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9,400 | Several artificial nucleobases have been synthesized, and successfully incorporated in the eight-base DNA analogue named Hachimoji DNA. Dubbed S, B, P, and Z, these artificial bases are capable of bonding with each other in a predictable way (S–B and P–Z), maintain the double helix structure of DNA, and be transcribed ... | https://en.wikipedia.org/wiki?curid=7955 |
9,401 | The phosphate groups of DNA give it similar acidic properties to phosphoric acid and it can be considered as a strong acid. It will be fully ionized at a normal cellular pH, releasing protons which leave behind negative charges on the phosphate groups. These negative charges protect DNA from breakdown by hydrolysis by ... | https://en.wikipedia.org/wiki?curid=7955 |
9,402 | The expression of genes is influenced by how the DNA is packaged in chromosomes, in a structure called chromatin. Base modifications can be involved in packaging, with regions that have low or no gene expression usually containing high levels of methylation of cytosine bases. DNA packaging and its influence on gene exp... | https://en.wikipedia.org/wiki?curid=7955 |
9,403 | For one example, cytosine methylation produces 5-methylcytosine, which is important for X-inactivation of chromosomes. The average level of methylation varies between organisms—the worm "Caenorhabditis elegans" lacks cytosine methylation, while vertebrates have higher levels, with up to 1% of their DNA containing 5-met... | https://en.wikipedia.org/wiki?curid=7955 |
9,404 | DNA can be damaged by many sorts of mutagens, which change the DNA sequence. Mutagens include oxidizing agents, alkylating agents and also high-energy electromagnetic radiation such as ultraviolet light and X-rays. The type of DNA damage produced depends on the type of mutagen. For example, UV light can damage DNA by p... | https://en.wikipedia.org/wiki?curid=7955 |
9,405 | Many mutagens fit into the space between two adjacent base pairs, this is called "intercalation". Most intercalators are aromatic and planar molecules; examples include ethidium bromide, acridines, daunomycin, and doxorubicin. For an intercalator to fit between base pairs, the bases must separate, distorting the DNA st... | https://en.wikipedia.org/wiki?curid=7955 |
9,406 | DNA usually occurs as linear chromosomes in eukaryotes, and circular chromosomes in prokaryotes. The set of chromosomes in a cell makes up its genome; the human genome has approximately 3 billion base pairs of DNA arranged into 46 chromosomes. The information carried by DNA is held in the sequence of pieces of DNA call... | https://en.wikipedia.org/wiki?curid=7955 |
9,407 | Genomic DNA is tightly and orderly packed in the process called DNA condensation, to fit the small available volumes of the cell. In eukaryotes, DNA is located in the cell nucleus, with small amounts in mitochondria and chloroplasts. In prokaryotes, the DNA is held within an irregularly shaped body in the cytoplasm cal... | https://en.wikipedia.org/wiki?curid=7955 |
9,408 | In many species, only a small fraction of the total sequence of the genome encodes protein. For example, only about 1.5% of the human genome consists of protein-coding exons, with over 50% of human DNA consisting of non-coding repetitive sequences. The reasons for the presence of so much noncoding DNA in eukaryotic gen... | https://en.wikipedia.org/wiki?curid=7955 |
9,409 | Some noncoding DNA sequences play structural roles in chromosomes. Telomeres and centromeres typically contain few genes but are important for the function and stability of chromosomes. An abundant form of noncoding DNA in humans are pseudogenes, which are copies of genes that have been disabled by mutation. These sequ... | https://en.wikipedia.org/wiki?curid=7955 |
9,410 | A gene is a sequence of DNA that contains genetic information and can influence the phenotype of an organism. Within a gene, the sequence of bases along a DNA strand defines a messenger RNA sequence, which then defines one or more protein sequences. The relationship between the nucleotide sequences of genes and the ami... | https://en.wikipedia.org/wiki?curid=7955 |
9,411 | In transcription, the codons of a gene are copied into messenger RNA by RNA polymerase. This RNA copy is then decoded by a ribosome that reads the RNA sequence by base-pairing the messenger RNA to transfer RNA, which carries amino acids. Since there are 4 bases in 3-letter combinations, there are 64 possible codons (4 ... | https://en.wikipedia.org/wiki?curid=7955 |
