Self-replicating machine

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Self-replicating_machine 

A simple form of machine self-replication

A self-replicating machine is a type of autonomous robot that is capable of reproducing itself autonomously using raw materials found in the environment, thus exhibiting self-replication in a way analogous to that found in nature. The concept of self-replicating machines has been advanced and examined by Homer Jacobson, Edward F. Moore, Freeman Dyson, John von Neumann, Konrad Zuse and in more recent times by K. Eric Drexler in his book on nanotechnology, Engines of Creation (coining the term clanking replicator for such machines) and by Robert Freitas and Ralph Merkle in their review Kinematic Self-Replicating Machines which provided the first comprehensive analysis of the entire replicator design space. The future development of such technology is an integral part of several plans involving the mining of moons and asteroid belts for ore and other materials, the creation of lunar factories, and even the construction of solar power satellites in space. The von Neumann probe is one theoretical example of such a machine. Von Neumann also worked on what he called the universal constructor, a self-replicating machine that would be able to evolve and which he formalized in a cellular automata environment. Notably, Von Neumann's Self-Reproducing Automata scheme posited that open-ended evolution requires inherited information to be copied and passed to offspring separately from the self-replicating machine, an insight that preceded the discovery of the structure of the DNA molecule by Watson and Crick and how it is separately translated and replicated in the cell.

A self-replicating machine is an artificial self-replicating system that relies on conventional large-scale technology and automation. The concept, first proposed by Von Neumann no later than the 1940s, has attracted a range of different approaches involving various types of technology. Certain idiosyncratic terms are occasionally found in the literature. For example, the term clanking replicator was once used by Drexler to distinguish macroscale replicating systems from the microscopic nanorobots or "assemblers" that nanotechnology may make possible, but the term is informal and is rarely used by others in popular or technical discussions. Replicators have also been called "von Neumann machines" after John von Neumann, who first rigorously studied the idea. However, the term "von Neumann machine" is less specific and also refers to a completely unrelated computer architecture that von Neumann proposed and so its use is discouraged where accuracy is important. Von Neumann used the term universal constructor to describe such self-replicating machines.

Historians of machine tools, even before the numerical control era, sometimes figuratively said that machine tools were a unique class of machines because they have the ability to "reproduce themselves" by copying all of their parts. Implicit in these discussions is that a human would direct the cutting processes (later planning and programming the machines), and would then assemble the parts. The same is true for RepRaps, which are another class of machines sometimes mentioned in reference to such non-autonomous "self-replication". Such discussions refer to collections of machine tools, and such collections have an ability to reproduce their own parts which is finite and low for one machine, and ascends to nearly 100% with collections of only about a dozen similarly made, but uniquely functioning machines, establishing what authors Frietas and Merkle refer to as matter or material closure. Energy closure is the next most difficult dimension to close, and control the most difficult, noting that there are no other dimensions to the problem. In contrast, machines that are truly autonomously self-replicating (like biological machines) are the main subject discussed here, and would have closure in each of the three dimensions.

History

The general concept of artificial machines capable of producing copies of themselves dates back at least several hundred years. An early reference is an anecdote regarding the philosopher René Descartes, who suggested to Queen Christina of Sweden that the human body could be regarded as a machine; she responded by pointing to a clock and ordering "see to it that it reproduces offspring." Several other variations on this anecdotal response also exist. Samuel Butler proposed in his 1872 novel Erewhon that machines were already capable of reproducing themselves but it was man who made them do so, and added that "machines which reproduce machinery do not reproduce machines after their own kind". In George Eliot's 1879 book Impressions of Theophrastus Such, a series of essays that she wrote in the character of a fictional scholar named Theophrastus, the essay "Shadows of the Coming Race" speculated about self-replicating machines, with Theophrastus asking "how do I know that they may not be ultimately made to carry, or may not in themselves evolve, conditions of self-supply, self-repair, and reproduction".

In 1802 William Paley formulated the first known teleological argument depicting machines producing other machines, suggesting that the question of who originally made a watch was rendered moot if it were demonstrated that the watch was able to manufacture a copy of itself. Scientific study of self-reproducing machines was anticipated by John Bernal as early as 1929 and by mathematicians such as Stephen Kleene who began developing recursion theory in the 1930s. Much of this latter work was motivated by interest in information processing and algorithms rather than physical implementation of such a system, however. In the course of the 1950s, suggestions of several increasingly simple mechanical systems capable of self-reproduction were made — notably by Lionel Penrose.

Von Neumann's kinematic model

A detailed conceptual proposal for a self-replicating machine was first put forward by mathematician John von Neumann in lectures delivered in 1948 and 1949, when he proposed a kinematic model of self-reproducing automata as a thought experiment. Von Neumann's concept of a physical self-replicating machine was dealt with only abstractly, with the hypothetical machine using a "sea" or stockroom of spare parts as its source of raw materials. The machine had a program stored on a memory tape that instructed it to retrieve parts from this "sea" using a manipulator, assemble them into a copy of itself, and then transfer the contents of its memory tape into the new duplicate. The machine was envisioned as consisting of as few as eight different types of components: four logic elements for sending and receiving stimuli and four mechanical elements for providing structural support and mobility. Although qualitatively sound, von Neumann was evidently dissatisfied with this self-replicating machine model due to the difficulty of analyzing it with mathematical precision. He went on to instead develop an even more abstract model self-replicator based on cellular automata. His original kinematic concept remained obscure until it was popularized in a 1955 issue of Scientific American.

Von Neumann's goal for his self-reproducing automata theory, as specified in his lectures at the University of Illinois in 1949, was to design a machine whose complexity could grow automatically akin to biological organisms under natural selection. He asked what is the threshold of complexity that must be crossed for machines to be able to evolve. His answer was to design an abstract machine which, when run, would replicate itself. Notably, his design implies that open-ended evolution requires inherited information to be copied and passed to offspring separately from the self-replicating machine, an insight that preceded the discovery of the structure of the DNA molecule by Watson and Crick and how it is separately translated and replicated in the cell.

Moore's artificial living plants

In 1956 mathematician Edward F. Moore proposed the first known suggestion for a practical real-world self-replicating machine, also published in Scientific American. Moore's "artificial living plants" were proposed as machines able to use air, water and soil as sources of raw materials and to draw its energy from sunlight via a solar battery or a steam engine. He chose the seashore as an initial habitat for such machines, giving them easy access to the chemicals in seawater, and suggested that later generations of the machine could be designed to float freely on the ocean's surface as self-replicating factory barges or to be placed in barren desert terrain that was otherwise useless for industrial purposes. The self-replicators would be "harvested" for their component parts, to be used by humanity in other non-replicating machines.

