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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 to RNA. Their existence could be seen as an indication that there is nothing special about the four natural nucleobases that evolved on Earth. On the other hand, DNA is tightly related to RNA which does not only act as a transcript of DNA but also performs as moleular machines many tasks in cells. For this purpose it has to fold into a structure. It has been shown that to allow to create all possible structures at least four bases are required for the corresponding RNA, while a higher number is also possible but this would be against the natural principle of least effort. | 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 repelling nucleophiles which could hydrolyze it. | 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 expression can also occur by covalent modifications of the histone protein core around which DNA is wrapped in the chromatin structure or else by remodeling carried out by chromatin remodeling complexes (see Chromatin remodeling). There is, further, crosstalk between DNA methylation and histone modification, so they can coordinately affect chromatin and gene expression. | 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-methylcytosine. Despite the importance of 5-methylcytosine, it can deaminate to leave a thymine base, so methylated cytosines are particularly prone to mutations. Other base modifications include adenine methylation in bacteria, the presence of 5-hydroxymethylcytosine in the brain, and the glycosylation of uracil to produce the "J-base" in kinetoplastids. | 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 producing thymine dimers, which are cross-links between pyrimidine bases. On the other hand, oxidants such as free radicals or hydrogen peroxide produce multiple forms of damage, including base modifications, particularly of guanosine, and double-strand breaks. A typical human cell contains about 150,000 bases that have suffered oxidative damage. Of these oxidative lesions, the most dangerous are double-strand breaks, as these are difficult to repair and can produce point mutations, insertions, deletions from the DNA sequence, and chromosomal translocations. These mutations can cause cancer. Because of inherent limits in the DNA repair mechanisms, if humans lived long enough, they would all eventually develop cancer. DNA damages that are naturally occurring, due to normal cellular processes that produce reactive oxygen species, the hydrolytic activities of cellular water, etc., also occur frequently. Although most of these damages are repaired, in any cell some DNA damage may remain despite the action of repair processes. These remaining DNA damages accumulate with age in mammalian postmitotic tissues. This accumulation appears to be an important underlying cause of aging. | 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 strands by unwinding of the double helix. This inhibits both transcription and DNA replication, causing toxicity and mutations. As a result, DNA intercalators may be carcinogens, and in the case of thalidomide, a teratogen. Others such as benzo["a"]pyrene diol epoxide and aflatoxin form DNA adducts that induce errors in replication. Nevertheless, due to their ability to inhibit DNA transcription and replication, other similar toxins are also used in chemotherapy to inhibit rapidly growing cancer cells. | 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 called genes. Transmission of genetic information in genes is achieved via complementary base pairing. For example, in transcription, when a cell uses the information in a gene, the DNA sequence is copied into a complementary RNA sequence through the attraction between the DNA and the correct RNA nucleotides. Usually, this RNA copy is then used to make a matching protein sequence in a process called translation, which depends on the same interaction between RNA nucleotides. In an alternative fashion, a cell may copy its genetic information in a process called DNA replication. The details of these functions are covered in other articles; here the focus is on the interactions between DNA and other molecules that mediate the function of the genome. | 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 called the nucleoid. The genetic information in a genome is held within genes, and the complete set of this information in an organism is called its genotype. A gene is a unit of heredity and is a region of DNA that influences a particular characteristic in an organism. Genes contain an open reading frame that can be transcribed, and regulatory sequences such as promoters and enhancers, which control transcription of the open reading frame. | 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 genomes and the extraordinary differences in genome size, or "C-value", among species, represent a long-standing puzzle known as the "C-value enigma". However, some DNA sequences that do not code protein may still encode functional non-coding RNA molecules, which are involved in the regulation of gene expression. | 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 sequences are usually just molecular fossils, although they can occasionally serve as raw genetic material for the creation of new genes through the process of gene duplication and divergence. | 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 amino-acid sequences of proteins is determined by the rules of translation, known collectively as the genetic code. The genetic code consists of three-letter 'words' called "codons" formed from a sequence of three nucleotides (e.g. ACT, CAG, TTT). | 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 combinations). These encode the twenty standard amino acids, giving most amino acids more