9,412 | Cell division is essential for an organism to grow, but, when a cell divides, it must replicate the DNA in its genome so that the two daughter cells have the same genetic information as their parent. The double-stranded structure of DNA provides a simple mechanism for DNA replication. Here, the two strands are separate... | https://en.wikipedia.org/wiki?curid=7955 |
9,413 | Naked extracellular DNA (eDNA), most of it released by cell death, is nearly ubiquitous in the environment. Its concentration in soil may be as high as 2 μg/L, and its concentration in natural aquatic environments may be as high at 88 μg/L. Various possible functions have been proposed for eDNA: it may be involved in h... | https://en.wikipedia.org/wiki?curid=7955 |
9,414 | Cell-free fetal DNA is found in the blood of the mother, and can be sequenced to determine a great deal of information about the developing fetus. | https://en.wikipedia.org/wiki?curid=7955 |
9,415 | Under the name of environmental DNA eDNA has seen increased use in the natural sciences as a survey tool for ecology, monitoring the movements and presence of species in water, air, or on land, and assessing an area's biodiversity. | https://en.wikipedia.org/wiki?curid=7955 |
9,416 | Neutrophil extracellular traps (NETs) are networks of extracellular fibers, primarily composed of DNA, which allow neutrophils, a type of white blood cell, to kill extracellular pathogens while minimizing damage to the host cells. | https://en.wikipedia.org/wiki?curid=7955 |
9,417 | All the functions of DNA depend on interactions with proteins. These protein interactions can be non-specific, or the protein can bind specifically to a single DNA sequence. Enzymes can also bind to DNA and of these, the polymerases that copy the DNA base sequence in transcription and DNA replication are particularly i... | https://en.wikipedia.org/wiki?curid=7955 |
9,418 | Structural proteins that bind DNA are well-understood examples of non-specific DNA-protein interactions. Within chromosomes, DNA is held in complexes with structural proteins. These proteins organize the DNA into a compact structure called chromatin. In eukaryotes, this structure involves DNA binding to a complex of sm... | https://en.wikipedia.org/wiki?curid=7955 |
9,419 | A distinct group of DNA-binding proteins is the DNA-binding proteins that specifically bind single-stranded DNA. In humans, replication protein A is the best-understood member of this family and is used in processes where the double helix is separated, including DNA replication, recombination, and DNA repair. These bin... | https://en.wikipedia.org/wiki?curid=7955 |
9,420 | In contrast, other proteins have evolved to bind to particular DNA sequences. The most intensively studied of these are the various transcription factors, which are proteins that regulate transcription. Each transcription factor binds to one particular set of DNA sequences and activates or inhibits the transcription of... | https://en.wikipedia.org/wiki?curid=7955 |
9,421 | As these DNA targets can occur throughout an organism's genome, changes in the activity of one type of transcription factor can affect thousands of genes. Consequently, these proteins are often the targets of the signal transduction processes that control responses to environmental changes or cellular differentiation a... | https://en.wikipedia.org/wiki?curid=7955 |
9,422 | Nucleases are enzymes that cut DNA strands by catalyzing the hydrolysis of the phosphodiester bonds. Nucleases that hydrolyse nucleotides from the ends of DNA strands are called exonucleases, while endonucleases cut within strands. The most frequently used nucleases in molecular biology are the restriction endonuclease... | https://en.wikipedia.org/wiki?curid=7955 |
9,423 | Enzymes called DNA ligases can rejoin cut or broken DNA strands. Ligases are particularly important in lagging strand DNA replication, as they join the short segments of DNA produced at the replication fork into a complete copy of the DNA template. They are also used in DNA repair and genetic recombination. | https://en.wikipedia.org/wiki?curid=7955 |