Dyson's replicating systems

The next major development of the concept of self-replicating machines was a series of thought experiments proposed by physicist Freeman Dyson in his 1970 Vanuxem Lecture. He proposed three large-scale applications of machine replicators. First was to send a self-replicating system to Saturn's moon Enceladus, which in addition to producing copies of itself would also be programmed to manufacture and launch solar sail-propelled cargo spacecraft. These spacecraft would carry blocks of Enceladean ice to Mars, where they would be used to terraform the planet. His second proposal was a solar-powered factory system designed for a terrestrial desert environment, and his third was an "industrial development kit" based on this replicator that could be sold to developing countries to provide them with as much industrial capacity as desired. When Dyson revised and reprinted his lecture in 1979 he added proposals for a modified version of Moore's seagoing artificial living plants that was designed to distill and store fresh water for human use and the "Astrochicken."

Advanced Automation for Space Missions

An artist's conception of a "self-growing" robotic lunar factory

In 1980, inspired by a 1979 "New Directions Workshop" held at Wood's Hole, NASA conducted a joint summer study with ASEE entitled Advanced Automation for Space Missions to produce a detailed proposal for self-replicating factories to develop lunar resources without requiring additional launches or human workers on-site. The study was conducted at Santa Clara University and ran from June 23 to August 29, with the final report published in 1982. The proposed system would have been capable of exponentially increasing productive capacity and the design could be modified to build self-replicating probes to explore the galaxy.

The reference design included small computer-controlled electric carts running on rails inside the factory, mobile "paving machines" that used large parabolic mirrors to focus sunlight on lunar regolith to melt and sinter it into a hard surface suitable for building on, and robotic front-end loaders for strip mining. Raw lunar regolith would be refined by a variety of techniques, primarily hydrofluoric acid leaching. Large transports with a variety of manipulator arms and tools were proposed as the constructors that would put together new factories from parts and assemblies produced by its parent.

Power would be provided by a "canopy" of solar cells supported on pillars. The other machinery would be placed under the canopy.

A "casting robot" would use sculpting tools and templates to make plaster molds. Plaster was selected because the molds are easy to make, can make precise parts with good surface finishes, and the plaster can be easily recycled afterward using an oven to bake the water back out. The robot would then cast most of the parts either from nonconductive molten rock (basalt) or purified metals. A carbon dioxide laser cutting and welding system was also included.

A more speculative, more complex microchip fabricator was specified to produce the computer and electronic systems, but the designers also said that it might prove practical to ship the chips from Earth as if they were "vitamins."

A 2004 study supported by NASA's Institute for Advanced Concepts took this idea further. Some experts are beginning to consider self-replicating machines for asteroid mining.

Much of the design study was concerned with a simple, flexible chemical system for processing the ores, and the differences between the ratio of elements needed by the replicator, and the ratios available in lunar regolith. The element that most limited the growth rate was chlorine, needed to process regolith for aluminium. Chlorine is very rare in lunar regolith.

Lackner-Wendt Auxon replicators

In 1995, inspired by Dyson's 1970 suggestion of seeding uninhabited deserts on Earth with self-replicating machines for industrial development, Klaus Lackner and Christopher Wendt developed a more detailed outline for such a system. They proposed a colony of cooperating mobile robots 10–30 cm in size running on a grid of electrified ceramic tracks around stationary manufacturing equipment and fields of solar cells. Their proposal didn't include a complete analysis of the system's material requirements, but described a novel method for extracting the ten most common chemical elements found in raw desert topsoil (Na, Fe, Mg, Si, Ca, Ti, Al, C, O₂ and H₂) using a high-temperature carbothermic process. This proposal was popularized in Discover magazine, featuring solar-powered desalination equipment used to irrigate the desert in which the system was based. They named their machines "Auxons", from the Greek word auxein which means "to grow".

Recent work

NIAC studies on self-replicating systems

In the spirit of the 1980 "Advanced Automation for Space Missions" study, the NASA Institute for Advanced Concepts began several studies of self-replicating system design in 2002 and 2003. Four phase I grants were awarded:

• Hod Lipson (Cornell University), "Autonomous Self-Extending Machines for Accelerating Space Exploration"
• Gregory Chirikjian (Johns Hopkins University), "Architecture for Unmanned Self-Replicating Lunar Factories"
• Paul Todd (Space Hardware Optimization Technology Inc.), "Robotic Lunar Ecopoiesis"
• Tihamer Toth-Fejel (General Dynamics), "Modeling Kinematic Cellular Automata: An Approach to Self-Replication" The study concluded that complexity of the development was equal to that of a Pentium 4, and promoted a design based on cellular automata.

Bootstrapping self-replicating factories in space

In 2012, NASA researchers Metzger, Muscatello, Mueller, and Mantovani argued for a so-called "bootstrapping approach" to start self-replicating factories in space. They developed this concept on the basis of In Situ Resource Utilization (ISRU) technologies that NASA has been developing to "live off the land" on the Moon or Mars. Their modeling showed that in just 20 to 40 years this industry could become self-sufficient then grow to large size, enabling greater exploration in space as well as providing benefits back to Earth. In 2014, Thomas Kalil of the White House Office of Science and Technology Policy published on the White House blog an interview with Metzger on bootstrapping solar system civilization through self-replicating space industry. Kalil requested the public submit ideas for how "the Administration, the private sector, philanthropists, the research community, and storytellers can further these goals." Kalil connected this concept to what former NASA Chief technologist Mason Peck has dubbed "Massless Exploration", the ability to make everything in space so that you do not need to launch it from Earth. Peck has said, "...all the mass we need to explore the solar system is already in space. It's just in the wrong shape." In 2016, Metzger argued that fully self-replicating industry can be started over several decades by astronauts at a lunar outpost for a total cost (outpost plus starting the industry) of about a third of the space budgets of the International Space Station partner nations, and that this industry would solve Earth's energy and environmental problems in addition to providing massless exploration.

New York University artificial DNA tile motifs

In 2011, a team of scientists at New York University created a structure called 'BTX' (bent triple helix) based around three double helix molecules, each made from a short strand of DNA. Treating each group of three double-helices as a code letter, they can (in principle) build up self-replicating structures that encode large quantities of information.