than one possible codon. There are also three 'stop' or 'nonsense' codons signifying the end of the coding region; these are the TAG, TAA, and TGA codons, (UAG, UAA, and UGA on the mRNA). | 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 separated and then each strand's complementary DNA sequence is recreated by an enzyme called DNA polymerase. This enzyme makes the complementary strand by finding the correct base through complementary base pairing and bonding it onto the original strand. As DNA polymerases can only extend a DNA strand in a 5′ to 3′ direction, different mechanisms are used to copy the antiparallel strands of the double helix. In this way, the base on the old strand dictates which base appears on the new strand, and the cell ends up with a perfect copy of its DNA. | 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 horizontal gene transfer; it may provide nutrients; and it may act as a buffer to recruit or titrate ions or antibiotics. Extracellular DNA acts as a functional extracellular matrix component in the biofilms of several bacterial species. It may act as a recognition factor to regulate the attachment and dispersal of specific cell types in the biofilm; it may contribute to biofilm formation; and it may contribute to the biofilm's physical strength and resistance to biological stress. | 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 important. | 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 small basic proteins called histones, while in prokaryotes multiple types of proteins are involved. The histones form a disk-shaped complex called a nucleosome, which contains two complete turns of double-stranded DNA wrapped around its surface. These non-specific interactions are formed through basic residues in the histones, making ionic bonds to the acidic sugar-phosphate backbone of the DNA, and are thus largely independent of the base sequence. Chemical modifications of these basic amino acid residues include methylation, phosphorylation, and acetylation. These chemical changes alter the strength of the interaction between the DNA and the histones, making the DNA more or less accessible to transcription factors and changing the rate of transcription. Other non-specific DNA-binding proteins in chromatin include the high-mobility group proteins, which bind to bent or distorted DNA. These proteins are important in bending arrays of nucleosomes and arranging them into the larger structures that make up chromosomes. | 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 binding proteins seem to stabilize single-stranded DNA and protect it from forming stem-loops or being degraded by nucleases. | 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 genes that have these sequences close to their promoters. The transcription factors do this in two ways. Firstly, they can bind the RNA polymerase responsible for transcription, either directly or through other mediator proteins; this locates the polymerase at the promoter and allows it to begin transcription. Alternatively, transcription factors can bind enzymes that modify the histones at the promoter. This changes the accessibility of the DNA template to the polymerase. | 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 and development. The specificity of these transcription factors' interactions with DNA come from the proteins making multiple contacts to the edges of the DNA bases, allowing them to "read" the DNA sequence. Most of these base-interactions are made in the major groove, where the bases are most accessible. | 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 endonucleases, which cut DNA at specific sequences. For instance, the EcoRV enzyme shown to the left recognizes the 6-base sequence 5′-GATATC-3′ and makes a cut at the horizontal line. In nature, these enzymes protect bacteria against phage infection by digesting the phage DNA when it enters the bacterial cell, acting as part of the restriction modification system. In technology, these sequence-specific nucleases are used in molecular cloning and DNA fingerprinting. | 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 enzymes are capable of cutting one DNA helix and then passing a second strand of DNA through this break, before rejoining the helix. Topoisomerases are required for many processes involving DNA, such as DNA replication and transcription. | 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 access the DNA bases. | 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 polynucleotide chain. As a consequence, all polymerases work in a 5′ to 3′ direction. In the active site of these enzymes, the incoming nucleoside triphosphate base-pairs to the template: this allows polymerases to accurately synthesize the complementary strand of their template. Polymerases are classified according to the type of template that they use. | 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, the polymerase recognizes the occasional mistakes in the synthesis reaction by the lack of base pairing between the mismatched nucleotides. If a mismatch is detected, a 3′ to 5′ exonuclease activity is activated and the incorrect base removed. In most organisms, DNA polymerases function in a large complex called the replisome that contains multiple accessory subunits, such as the DNA clamp or helicases. | 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 reverse transcriptase is an enzyme for AIDS virus replication. Telomerase is an unusual polymerase because it contains its own RNA template as part of its structure. It synthesizes telomeres at the ends of chromosomes. Telomeres prevent fusion of the ends of neighboring chromosomes and protect chromosome ends from damage. | 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 a region of DNA called the terminator, where