9,424 | Topoisomerases are enzymes with both nuclease and ligase activity. These proteins change the amount of supercoiling in DNA. Some of these enzymes work by cutting the DNA helix and allowing one section to rotate, thereby reducing its level of supercoiling; the enzyme then seals the DNA break. Other types of these enzyme... | https://en.wikipedia.org/wiki?curid=7955 |
9,425 | Helicases are proteins that are a type of molecular motor. They use the chemical energy in nucleoside triphosphates, predominantly adenosine triphosphate (ATP), to break hydrogen bonds between bases and unwind the DNA double helix into single strands. These enzymes are essential for most processes where enzymes need to... | https://en.wikipedia.org/wiki?curid=7955 |
9,426 | Polymerases are enzymes that synthesize polynucleotide chains from nucleoside triphosphates. The sequence of their products is created based on existing polynucleotide chains—which are called "templates". These enzymes function by repeatedly adding a nucleotide to the 3′ hydroxyl group at the end of the growing polynuc... | https://en.wikipedia.org/wiki?curid=7955 |
9,427 | In DNA replication, DNA-dependent DNA polymerases make copies of DNA polynucleotide chains. To preserve biological information, it is essential that the sequence of bases in each copy are precisely complementary to the sequence of bases in the template strand. Many DNA polymerases have a proofreading activity. Here, th... | https://en.wikipedia.org/wiki?curid=7955 |
9,428 | RNA-dependent DNA polymerases are a specialized class of polymerases that copy the sequence of an RNA strand into DNA. They include reverse transcriptase, which is a viral enzyme involved in the infection of cells by retroviruses, and telomerase, which is required for the replication of telomeres. For example, HIV reve... | https://en.wikipedia.org/wiki?curid=7955 |
9,429 | Transcription is carried out by a DNA-dependent RNA polymerase that copies the sequence of a DNA strand into RNA. To begin transcribing a gene, the RNA polymerase binds to a sequence of DNA called a promoter and separates the DNA strands. It then copies the gene sequence into a messenger RNA transcript until it reaches... | https://en.wikipedia.org/wiki?curid=7955 |
9,430 | A DNA helix usually does not interact with other segments of DNA, and in human cells, the different chromosomes even occupy separate areas in the nucleus called "chromosome territories". This physical separation of different chromosomes is important for the ability of DNA to function as a stable repository for informat... | https://en.wikipedia.org/wiki?curid=7955 |
9,431 | Recombination allows chromosomes to exchange genetic information and produces new combinations of genes, which increases the efficiency of natural selection and can be important in the rapid evolution of new proteins. Genetic recombination can also be involved in DNA repair, particularly in the cell's response to doubl... | https://en.wikipedia.org/wiki?curid=7955 |
9,432 | The most common form of chromosomal crossover is homologous recombination, where the two chromosomes involved share very similar sequences. Non-homologous recombination can be damaging to cells, as it can produce chromosomal translocations and genetic abnormalities. The recombination reaction is catalyzed by enzymes kn... | https://en.wikipedia.org/wiki?curid=7955 |
9,433 | DNA contains the genetic information that allows all forms of life to function, grow and reproduce. However, it is unclear how long in the 4-billion-year history of life DNA has performed this function, as it has been proposed that the earliest forms of life may have used RNA as their genetic material. RNA may have act... | https://en.wikipedia.org/wiki?curid=7955 |
9,434 | Building blocks of DNA (adenine, guanine, and related organic molecules) may have been formed extraterrestrially in outer space. Complex DNA and RNA organic compounds of life, including uracil, cytosine, and thymine, have also been formed in the laboratory under conditions mimicking those found in outer space, using st... | https://en.wikipedia.org/wiki?curid=7955 |
9,435 | In February 2021, scientists reported, for the first time, the sequencing of DNA from animal remains, a mammoth in this instance over a million years old, the oldest DNA sequenced to date. | https://en.wikipedia.org/wiki?curid=7955 |