Self-replication of magnetic polymers

In 2001, Jarle Breivik at University of Oslo created a system of magnetic building blocks, which in response to temperature fluctuations, spontaneously form self-replicating polymers.

Self-replication of neural circuits

In 1968, Zellig Harris wrote that "the metalanguage is in the language," suggesting that self-replication is part of language. In 1977 Niklaus Wirth formalized this proposition by publishing a self-replicating deterministic context-free grammar. Adding to it probabilities, Bertrand du Castel published in 2015 a self-replicating stochastic grammar and presented a mapping of that grammar to neural networks, thereby presenting a model for a self-replicating neural circuit.

Harvard Wyss Institute

November 29, 2021 a team at Harvard Wyss Institute built the first living robots that can reproduce.

Self-replicating spacecraft

Main article: Self-replicating spacecraft

The idea of an automated spacecraft capable of constructing copies of itself was first proposed in scientific literature in 1974 by Michael A. Arbib, but the concept had appeared earlier in science fiction such as the 1967 novel Berserker by Fred Saberhagen or the 1950 novellette trilogy The Voyage of the Space Beagle by A. E. van Vogt. The first quantitative engineering analysis of a self-replicating spacecraft was published in 1980 by Robert Freitas, in which the non-replicating Project Daedalus design was modified to include all subsystems necessary for self-replication. The design's strategy was to use the probe to deliver a "seed" factory with a mass of about 443 tons to a distant site, have the seed factory replicate many copies of itself there to increase its total manufacturing capacity, and then use the resulting automated industrial complex to construct more probes with a single seed factory on board each.

Prospects for implementation

As the use of industrial automation has expanded over time, some factories have begun to approach a semblance of self-sufficiency that is suggestive of self-replicating machines. However, such factories are unlikely to achieve "full closure" until the cost and flexibility of automated machinery comes close to that of human labour and the manufacture of spare parts and other components locally becomes more economical than transporting them from elsewhere. As Samuel Butler has pointed out in Erewhon, replication of partially closed universal machine tool factories is already possible. Since safety is a primary goal of all legislative consideration of regulation of such development, future development efforts may be limited to systems which lack either control, matter, or energy closure. Fully capable machine replicators are most useful for developing resources in dangerous environments which are not easily reached by existing transportation systems (such as outer space).

An artificial replicator can be considered to be a form of artificial life. Depending on its design, it might be subject to evolution over an extended period of time. However, with robust error correction, and the possibility of external intervention, the common science fiction scenario of robotic life run amok will remain extremely unlikely for the foreseeable future.

In fiction

Authors who have used self-replicating machine in works of fiction include: Philip K. Dick, Arthur C. Clarke, Karel Čapek: (R.U.R.: Rossum’s Universal Robots (1920)), John Sladek (The Reproductive System), Samuel Butler (Erewhon), Dennis E. Taylor and E. M. Forster (The Machine Stops (1909)).

Other sources

• A number of patents have been granted for self-replicating machine concepts. U.S. patent 5,659,477 "Self reproducing fundamental fabricating machines (F-Units)" Inventor: Collins; Charles M. (Burke, Va.) (August 1997), U.S. patent 5,764,518 " Self reproducing fundamental fabricating machine system" Inventor: Collins; Charles M. (Burke, Va.)(June 1998); and Collins' PCT patent WO 96/20453: "Method and system for self-replicating manufacturing stations" Inventors: Merkle; Ralph C. (Sunnyvale, Calif.), Parker; Eric G. (Wylie, Tex.), Skidmore; George D. (Plano, Tex.) (January 2003).
• Macroscopic replicators are mentioned briefly in the fourth chapter of K. Eric Drexler's 1986 book Engines of Creation.
• In 1995, Nick Szabo proposed a challenge to build a macroscale replicator from Lego robot kits and similar basic parts. Szabo wrote that this approach was easier than previous proposals for macroscale replicators, but successfully predicted that even this method would not lead to a macroscale replicator within ten years.
• In 2004, Robert Freitas and Ralph Merkle published the first comprehensive review of the field of self-replication (from which much of the material in this article is derived, with permission of the authors), in their book Kinematic Self-Replicating Machines, which includes 3000+ literature references. This book included a new molecular assembler design, a primer on the mathematics of replication, and the first comprehensive analysis of the entire replicator design space.

Right to property

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Right_to_property

The right to property, or the right to own property (cf. ownership), is often classified as a human right for natural persons regarding their private property. The Fourth Amendment to the United States Constitution is credited as a significant precedent for the legal protection of individual property rights.

A right to property is specified in Article 17 of the 1948 Universal Declaration of Human Rights, but it is not recognised in the 1966 International Covenant on Civil and Political Rights or in the 1966 International Covenant on Economic, Social and Cultural Rights. The 1950 European Convention on Human Rights acknowledges a right for a natural or legal person to "peaceful enjoyment of his possessions", subject to the "general interest or to secure the payment of taxes."

Definition

Article 17 of the Universal Declaration of Human Rights (UDHR) enshrines the right to property as follows:

(1) Everyone has the right to own property alone as well as in association with others. (2) No one shall be arbitrarily deprived of his or her property.

The object of the right to property as it is usually understood nowadays consists of property already owned or possessed, or of property acquired or to be acquired by a person through lawful means. Not in opposition but in contrast to this, some proposals also defend a universal right to private property, in the sense of a right of every person to effectively receive a certain amount of property, grounded in a claim to Earth's natural resources or other theories of justice.

The right to property is one of the most controversial human rights, both in terms of its existence and interpretation. The controversy about the definition of the right meant that it was not included in the International Covenant on Civil and Political Rights or the International Covenant on Economic, Social and Cultural Rights. Controversy centres upon who is deemed to have property rights protected (e.g. human beings or also corporations), the type of property which is protected (property used for the purpose of consumption or production) and the reasons for which property can be restricted (for instance, for regulations, taxation or nationalisation in the public interest). In all human rights instruments, either implicit or express restrictions exist on the extent to which property is protected.

Africa

See also: African Charter on Human and Peoples' Rights

The African Charter on Human and Peoples' Rights (ACHPR) protects the right to property most explicitly in Article 14, stating:

The right to property shall be guaranteed. It may only be encroached upon in the interest of public need or in the general interest of the community and in accordance with the provisions of appropriate laws.