it halts and detaches from the DNA. As with human DNA-dependent DNA polymerases, RNA polymerase II, the enzyme that transcribes most of the genes in the human genome, operates as part of a large protein complex with multiple regulatory and accessory subunits. | 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 information, as one of the few times chromosomes interact is in chromosomal crossover which occurs during sexual reproduction, when genetic recombination occurs. Chromosomal crossover is when two DNA helices break, swap a section and then rejoin. | 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 double-strand breaks. | 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 known as recombinases, such as RAD51. The first step in recombination is a double-stranded break caused by either an endonuclease or damage to the DNA. A series of steps catalyzed in part by the recombinase then leads to joining of the two helices by at least one Holliday junction, in which a segment of a single strand in each helix is annealed to the complementary strand in the other helix. The Holliday junction is a tetrahedral junction structure that can be moved along the pair of chromosomes, swapping one strand for another. The recombination reaction is then halted by cleavage of the junction and re-ligation of the released DNA. Only strands of like polarity exchange DNA during recombination. There are two types of cleavage: east-west cleavage and north–south cleavage. The north–south cleavage nicks both strands of DNA, while the east–west cleavage has one strand of DNA intact. The formation of a Holliday junction during recombination makes it possible for genetic diversity, genes to exchange on chromosomes, and expression of wild-type viral genomes. | 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 acted as the central part of early cell metabolism as it can both transmit genetic information and carry out catalysis as part of ribozymes. This ancient RNA world where nucleic acid would have been used for both catalysis and genetics may have influenced the evolution of the current genetic code based on four nucleotide bases. This would occur, since the number of different bases in such an organism is a trade-off between a small number of bases increasing replication accuracy and a large number of bases increasing the catalytic efficiency of ribozymes. However, there is no direct evidence of ancient genetic systems, as recovery of DNA from most fossils is impossible because DNA survives in the environment for less than one million years, and slowly degrades into short fragments in solution. Claims for older DNA have been made, most notably a report of the isolation of a viable bacterium from a salt crystal 250 million years old, but these claims are controversial. | 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 starting chemicals, such as pyrimidine, found in meteorites. Pyrimidine, like polycyclic aromatic hydrocarbons (PAHs), the most carbon-rich chemical found in the universe, may have been formed in red giants or in interstellar cosmic dust and gas clouds. | 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 a man-made DNA sequence that has been assembled from other DNA sequences. They can be transformed into organisms in the form of plasmids or in the appropriate format, by using a viral vector. The genetically modified organisms produced can be used to produce products such as recombinant proteins, used in medical research, or be grown in agriculture. | 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 short tandem repeats and minisatellites, are compared between people. This method is usually an extremely reliable technique for identifying a matching DNA. However, identification can be complicated if the scene is contaminated with DNA from several people. DNA profiling was developed in 1984 by British geneticist Sir Alec Jeffreys, and first used in forensic science to convict Colin Pitchfork in the 1988 Enderby murders case. | 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 jeopardy law in some places, this can allow cases to be reopened where prior trials have failed to produce sufficient evidence to convince a jury. People charged with serious crimes may be required to provide a sample of DNA for matching purposes. The most obvious defense to DNA matches obtained forensically is to claim that cross-contamination of evidence has occurred. This has resulted in meticulous strict handling procedures with new cases of serious crime. | 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 to test paternity while a mother is still pregnant. | 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 catalyze variety of chemical reactions including RNA-DNA cleavage, RNA-DNA ligation, amino acids phosphorylation-dephosphorylation, carbon-carbon bond formation, etc. DNAzymes can enhance catalytic rate of chemical reactions up to 100,000,000,000-fold over the uncatalyzed reaction. The most extensively studied class of DNAzymes is RNA-cleaving types which have been used to detect different metal ions and designing therapeutic agents. Several metal-specific DNAzymes have been reported including the GR-5 DNAzyme (lead-specific), the CA1-3 DNAzymes (copper-specific), the 39E DNAzyme (uranyl-specific) and the NaA43 DNAzyme (sodium-specific). The NaA43 DNAzyme, which is reported to be more than 10,000-fold selective for sodium over other metal ions, was used to make a real-time sodium sensor in cells. | 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 matching algorithms, which find an occurrence of a sequence of letters inside a larger sequence of letters, were developed to search for specific sequences of nucleotides. The DNA sequence may be aligned with other DNA sequences to identify homologous sequences