9,436 | Methods have been developed to purify DNA from organisms, such as phenol-chloroform extraction, and to manipulate it in the laboratory, such as restriction digests and the polymerase chain reaction. Modern biology and biochemistry make intensive use of these techniques in recombinant DNA technology. Recombinant DNA is ... | https://en.wikipedia.org/wiki?curid=7955 |
9,437 | Forensic scientists can use DNA in blood, semen, skin, saliva or hair found at a crime scene to identify a matching DNA of an individual, such as a perpetrator. This process is formally termed DNA profiling, also called "DNA fingerprinting". In DNA profiling, the lengths of variable sections of repetitive DNA, such as ... | https://en.wikipedia.org/wiki?curid=7955 |
9,438 | The development of forensic science and the ability to now obtain genetic matching on minute samples of blood, skin, saliva, or hair has led to re-examining many cases. Evidence can now be uncovered that was scientifically impossible at the time of the original examination. Combined with the removal of the double jeopa... | https://en.wikipedia.org/wiki?curid=7955 |
9,439 | DNA profiling is also used successfully to positively identify victims of mass casualty incidents, bodies or body parts in serious accidents, and individual victims in mass war graves, via matching to family members. | https://en.wikipedia.org/wiki?curid=7955 |
9,440 | DNA profiling is also used in DNA paternity testing to determine if someone is the biological parent or grandparent of a child with the probability of parentage is typically 99.99% when the alleged parent is biologically related to the child. Normal DNA sequencing methods happen after birth, but there are new methods t... | https://en.wikipedia.org/wiki?curid=7955 |
9,441 | Deoxyribozymes, also called DNAzymes or catalytic DNA, were first discovered in 1994. They are mostly single stranded DNA sequences isolated from a large pool of random DNA sequences through a combinatorial approach called in vitro selection or systematic evolution of ligands by exponential enrichment (SELEX). DNAzymes... | https://en.wikipedia.org/wiki?curid=7955 |
9,442 | Bioinformatics involves the development of techniques to store, data mine, search and manipulate biological data, including DNA nucleic acid sequence data. These have led to widely applied advances in computer science, especially string searching algorithms, machine learning, and database theory. String searching or ma... | https://en.wikipedia.org/wiki?curid=7955 |
9,443 | DNA nanotechnology uses the unique molecular recognition properties of DNA and other nucleic acids to create self-assembling branched DNA complexes with useful properties. DNA is thus used as a structural material rather than as a carrier of biological information. This has led to the creation of two-dimensional period... | https://en.wikipedia.org/wiki?curid=7955 |
9,444 | Because DNA collects mutations over time, which are then inherited, it contains historical information, and, by comparing DNA sequences, geneticists can infer the evolutionary history of organisms, their phylogeny. This field of phylogenetics is a powerful tool in evolutionary biology. If DNA sequences within a species... | https://en.wikipedia.org/wiki?curid=7955 |
9,445 | DNA as a storage device for information has enormous potential since it has much higher storage density compared to electronic devices. However, high costs, slow read and write times (memory latency), and insufficient reliability has prevented its practical use. | https://en.wikipedia.org/wiki?curid=7955 |
9,446 | DNA was first isolated by the Swiss physician Friedrich Miescher who, in 1869, discovered a microscopic substance in the pus of discarded surgical bandages. As it resided in the nuclei of cells, he called it "nuclein". In 1878, Albrecht Kossel isolated the non-protein component of "nuclein", nucleic acid, and later iso... | https://en.wikipedia.org/wiki?curid=7955 |
9,447 | In 1909, Phoebus Levene identified the base, sugar, and phosphate nucleotide unit of the RNA (then named "yeast nucleic acid"). In 1929, Levene identified deoxyribose sugar in "thymus nucleic acid" (DNA). Levene suggested that DNA consisted of a string of four nucleotide units linked together through the phosphate grou... | https://en.wikipedia.org/wiki?curid=7955 |
9,448 | In 1933, while studying virgin sea urchin eggs, Jean Brachet suggested that DNA is found in the cell nucleus and that RNA is present exclusively in the cytoplasm. At the time, "yeast nucleic acid" (RNA) was thought to occur only in plants, while "thymus nucleic acid" (DNA) only in animals. The latter was thought to be ... | https://en.wikipedia.org/wiki?curid=7955 |