Property rights are furthermore recognised in Article 13 of the ACHPR, which states that every citizen has the right to participate freely in the government of his country, the right to equal access to public services and "the right of access to public property and services in strict equality of all persons before the law". Article 21 of the ACHPR recognises the right of all peoples to freely dispose of their wealth and natural resources and that this right shall be exercised in the exclusive interest of the people, who may not be deprived of this right. Article 21 also provides that "in case of spoliation the dispossessed people shall have the right to the lawful recovery of its property as well as to adequate compensation".

Americas

See also: American Declaration of the Rights and Duties of Man and American Convention on Human Rights

When the text of the UDHR was negotiated, other states in the Americas argued that the right to property should be limited to the protection of private property necessary for subsistence. Their suggestion was opposed, but was enshrined in the American Declaration of the Rights and Duties of Man, which was negotiated at the same time and adopted one year before the UDHR in 1948. Article 23 of the declaration states:

Every Person has the right to own such private property as meets the essential needs of decent living and helps to maintain the dignity of the individual and of the home.

The definition of the right to property is heavily influenced by Western concepts of property rights, but because property rights vary considerably in different legal systems it has not been possible to establish international standards on property rights. The regional human rights instruments of Europe, Africa and the Americas recognise the right to protection of property to varying degrees.

The American Convention on Human Rights (ACHR) recognises the right to protection of property, including the right to "just compensation". The ACHR also prohibits usury and other exploitation, which is unique amongst human rights instruments. Article 21 of the ACHR states:

(1) Everyone has the right to the use and enjoyment of his property. The law may subordinate such use and enjoyment to the interest of society.

(2) No one shall be deprived of his property except upon payment of just compensation, for reasons of public utility or social interest, and in the cases and according to the forms established by law.

(3) Usury and any other form of exploitation of man by man shall be prohibited by law.

Europe

Main article: European Convention on Human Rights

After failed attempts to include the right to protection of property in the European Convention on Human Rights (ECHR), European states enshrined the right to protection of property in Article 1 of Protocol I to the ECHR as the "right to peaceful enjoyment of possessions", where the right to protection of property is defined as such:

(1) Every natural or legal person is entitled to the peaceful enjoyment of his possessions. No one shall be deprived of his possessions except in the public interest and subject to the conditions provided for by law and by the general principles of international law. (2) The preceding provisions shall not, however, in any way impair the right of a State to enforce such laws as it deems necessary to control the use of property in accordance with the general interest or to secure the payment of taxes or other contributions or penalties.

Therefore, European human rights law recognises the right to peaceful enjoyment of property, makes deprivation of possessions subject to certain conditions and recognises that states can balance the right to peaceful possession of property against the public interest. The European Court of Human Rights has interpreted "possessions" to include not only tangible property, but also economic interests, contractual agreements with economic value, compensation claims against the state and public law related claims such as pensions. The European Court of Human Rights has held that the right to property is not absolute and states have a wide degree of discretion to limit the rights. As such, the right to property is regarded as a more flexible right than other human rights. States' degree of discretion is defined in Handyside v. United Kingdom, heard by the European Court of Human Rights in 1976. Notable cases where the European Court of Human Rights has found the right to property having been violated include Sporrong and Lonnroth v. Sweden, heard in 1982, where Swedish law kept property under the threat of expropriation for an extended period of time. The highest economic compensation following a judgment of the Strasbourg Court on this matter was given (1,3 million euro) in case Beyeler v. Italy.

India

In India property rights (Article 31) was one of the fundamental rights of citizens until 1978, and it became a legal right through the 44th Amendment to the Constitution in 1978. The amendment was introduced by the Morarji Desai government as part of land reform policies. In 2020, the Supreme Court of India has stated that, even though property rights are not part of a citizen's fundamental right, it should be considered as one of the human rights promised by the Constitution. The Supreme Court also ruled that the states cannot acquire individual land unless there is a clear legal framework.

International conventions

Property rights are also recognised in the International Convention on the Elimination of All Forms of Racial Discrimination which states in Article 5 that everyone has the right to equality before the law without distinction as to race, colour and national or ethnic origin, including the "right to own property alone as well as in association with others" and "the right to inherit". The Convention on the Elimination of All Forms of Discrimination against Women recognises the property rights in Article 16, which establishes the same right for both spouses to ownership, acquisition, management, administration, enjoyment and disposition of property and Article 15, which establishes women's right to conclude contracts.

Property rights are also enshrined in the Convention Relating to the Status of Refugees and the Convention on the Protection of the Rights of All Migrant Workers and Members of Their Families. These international human rights instruments for minorities do not establish a separate right to property, but prohibit discrimination in relation to property rights where such rights are guaranteed.

Relationship to other rights

The right to private property was a crucial demand in early quests for political freedom and equality and against feudal control of property. Property can serve as the basis for the entitlements that ensure the realisation of the right to an adequate standard of living and it was only property owners which were initially granted civil and political rights, such as the right to vote. Because not everybody is a property owner, the right to work was enshrined to allow everybody to attain an adequate standard of living. Today, discrimination on the basis of property ownership is commonly seen as a serious threat to the equal enjoyment of human rights by all and non-discrimination clauses in international human rights instruments frequently include property as a ground on the basis of which discrimination is prohibited (see the right to equality before the law). The protection of private property may come into conflict with economic, social and cultural rights and civil and political rights, such as the right to freedom of expression. To mitigate this, the right to property is commonly limited to protect the public interest. Many states also maintain systems of communal and collective ownership. Property rights have frequently been regarded as preventing the realisation of human rights for all, through for example slavery and the exploitation of others. Unequal distribution of wealth often follows line of sex, race and minorities, therefore property rights may appear to be part of the problem, rather than as an interest that merits protection. Property rights have been at the centre of recent human rights debates on land reform, the return of cultural artifacts by collectors and museums to indigenous peoples and the popular sovereignty of peoples over natural resources.

History

The Roman law defined property as "the right to use and abuse one's own within the limits of the law" — jus utendi et abutendi re suâ, guatenus juris ratio patitur. Second, salus populi suprema lex esto, or "the safety of the people shall be the supreme law," was stipulated as early as the Law of the Twelve Tables. The notion of private property and property rights was elaborated further in the Renaissance as international trade by merchants gave rise to mercantilist ideas. In 16th-century Europe, Lutheranism and the Protestant Reformation advanced property rights using biblical terminology. The Protestant work ethic and views on man's destiny came to underline social views in emerging capitalist economies in early modern Europe. The right to private property emerged as a radical demand for human rights vis-a-vis the state in 17th-century revolutionary Europe, but in the 18th and 19th centuries the right to property as a human right became subject of intense controversy.

