bonding increasing the electrostatic attraction between molecules and thus to higher boiling and melting points than found for compounds that lack this functional group. Organic compounds, which are often poorly soluble in water, become water-soluble when they contain two or more hydroxy groups, as illustrated by sugars and amino acid.

Occurrence
The hydroxy group is pervasive in chemistry and biochemistry. Many inorganic compounds contain hydroxyl groups, including sulfuric acid, the chemical compound produced on the largest scale industrially.

Hydroxy groups participate in the dehydration reactions that link simple biological molecules into long chains. The joining of a fatty acid to glycerol to form a triacylglycerol removes the −OH from the carboxy end of the fatty acid. The joining of two aldehyde sugars to form a disaccharide removes the −OH from the carboxy group at the aldehyde end of one sugar. The creation of a peptide bond to link two amino acids to make a protein removes the −OH from the carboxy group of one amino acid.

Hydroxyl radical
Hydroxyl radicals are highly reactive and undergo chemical reactions that make them short-lived. When biological systems are exposed to hydroxyl radicals, they can cause damage to cells, including those in humans, where they can react with DNA, lipids, and proteins.

Airglow of the Earth
The Earth's night sky is illuminated by diffuse light, called airglow, that is produced by radiative transitions of atoms and molecules. Among the most intense such features observed in the Earth's night sky is a group of infrared transitions at wavelengths between 700 nanometers and 900 nanometers. In 1950, Aden Meinel showed that these were transitions of the hydroxyl molecule, OH.

Surface of the Moon
In 2009, India's Chandrayaan-1 satellite and the National Aeronautics and Space Administration (NASA) Cassini spacecraft and Deep Impact probe each detected evidence of water by evidence of hydroxyl fragments on the Moon. As reported by Richard Kerr, "A spectrometer [the Moon Mineralogy Mapper, also known as "M3"] detected an infrared absorption at a wavelength of 3.0 micrometers that only water or hydroxyl—a hydrogen and an oxygen bound together—could have created." NASA also reported in 2009 that the LCROSS probe revealed an ultraviolet emission spectrum consistent with hydroxyl presence.

On 26 October 2020, NASA reported definitive evidence of water on the sunlit surface of the Moon, in the vicinity of the crater Clavius (crater), obtained by the Stratospheric Observatory for Infrared Astronomy (SOFIA). The SOFIA Faint Object infrared Camera for the SOFIA Telescope (FORCAST) detected emission bands at a wavelength of 6.1 micrometers that are present in water but not in hydroxyl. The abundance of water on the Moon's surface was inferred to be equivalent to the contents of a 12-ounce bottle of water per cubic meter of lunar soil.

The Chang'e 5 probe, which landed on the Moon on 1 December 2020, carried a mineralogical spectrometer that could measure infrared reflectance spectra of lunar rock and regolith. The reflectance spectrum of a rock sample at a wavelength of 2.85 micrometers indicated localized water/hydroxyl concentrations as high as 180 parts per million.

Atmosphere of Venus
The Venus Express orbiter collected Venus science data from April 2006 until December 2014. In 2008, Piccioni, et al. reported measurements of night-side airglow emission in the atmosphere of Venus made with the Visible and Infrared Thermal Imaging Spectrometer (VIRTIS) on Venus Express. They attributed emission bands in wavelength ranges of 1.40 - 1.49 micrometers and 2.6 - 3.14 micrometers to vibrational transitions of OH. This was the first evidence for OH in the atmosphere of any planet other than Earth's.

Atmosphere of Mars
In 2013, OH near-infrared spectra were observed in the night glow in the polar winter atmosphere of Mars by use of the Compact Reconnaissance Imaging Spectrometer for Mars (CRISM).

Exoplanets
In 2021, evidence for OH in the dayside atmosphere of the exoplanet WASP-33b was found in its emission spectrum at wavelengths between 1 and 2 micrometers. Evidence for OH in the atmosphere of exoplanet WASP-76b was subsequently found. Both WASP-33b and WASP-76b are ultra-hot Jupiters and it is likely that any water in their atmospheres is present as dissociated ions.

title: Hydroxy group