9,449 | In 1937, William Astbury produced the first X-ray diffraction patterns that showed that DNA had a regular structure. | https://en.wikipedia.org/wiki?curid=7955 |
9,450 | In 1943, Oswald Avery, along with co-workers Colin MacLeod and Maclyn McCarty, identified DNA as the transforming principle, supporting Griffith's suggestion (Avery–MacLeod–McCarty experiment). Erwin Chargaff developed and published observations now known as Chargaff's rules, stating that in DNA from any species of any... | https://en.wikipedia.org/wiki?curid=7955 |
9,451 | In May 1952, Raymond Gosling, a graduate student working under the supervision of Rosalind Franklin, took an X-ray diffraction image, labeled as "Photo 51", at high hydration levels of DNA. This photo was given to Watson and Crick by Maurice Wilkins and was critical to their obtaining the correct structure of DNA. Fran... | https://en.wikipedia.org/wiki?curid=7955 |
9,452 | The 25 April 1953 issue of the journal "Nature" published a series of five articles giving the Watson and Crick double-helix structure DNA and evidence supporting it. The structure was reported in a letter titled ""MOLECULAR STRUCTURE OF NUCLEIC ACIDS A Structure for Deoxyribose Nucleic Acid", in which they said, "It h... | https://en.wikipedia.org/wiki?curid=7955 |
9,453 | In 1962, after Franklin's death, Watson, Crick, and Wilkins jointly received the Nobel Prize in Physiology or Medicine. Nobel Prizes are awarded only to living recipients. A debate continues about who should receive credit for the discovery. | https://en.wikipedia.org/wiki?curid=7955 |
9,454 | In an influential presentation in 1957, Crick laid out the central dogma of molecular biology, which foretold the relationship between DNA, RNA, and proteins, and articulated the "adaptor hypothesis". Final confirmation of the replication mechanism that was implied by the double-helical structure followed in 1958 throu... | https://en.wikipedia.org/wiki?curid=7955 |
9,455 | Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, quantum field theory, quantum technology, and quantum information science. | https://en.wikipedia.org/wiki?curid=25202 |
9,456 | Classical physics, the collection of theories that existed before the advent of quantum mechanics, describes many aspects of nature at an ordinary (macroscopic) scale, but is not sufficient for describing them at small (atomic and subatomic) scales. Most theories in classical physics can be derived from quantum mechani... | https://en.wikipedia.org/wiki?curid=25202 |
9,457 | Quantum mechanics differs from classical physics in that energy, momentum, angular momentum, and other quantities of a bound system are restricted to discrete values (quantization); objects have characteristics of both particles and waves (wave–particle duality); and there are limits to how accurately the value of a ph... | https://en.wikipedia.org/wiki?curid=25202 |
9,458 | Quantum mechanics arose gradually from theories to explain observations which could not be reconciled with classical physics, such as Max Planck's solution in 1900 to the black-body radiation problem, and the correspondence between energy and frequency in Albert Einstein's 1905 paper which explained the photoelectric e... | https://en.wikipedia.org/wiki?curid=25202 |
9,459 | Quantum mechanics allows the calculation of properties and behaviour of physical systems. It is typically applied to microscopic systems: molecules, atoms and sub-atomic particles. It has been demonstrated to hold for complex molecules with thousands of atoms, but its application to human beings raises philosophical pr... | https://en.wikipedia.org/wiki?curid=25202 |
9,460 | A fundamental feature of the theory is that it usually cannot predict with certainty what will happen, but only give probabilities. Mathematically, a probability is found by taking the square of the absolute value of a complex number, known as a probability amplitude. This is known as the Born rule, named after physici... | https://en.wikipedia.org/wiki?curid=25202 |
9,461 | One consequence of the mathematical rules of quantum mechanics is a tradeoff in predictability between different measurable quantities. The most famous form of this uncertainty principle says that no matter how a quantum particle is prepared or how carefully experiments upon it are arranged, it is impossible to have a ... | https://en.wikipedia.org/wiki?curid=25202 |