English Civil War

See also: English Civil War

The arguments advanced by the Levellers during the English Civil War on property and civil and political rights, such as the right to vote, informed subsequent debates in other countries. The Levellers emerged as a political movement in mid-17th century England in the aftermath of the Protestant Reformation. They believed that property which had been earned as the fruit of one's labour was sacred under the Bible's commandment "thou shall not steal". As such, they believed that the right to acquire property from one's work was sacred. Levellers' views on the right to property and the right not to be deprived of property as a civil and political right were developed by the pamphleteer Richard Overton. In "An Arrow against all Tyrants" (1646), Overton argued:

To every individual in nature is given an individual property by nature not to be invaded or usurped by any. For everyone, as he is himself, so he has a self propertiety, else he could not be himself; and of this no second may presume to deprive of without manifest violation and affront to the very principles of nature of the rules of equity and justice between man and man. Mine and thine cannot be, except this. No man has power over my rights and liberties, and I over no man.

The views of the Levellers, who enjoyed support amongst small-scale property-owners and craftsmen, were not shared by all revolutionary parties of the English Civil War. At the 1647 General Council, Oliver Cromwell and Henry Ireton argued against equating the right to life with the right to property. They argued that doing so would establish the right to take anything that one may want, irrespective of the rights of others. The Leveller Thomas Rainsborough responded, relying on Overton's arguments, that the Levellers required respect for others' natural rights. The definition of property and whether it was acquired as the fruit of one's labour and as such a natural right was subject to intense debate because the right to vote depended on property ownership. Political freedom was at the time associated with property ownership and individual independence. Cromwell and Ireton maintained that only property in freehold land or chartered trading rights gave a man the right to vote. They argued that this type of property ownership constituted a "stake in society", which entitles men to political power. In contrast, Levellers argued that all men who are not servants, alms-recipients or beggars should be considered as property owners and be given voting rights. They believed that political freedom could only be secured by individuals, such as craftsmen, engaging in independent economic activity.

Levellers were primarily concerned with the civil and political rights of small-scale property owners and workers, whereas the Diggers, a smaller revolutionary group led by Gerrard Winstanley, focused on the rights of the rural poor who worked on landed property. The Diggers argued that private property was not consistent with justice and that the land that had been confiscated from the Crown and Church should be turned into communal land to be cultivated by the poor. According to the Diggers, the right to vote should be extended to all and everybody had the right to an adequate standard of living. With the Restoration of the English monarchy in 1660, all confiscated land returned to the Crown and Church. Some property rights were recognised and limited voting rights were established. The ideas of the Levellers on property and civil and political rights remained influential and were advanced in the subsequent 1688 Glorious Revolution, but restrictions on the right to vote based on property meant that only a fraction of the British population had the suffrage. In 1780 only 214,000 property-owning men were entitled to vote in England and Wales, less than 3 percent of the population of 8 million. The Reform Act 1832 restricted the right to vote to men who owned property with an annual value of £10, giving approximately 4 percent of the adult male population the right to vote. The reforms of 1867 extended the right to vote to approximately 8 percent. The working class (which increased dramatically with the Industrial Revolution) and industrialists remained effectively excluded from the political system.

John Locke and the American and French revolutions

John Locke's 1689 Two Treatises of Government in which Locke calls "lives, liberties and estates" the "property" of individuals

The English philosopher John Locke (1632–1704) developed the ideas of property, civil and political rights further. In his Second Treatise on Civil Government (1689), Locke proclaimed that "everyman has a property in his person; this nobody has a right to but himself. The labor of his body and the work of his hand, we may say, are properly his". He argued that property ownership derives from one's labor, though those who do not own property and only have their labor to sell should not be given the same political power as those who owned property. Labourers, small-scale property owners and large-scale property owners should have civil and political rights in proportion to the property they owned. According to Locke, the right to property and the right to life were inalienable rights and that it was the duty of the state to secure these rights for individuals. Locke argued that the safeguarding of natural rights, such as the right to property, along with the separation of powers and other checks and balances, would help to curtail political abuses by the state.

Locke's labor theory of property and the separation of powers greatly influenced the American Revolution and the French Revolution. The entitlement to civil and political rights, such as the right to vote, was tied to the question of property in both revolutions. American revolutionaries, such as Benjamin Franklin and Thomas Jefferson, opposed universal suffrage, advocating votes only for those who owned a "stake" in society. James Madison argued that extending the right to vote to all could lead in the right to property and justice being "overruled by a majority without property". While it was initially suggested to establish the right to vote for all men, eventually the right to vote in the nascent United States was extended to white men who owned a specified amount of real estate and personal property.

French revolutionaries recognised property rights in Article 17 of the Declaration of the Rights of Man and of the Citizen (1791), which stated that no one "may be deprived of property rights unless a legally established public necessity required it and upon condition of a just and previous indemnity". Articles 3 and 6 declared that "all citizens have the right to contribute personally or through their representatives" in the political system and that "all citizens being equal before [the law], are equally admissible to all public offices, positions and employment according to their capacity, and without other distinction than that of virtues and talents". However, in practice the French revolutionaries did not extend civil and political rights to all, although the property qualification required for such rights was lower than that established by the American revolutionaries.

According to the French revolutionary Abbé Sieyès, "all the inhabitants of a country should enjoy the right of a passive citizen... but those alone who contribute to the public establishment are like the true shareholders in the great social enterprise. They alone are the true active citizens, the true members of the association". Three months after the Declaration had been adopted, domestic servants, women and those who did not pay taxes equal to three days of labor were declared "passive citizens". Sieyes wanted to see the rapid expansion of commercial activities and favoured the unrestricted accumulation of property. In contrast, Maximilien Robespierre warned that the free accumulation of wealth ought to be limited and that the right to property should not be permitted to violate the rights of others, particularly poorer citizens, including the working poor and peasants. Robespierre's views were eventually excluded from the French Constitution of 1793 and a property qualification for civil and political rights was maintained.

Hamiltonian (quantum mechanics)

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/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 ${\displaystyle \hat{H}}$, where the hat indicates that it is an operator. It can also be written as ${\displaystyle H}$ or ${\displaystyle \check{H}}$.

Introduction

Main article: Operator (physics) § Operators in quantum mechanics

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.

Schrödinger Hamiltonian

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

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: $${\displaystyle \begin{array}{rl}\hat{H}& =\hat{T}+\hat{V}\\ & =\frac{\hat{\mathbf{p}}\cdot \hat{\mathbf{p}}}{2m}+V(\mathbf{r},t)\\ & =-\frac{{\hslash }^2}{2m}{\mathrm{\nabla }}^2+V(\mathbf{r},t)\end{array}}$$ which allows one to apply the Hamiltonian to systems described by a wave function ${\displaystyle \mathrm{\Psi }(\mathbf{r},t)}$. 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: $${\displaystyle \begin{array}{rl}T& =-\frac{{\hslash }^2}{2m}{\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}{\psi }^{\ast }\frac{d^2\psi }{d{x}^2}dx\\ & =-\frac{{\hslash }^2}{2m}\left({\left[{\psi }^{\prime }(x){\psi }^{\ast }(x)\right]}_{-\mathrm{\infty }}^{+\mathrm{\infty }}-{\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}\frac{d\psi }{dx}\frac{d{\psi }^{\ast }}{dx}dx\right)\\ & =\frac{{\hslash }^2}{2m}{\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}{\left|\frac{d\psi }{dx}\right|}^{2}dx\ge 0\end{array}}$$

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: $${\displaystyle E=T+⟨V(x)⟩=T+{\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}V(x)|\psi (x){|}^{2}dx\ge V_{\text{min}}(x){\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}|\psi (x){|}^{2}dx\ge V_{\text{min}}(x)}$$ which complete the proof. Similarly, the condition can be generalized to any higher dimensions using the divergence theorem.

Many particles

The formalism can be extended to ${\displaystyle N}$ particles: $${\displaystyle \hat{H}=\sum _{n=1}^N{\hat{T}}_n+\hat{V}}$$ where $${\displaystyle \hat{V}=V({\mathbf{r}}_1,{\mathbf{r}}_2,\dots ,{\mathbf{r}}_N,t),}$$ 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 $${\displaystyle {\hat{T}}_n=\frac{{\hat{\mathbf{p}}}_n\cdot {\hat{\mathbf{p}}}_n}{2m_n}=-\frac{{\hslash }^2}{2m_n}{\mathrm{\nabla }}_n^2}$$ is the kinetic energy operator of particle ${\displaystyle n}$, ${\displaystyle {\mathrm{\nabla }}_n}$ is the gradient for particle ${\displaystyle n}$, and ${\displaystyle {\mathrm{\nabla }}_n^2}$ is the Laplacian for particle n: $${\displaystyle {\mathrm{\nabla }}_n^2=\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{x}_n^2}+\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{y}_n^2}+\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{z}_n^2},}$$

Combining these yields the Schrödinger Hamiltonian for the ${\displaystyle N}$-particle case: $${\displaystyle \begin{array}{rl}\hat{H}& =\sum _{n=1}^N{\hat{T}}_n+\hat{V}\\ & =\sum _{n=1}^N\frac{{\hat{\mathbf{p}}}_n\cdot {\hat{\mathbf{p}}}_n}{2m_n}+V({\mathbf{r}}_1,{\mathbf{r}}_2,\dots ,{\mathbf{r}}_N,t)\\ & =-\frac{{\hslash }^2}{2}\sum _{n=1}^N\frac{1}{m_n}{\mathrm{\nabla }}_n^2+V({\mathbf{r}}_1,{\mathbf{r}}_2,\dots ,{\mathbf{r}}_N,t)\end{array}}$$

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: $${\displaystyle -\frac{{\hslash }^2}{2M}{\mathrm{\nabla }}_i\cdot {\mathrm{\nabla }}_j}$$ where ${\displaystyle M}$ 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 ${\displaystyle N}$ interacting particles, i.e. particles which interact mutually and constitute a many-body situation, the potential energy function ${\displaystyle V}$ 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,^[1] that is $${\displaystyle V=\sum _{i=1}^{N}V({\mathbf{r}}_i,t)=V({\mathbf{r}}_1,t)+V({\mathbf{r}}_2,t)+\cdots +V({\mathbf{r}}_N,t)}$$

The general form of the Hamiltonian in this case is: $${\displaystyle \begin{array}{rl}\hat{H}& =-\frac{{\hslash }^2}{2}\sum _{i=1}^N\frac{1}{m_i}{\mathrm{\nabla }}_i^2+\sum _{i=1}^N{V}_i\\ & =\sum _{i=1}^N\left(-\frac{{\hslash }^2}{2m_i}{\mathrm{\nabla }}_i^2+V_i\right)\\ & =\sum _{i=1}^N{\hat{H}}_i\end{array}}$$ 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

Main article: Schrödinger equation

The Hamiltonian generates the time evolution of quantum states. If ${\displaystyle |\psi (t)⟩}$ is the state of the system at time ${\displaystyle t}$, then $${\displaystyle H|\psi (t)⟩=i\hslash \frac{d}{dt}|\psi (t)⟩.}$$

This equation is the Schrödinger equation. It takes the same form as the Hamilton–Jacobi equation, which is one of the reasons ${\displaystyle H}$ is also called the Hamiltonian. Given the state at some initial time (${\displaystyle t=0}$), we can solve it to obtain the state at any subsequent time. In particular, if ${\displaystyle H}$ is independent of time, then $${\displaystyle |\psi (t)⟩=e^{-iHt/\hslash }|\psi (0)⟩.}$$

The exponential operator on the right hand side of the Schrödinger equation is usually defined by the corresponding power series in ${\displaystyle H}$. 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 $${\displaystyle U=e^{-iHt/\hslash }}$$ is a unitary operator. It is the time evolution operator or propagator of a closed quantum system. If the Hamiltonian is time-independent, ${\displaystyle \{U(t)\}}$ 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 of ${\displaystyle H}$, denoted ${\displaystyle |a⟩}$, 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 ${\displaystyle \{E_a\}}$, solving the equation: $${\displaystyle H|a⟩=E_a|a⟩.}$$

Since ${\displaystyle H}$ 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 ${\displaystyle m}$, and charges by ${\displaystyle q}$.