9,462 | Another consequence of the mathematical rules of quantum mechanics is the phenomenon of quantum interference, which is often illustrated with the double-slit experiment. In the basic version of this experiment, a coherent light source, such as a laser beam, illuminates a plate pierced by two parallel slits, and the lig... | https://en.wikipedia.org/wiki?curid=25202 |
9,463 | Another counter-intuitive phenomenon predicted by quantum mechanics is quantum tunnelling: a particle that goes up against a potential barrier can cross it, even if its kinetic energy is smaller than the maximum of the potential. In classical mechanics this particle would be trapped. Quantum tunnelling has several impo... | https://en.wikipedia.org/wiki?curid=25202 |
9,464 | When quantum systems interact, the result can be the creation of quantum entanglement: their properties become so intertwined that a description of the whole solely in terms of the individual parts is no longer possible. Erwin Schrödinger called entanglement "..."the" characteristic trait of quantum mechanics, the one ... | https://en.wikipedia.org/wiki?curid=25202 |
9,465 | Another possibility opened by entanglement is testing for "hidden variables", hypothetical properties more fundamental than the quantities addressed in quantum theory itself, knowledge of which would allow more exact predictions than quantum theory can provide. A collection of results, most significantly Bell's theorem... | https://en.wikipedia.org/wiki?curid=25202 |
9,466 | It is not possible to present these concepts in more than a superficial way without introducing the actual mathematics involved; understanding quantum mechanics requires not only manipulating complex numbers, but also linear algebra, differential equations, group theory, and other more advanced subjects. Accordingly, t... | https://en.wikipedia.org/wiki?curid=25202 |
9,467 | In the mathematically rigorous formulation of quantum mechanics, the state of a quantum mechanical system is a vector formula_1 belonging to a (separable) complex Hilbert space formula_2. This vector is postulated to be normalized under the Hilbert space inner product, that is, it obeys formula_3, and it is well-define... | https://en.wikipedia.org/wiki?curid=25202 |
9,468 | Physical quantities of interestposition, momentum, energy, spinare represented by observables, which are Hermitian (more precisely, self-adjoint) linear operators acting on the Hilbert space. A quantum state can be an eigenvector of an observable, in which case it is called an eigenstate, and the associated eigenvalue ... | https://en.wikipedia.org/wiki?curid=25202 |
9,469 | After the measurement, if result formula_8 was obtained, the quantum state is postulated to collapse to formula_10, in the non-degenerate case, or to formula_15, in the general case. The probabilistic nature of quantum mechanics thus stems from the act of measurement. This is one of the most difficult aspects of quantu... | https://en.wikipedia.org/wiki?curid=25202 |
9,470 | Here formula_17 denotes the Hamiltonian, the observable corresponding to the total energy of the system, and formula_18 is the reduced Planck constant. The constant formula_19 is introduced so that the Hamiltonian is reduced to the classical Hamiltonian in cases where the quantum system can be approximated by a classic... | https://en.wikipedia.org/wiki?curid=25202 |
9,471 | The operator formula_21 is known as the time-evolution operator, and has the crucial property that it is unitary. This time evolution is deterministic in the sense that – given an initial quantum state formula_22 – it makes a definite prediction of what the quantum state formula_23 will be at any later time. | https://en.wikipedia.org/wiki?curid=25202 |
9,472 | Some wave functions produce probability distributions that are independent of time, such as eigenstates of the Hamiltonian. Many systems that are treated dynamically in classical mechanics are described by such "static" wave functions. For example, a single electron in an unexcited atom is pictured classically as a par... | https://en.wikipedia.org/wiki?curid=25202 |