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: $${\displaystyle \hat{H}=-\frac{{\hslash }^2}{2m}\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{x}^2}}$$ and in higher dimensions: $${\displaystyle \hat{H}=-\frac{{\hslash }^2}{2m}{\mathrm{\nabla }}^2}$$

Constant-potential well

For a particle in a region of constant potential ${\displaystyle V=V_0}$ (no dependence on space or time), in one dimension, the Hamiltonian is: $${\displaystyle \hat{H}=-\frac{{\hslash }^2}{2m}\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{x}^2}+V_0}$$ in three dimensions $${\displaystyle \hat{H}=-\frac{{\hslash }^2}{2m}{\mathrm{\nabla }}^2+V_0}$$

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: $${\displaystyle V=\frac{k}{2}{x}^2=\frac{m{\omega }^2}{2}{x}^2}$$ where the angular frequency ${\displaystyle \omega }$, effective spring constant ${\displaystyle k}$, and mass ${\displaystyle m}$ of the oscillator satisfy: $${\displaystyle {\omega }^2=\frac{k}{m}}$$ so the Hamiltonian is: $${\displaystyle \hat{H}=-\frac{{\hslash }^2}{2m}\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{x}^2}+\frac{m{\omega }^2}{2}{x}^2}$$

For three dimensions, this becomes $${\displaystyle \hat{H}=-\frac{{\hslash }^2}{2m}{\mathrm{\nabla }}^2+\frac{m{\omega }^2}{2}{r}^2}$$ where the three-dimensional position vector ${\displaystyle \mathbf{r}}$ using Cartesian coordinates is ${\displaystyle (x,y,z)}$, its magnitude is $${\displaystyle r^2=\mathbf{r}\cdot \mathbf{r}=|\mathbf{r}{|}^2=x^2+y^2+z^2}$$

Writing the Hamiltonian out in full shows it is simply the sum of the one-dimensional Hamiltonians in each direction: $${\displaystyle \begin{array}{rl}\hat{H}& =-\frac{{\hslash }^2}{2m}\left(\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{x}^2}+\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{y}^2}+\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{z}^2}\right)+\frac{m{\omega }^2}{2}\left(x^2+y^2+z^2\right)\\ & =\left(-\frac{{\hslash }^2}{2m}\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{x}^2}+\frac{m{\omega }^2}{2}{x}^2\right)+\left(-\frac{{\hslash }^2}{2m}\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{y}^2}+\frac{m{\omega }^2}{2}{y}^2\right)+\left(-\frac{{\hslash }^2}{2m}\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{z}^2}+\frac{m{\omega }^2}{2}{z}^2\right)\end{array}}$$

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: $${\displaystyle \hat{H}=-\frac{{\hslash }^2}{2I_{xx}}{\hat{J}}_x^2-\frac{{\hslash }^2}{2I_{yy}}{\hat{J}}_y^2-\frac{{\hslash }^2}{2I_{zz}}{\hat{J}}_z^2}$$ where ${\displaystyle I_{xx}}$, ${\displaystyle I_{yy}}$, and ${\displaystyle I_{zz}}$ are the moment of inertia components (technically the diagonal elements of the moment of inertia tensor), and ${\displaystyle {\hat{J}}_x}$, ${\displaystyle {\hat{J}}_y}$, and ${\displaystyle {\hat{J}}_z}$ are the total angular momentum operators (components), about the ${\displaystyle x}$, ${\displaystyle y}$, and ${\displaystyle z}$ axes respectively.

Electrostatic (Coulomb) potential

The Coulomb potential energy for two point charges ${\displaystyle q_1}$ and ${\displaystyle q_2}$ (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): $${\displaystyle V=\frac{q_1{q}_2}{4\pi {\epsilon }_0|\mathbf{r}|}}$$

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 ${\displaystyle N}$ charges, the potential energy of charge ${\displaystyle q_j}$ due to all other charges is (see also Electrostatic potential energy stored in a configuration of discrete point charges): $${\displaystyle V_j=\frac{1}{2}\sum _{i\ne j}{q}_i\varphi ({\mathbf{r}}_i)=\frac{1}{8\pi {\epsilon }_0}\sum _{i\ne j}\frac{q_i{q}_j}{|{\mathbf{r}}_i-{\mathbf{r}}_j|}}$$ where ${\displaystyle \varphi ({\mathbf{r}}_i)}$ is the electrostatic potential of charge ${\displaystyle q_j}$ at ${\displaystyle {\mathbf{r}}_i}$. The total potential of the system is then the sum over ${\displaystyle j}$: $${\displaystyle V=\frac{1}{8\pi {\epsilon }_0}\sum _{j=1}^N\sum _{i\ne j}\frac{q_i{q}_j}{|{\mathbf{r}}_i-{\mathbf{r}}_j|}}$$ so the Hamiltonian is: $${\displaystyle \begin{array}{rl}\hat{H}& =-\frac{{\hslash }^2}{2}\sum _{j=1}^N\frac{1}{m_j}{\mathrm{\nabla }}_j^2+\frac{1}{8\pi {\epsilon }_0}\sum _{j=1}^N\sum _{i\ne j}\frac{q_i{q}_j}{|{\mathbf{r}}_i-{\mathbf{r}}_j|}\\ & =\sum _{j=1}^N\left(-\frac{{\hslash }^2}{2m_j}{\mathrm{\nabla }}_j^2+\frac{1}{8\pi {\epsilon }_0}\sum _{i\ne j}\frac{q_i{q}_j}{|{\mathbf{r}}_i-{\mathbf{r}}_j|}\right)\end{array}}$$

Electric dipole in an electric field

For an electric dipole moment ${\displaystyle \mathbf{d}}$ constituting charges of magnitude ${\displaystyle q}$, in a uniform, electrostatic field (time-independent) ${\displaystyle \mathbf{E}}$, positioned in one place, the potential is: $${\displaystyle V=-\hat{\mathbf{d}}\cdot \mathbf{E}}$$ the dipole moment itself is the operator $${\displaystyle \hat{\mathbf{d}}=q\hat{\mathbf{r}}}$$

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

Magnetic dipole in a magnetic field

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

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: $${\displaystyle \hat{H}=-\mathit{\mu }\cdot \mathbf{B}}$$

For a spin-1⁄2 particle, the corresponding spin magnetic moment is:^[4] $${\displaystyle {\mathit{\mu }}_S=\frac{g_{s}e}{2m}\mathbf{S}}$$ where ${\displaystyle g_s}$ is the "spin g-factor" (not to be confused with the gyromagnetic ratio), ${\displaystyle e}$ is the electron charge, ${\displaystyle \mathbf{S}}$ is the spin operator vector, whose components are the Pauli matrices, hence $${\displaystyle \hat{H}=\frac{g_{s}e}{2m}\mathbf{S}\cdot \mathbf{B}}$$