9,473 | Analytic solutions of the Schrödinger equation are known for very few relatively simple model Hamiltonians including the quantum harmonic oscillator, the particle in a box, the dihydrogen cation, and the hydrogen atom. Even the helium atom – which contains just two electrons – has defied all attempts at a fully analyti... | https://en.wikipedia.org/wiki?curid=25202 |
9,474 | However, there are techniques for finding approximate solutions. One method, called perturbation theory, uses the analytic result for a simple quantum mechanical model to create a result for a related but more complicated model by (for example) the addition of a weak potential energy. Another method is called "semi-cla... | https://en.wikipedia.org/wiki?curid=25202 |
9,475 | One consequence of the basic quantum formalism is the uncertainty principle. In its most familiar form, this states that no preparation of a quantum particle can imply simultaneously precise predictions both for a measurement of its position and for a measurement of its momentum. Both position and momentum are observab... | https://en.wikipedia.org/wiki?curid=25202 |
9,476 | Given a quantum state, the Born rule lets us compute expectation values for both formula_27 and formula_28, and moreover for powers of them. Defining | https://en.wikipedia.org/wiki?curid=25202 |
9,477 | Either standard deviation can in principle be made arbitrarily small, but not both simultaneously. This inequality generalizes to arbitrary pairs of self-adjoint operators formula_32 and formula_33. The commutator of these two operators is | https://en.wikipedia.org/wiki?curid=25202 |
9,478 | Another consequence of the canonical commutation relation is that the position and momentum operators are Fourier transforms of each other, so that a description of an object according to its momentum is the Fourier transform of its description according to its position. The fact that dependence in momentum is the Four... | https://en.wikipedia.org/wiki?curid=25202 |
9,479 | When two different quantum systems are considered together, the Hilbert space of the combined system is the tensor product of the Hilbert spaces of the two components. For example, let and be two quantum systems, with Hilbert spaces formula_40 and formula_41, respectively. The Hilbert space of the composite system is t... | https://en.wikipedia.org/wiki?curid=25202 |
9,480 | If the state for the first system is the vector formula_43 and the state for the second system is formula_44, then the state of the composite system is | https://en.wikipedia.org/wiki?curid=25202 |
9,481 | Not all states in the joint Hilbert space formula_46 can be written in this form, however, because the superposition principle implies that linear combinations of these "separable" or "product states" are also valid. For example, if formula_43 and formula_48 are both possible states for system formula_32, and likewise ... | https://en.wikipedia.org/wiki?curid=25202 |
9,482 | If the state for a composite system is entangled, it is impossible to describe either component system or system by a state vector. One can instead define reduced density matrices that describe the statistics that can be obtained by making measurements on either component system alone. This necessarily causes a loss of... | https://en.wikipedia.org/wiki?curid=25202 |
9,483 | As described above, entanglement is a key feature of models of measurement processes in which an apparatus becomes entangled with the system being measured. Systems interacting with the environment in which they reside generally become entangled with that environment, a phenomenon known as quantum decoherence. This can... | https://en.wikipedia.org/wiki?curid=25202 |
9,484 | There are many mathematically equivalent formulations of quantum mechanics. One of the oldest and most common is the "transformation theory" proposed by Paul Dirac, which unifies and generalizes the two earliest formulations of quantum mechanics – matrix mechanics (invented by Werner Heisenberg) and wave mechanics (inv... | https://en.wikipedia.org/wiki?curid=25202 |
9,485 | The Hamiltonian formula_17 is known as the "generator" of time evolution, since it defines a unitary time-evolution operator formula_55 for each value of formula_56. From this relation between formula_57 and formula_17, it follows that any observable formula_32 that commutes with formula_17 will be "conserved": its exp... | https://en.wikipedia.org/wiki?curid=25202 |