Charged particle in an electromagnetic field

For a particle with mass ${\displaystyle m}$ and charge ${\displaystyle q}$ in an electromagnetic field, described by the scalar potential ${\displaystyle \varphi }$ and vector potential ${\displaystyle \mathbf{A}}$, there are two parts to the Hamiltonian to substitute for. The canonical momentum operator ${\displaystyle \hat{\mathbf{p}}}$, which includes a contribution from the ${\displaystyle \mathbf{A}}$ field and fulfils the canonical commutation relation, must be quantized; $${\displaystyle \hat{\mathbf{p}}=m\dot{\mathbf{r}}+q\mathbf{A},}$$ where ${\displaystyle m\dot{\mathbf{r}}}$ is the kinetic momentum. The quantization prescription reads $${\displaystyle \hat{\mathbf{p}}=-i\hslash \mathrm{\nabla },}$$ so the corresponding kinetic energy operator is $${\displaystyle \hat{T}=\frac{1}{2}m\dot{\mathbf{r}}\cdot \dot{\mathbf{r}}=\frac{1}{2m}{\left(\hat{\mathbf{p}}-q\mathbf{A}\right)}^2}$$ and the potential energy, which is due to the ${\displaystyle \varphi }$ field, is given by $${\displaystyle \hat{V}=q\varphi .}$$

Casting all of these into the Hamiltonian gives $${\displaystyle \hat{H}=\frac{1}{2m}{\left(-i\hslash \mathrm{\nabla }-q\mathbf{A}\right)}^2+q\varphi .}$$

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 ${\displaystyle x}$ direction is a different state from one propagating in the ${\displaystyle y}$ 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 ${\displaystyle U}$ commutes with the Hamiltonian. To see this, suppose that ${\displaystyle |a⟩}$ is an energy eigenket. Then ${\displaystyle U|a⟩}$ is an energy eigenket with the same eigenvalue, since $${\displaystyle UH|a⟩=U{E}_a|a⟩=E_a(U|a⟩)=H(U|a⟩).}$$

Since ${\displaystyle U}$ is nontrivial, at least one pair of ${\displaystyle |a⟩}$ and ${\displaystyle U|a⟩}$ must represent distinct states. Therefore, ${\displaystyle H}$ 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 ${\displaystyle G}$ be the Hermitian generator of ${\displaystyle U}$: $${\displaystyle U=I-i\epsilon G+O({\epsilon }^2)}$$

It is straightforward to show that if ${\displaystyle U}$ commutes with ${\displaystyle H}$, then so does ${\displaystyle G}$: $${\displaystyle [H,G]=0}$$

Therefore, $${\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }t}⟨\psi (t)|G|\psi (t)⟩=\frac{1}{i\hslash }⟨\psi (t)|[G,H]|\psi (t)⟩=0.}$$

In obtaining this result, we have used the Schrödinger equation, as well as its dual, $${\displaystyle ⟨\psi (t)|H=-i\hslash \frac{d}{dt}⟨\psi (t)|.}$$

Thus, the expected value of the observable ${\displaystyle G}$ 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 ${\displaystyle \left\{|n⟩\right\}}$, which need not necessarily be eigenstates of the energy. For simplicity, we assume that they are discrete, and that they are orthonormal, i.e., $${\displaystyle ⟨n^{\prime }|n⟩={\delta }_{n{n}^{\prime }}}$$

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 ${\displaystyle t}$, ${\displaystyle |\psi \left(t\right)⟩}$, can be expanded in terms of these basis states: $${\displaystyle |\psi (t)⟩=\sum _n{a}_n(t)|n⟩}$$ where $${\displaystyle a_n(t)=⟨n|\psi (t)⟩.}$$

The coefficients ${\displaystyle a_n(t)}$ 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 $${\displaystyle ⟨H(t)⟩\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}⟨\psi (t)|H|\psi (t)⟩=\sum _{n{n}^{\prime }}{a}_{n^{\prime }}^{\ast }{a}_n⟨n^{\prime }|H|n⟩}$$ where the last step was obtained by expanding ${\displaystyle |\psi \left(t\right)⟩}$ in terms of the basis states.

Each ${\displaystyle a_n(t)}$ 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 ${\displaystyle a_n(t)}$ and its complex conjugate ${\displaystyle a_n^{\ast }(t)}$. With this choice of independent variables, we can calculate the partial derivative $${\displaystyle \frac{\mathrm{\partial }⟨H⟩}{\mathrm{\partial }{a}_{n^{\prime }}^{\ast }}=\sum _n{a}_n⟨n^{\prime }|H|n⟩=⟨n^{\prime }|H|\psi ⟩}$$

By applying the Schrödinger equation and using the orthonormality of the basis states, this further reduces to $${\displaystyle \frac{\mathrm{\partial }⟨H⟩}{\mathrm{\partial }{a}_{n^{\prime }}^{\ast }}=i\hslash \frac{\mathrm{\partial }{a}_{n^{\prime }}}{\mathrm{\partial }t}}$$

Similarly, one can show that $${\displaystyle \frac{\mathrm{\partial }⟨H⟩}{\mathrm{\partial }{a}_n}=-i\hslash \frac{\mathrm{\partial }{a}_n^{\ast }}{\mathrm{\partial }t}}$$

If we define "conjugate momentum" variables ${\displaystyle {\pi }_n}$ by $${\displaystyle {\pi }_n(t)=i\hslash a_n^{\ast }(t)}$$ then the above equations become $${\displaystyle \frac{\mathrm{\partial }⟨H⟩}{\mathrm{\partial }{\pi }_n}=\frac{\mathrm{\partial }{a}_n}{\mathrm{\partial }t},\frac{\mathrm{\partial }⟨H⟩}{\mathrm{\partial }{a}_n}=-\frac{\mathrm{\partial }{\pi }_n}{\mathrm{\partial }t}}$$ which is precisely the form of Hamilton's equations, with the ${\displaystyle a_n}$s as the generalized coordinates, the ${\displaystyle {\pi }_n}$s as the conjugate momenta, and ${\displaystyle ⟨H⟩}$ taking the place of the classical Hamiltonian.