9,486 | The simplest example of a quantum system with a position degree of freedom is a free particle in a single spatial dimension. A free particle is one which is not subject to external influences, so that its Hamiltonian consists only of its kinetic energy: | https://en.wikipedia.org/wiki?curid=25202 |
9,487 | which is a superposition of all possible plane waves formula_72, which are eigenstates of the momentum operator with momentum formula_73. The coefficients of the superposition are formula_74, which is the Fourier transform of the initial quantum state formula_75. | https://en.wikipedia.org/wiki?curid=25202 |
9,488 | It is not possible for the solution to be a single momentum eigenstate, or a single position eigenstate, as these are not normalizable quantum states. Instead, we can consider a Gaussian wave packet: | https://en.wikipedia.org/wiki?curid=25202 |
9,489 | We see that as we make formula_78 smaller the spread in position gets smaller, but the spread in momentum gets larger. Conversely, by making formula_78 larger we make the spread in momentum smaller, but the spread in position gets larger. This illustrates the uncertainty principle. | https://en.wikipedia.org/wiki?curid=25202 |
9,490 | As we let the Gaussian wave packet evolve in time, we see that its center moves through space at a constant velocity (like a classical particle with no forces acting on it). However, the wave packet will also spread out as time progresses, which means that the position becomes more and more uncertain. The uncertainty i... | https://en.wikipedia.org/wiki?curid=25202 |
9,491 | The particle in a one-dimensional potential energy box is the most mathematically simple example where restraints lead to the quantization of energy levels. The box is defined as having zero potential energy everywhere "inside" a certain region, and therefore infinite potential energy everywhere "outside" that region. ... | https://en.wikipedia.org/wiki?curid=25202 |
9,492 | with state formula_1 in this case having energy formula_85 coincident with the kinetic energy of the particle. | https://en.wikipedia.org/wiki?curid=25202 |
9,493 | The infinite potential walls of the box determine the values of formula_88 and formula_89 at formula_90 and formula_91 where formula_1 must be zero. Thus, at formula_90, | https://en.wikipedia.org/wiki?curid=25202 |
9,494 | in which formula_98 cannot be zero as this would conflict with the postulate that formula_1 has norm 1. Therefore, since formula_100, formula_101 must be an integer multiple of formula_102, | https://en.wikipedia.org/wiki?curid=25202 |
9,495 | A finite potential well is the generalization of the infinite potential well problem to potential wells having finite depth. The finite potential well problem is mathematically more complicated than the infinite particle-in-a-box problem as the wave function is not pinned to zero at the walls of the well. Instead, the ... | https://en.wikipedia.org/wiki?curid=25202 |
9,496 | This problem can either be treated by directly solving the Schrödinger equation, which is not trivial, or by using the more elegant "ladder method" first proposed by Paul Dirac. The eigenstates are given by | https://en.wikipedia.org/wiki?curid=25202 |
9,497 | The Mach–Zehnder interferometer (MZI) illustrates the concepts of superposition and interference with linear algebra in dimension 2, rather than differential equations. It can be seen as a simplified version of the double-slit experiment, but it is of interest in its own right, for example in the delayed choice quantum... | https://en.wikipedia.org/wiki?curid=25202 |
9,498 | We can model a photon going through the interferometer by considering that at each point it can be in a superposition of only two paths: the "lower" path which starts from the left, goes straight through both beam splitters, and ends at the top, and the "upper" path which starts from the bottom, goes straight through b... | https://en.wikipedia.org/wiki?curid=25202 |
9,499 | Both beam splitters are modelled as the unitary matrix formula_118, which means that when a photon meets the beam splitter it will either stay on the same path with a probability amplitude of formula_119, or be reflected to the other path with a probability amplitude of formula_120. The phase shifter on the upper arm i... | https://en.wikipedia.org/wiki?curid=25202 |
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