and locate the specific mutations that make them distinct. These techniques, especially multiple sequence alignment, are used in studying phylogenetic relationships and protein function. Data sets representing entire genomes' worth of DNA sequences, such as those produced by the Human Genome Project, are difficult to use without the annotations that identify the locations of genes and regulatory elements on each chromosome. Regions of DNA sequence that have the characteristic patterns associated with protein- or RNA-coding genes can be identified by gene finding algorithms, which allow researchers to predict the presence of particular gene products and their possible functions in an organism even before they have been isolated experimentally. Entire genomes may also be compared, which can shed light on the evolutionary history of particular organism and permit the examination of complex evolutionary events. | 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 periodic lattices (both tile-based and using the "DNA origami" method) and three-dimensional structures in the shapes of polyhedra. Nanomechanical devices and algorithmic self-assembly have also been demonstrated, and these DNA structures have been used to template the arrangement of other molecules such as gold nanoparticles and streptavidin proteins. DNA and other nucleic acids are the basis of aptamers, synthetic oligonucleotide ligands for specific target molecules used in a range of biotechnology and biomedical applications. | 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 are compared, population geneticists can learn the history of particular populations. This can be used in studies ranging from ecological genetics to anthropology. | 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 isolated its five primary nucleobases. | 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 groups ("tetranucleotide hypothesis"). Levene thought the chain was short and the bases repeated in a fixed order. In 1927, Nikolai Koltsov proposed that inherited traits would be inherited via a "giant hereditary molecule" made up of "two mirror strands that would replicate in a semi-conservative fashion using each strand as a template". In 1928, Frederick Griffith in his experiment discovered that traits of the "smooth" form of "Pneumococcus" could be transferred to the "rough" form of the same bacteria by mixing killed "smooth" bacteria with the live "rough" form. This system provided the first clear suggestion that DNA carries genetic information. | 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 a tetramer, with the function of buffering cellular pH. | 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 organism, the amount of guanine should be equal to cytosine and the amount of adenine should be equal to thymine. Late in 1951, Francis Crick started working with James Watson at the Cavendish Laboratory within the University of Cambridge. DNA's role in heredity was confirmed in 1952 when Alfred Hershey and Martha Chase in the Hershey–Chase experiment showed that DNA is the genetic material of the enterobacteria phage T2. | 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. Franklin told Crick and Watson that the backbones had to be on the outside. Before then, Linus Pauling, and Watson and Crick, had erroneous models with the chains inside and the bases pointing outwards. Franklin's identification of the space group for DNA crystals revealed to Crick that the two DNA strands were antiparallel. In February 1953, Linus Pauling and Robert Corey proposed a model for nucleic acids containing three intertwined chains, with the phosphates near the axis, and the bases on the outside. Watson and Crick completed their model, which is now accepted as the first correct model of the double helix of DNA. On 28 February 1953 Crick interrupted patrons' lunchtime at The Eagle pub in Cambridge to announce that he and Watson had "discovered the secret of life". | 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 has not escaped our notice that the specific pairing we have postulated immediately suggests a possible copying mechanism for the genetic material." This letter was followed by a letter from Franklin and Gosling, which was the first publication of their own X-ray diffraction data and of their original analysis method. Then followed a letter by Wilkins and two of his colleagues, which contained an analysis of "in vivo" B-DNA X-ray patterns, and which supported the presence "in vivo" of the Watson and Crick structure. | 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 through the Meselson–Stahl experiment. Further work by Crick and co-workers showed that the genetic code was based on non-overlapping triplets of bases, called codons, allowing Har Gobind Khorana, Robert W. Holley, and Marshall Warren Nirenberg to decipher the genetic code. These findings represent the birth of molecular biology. | 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 mechanics as an approximation valid at large (macroscopic) scale. | 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 physical quantity can be predicted prior to its measurement, given a complete set of initial conditions (the uncertainty principle). | 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 effect. These early attempts to understand microscopic phenomena, now known as the "old quantum theory", led to the full development of quantum mechanics in the mid-1920s by Niels Bohr, Erwin Schrödinger, Werner Heisenberg, Max Born, Paul Dirac and others. The modern theory is formulated in various specially developed mathematical formalisms. In one of them, a mathematical entity called the wave function provides information, in the form of probability amplitudes, about what measurements of a particle's energy, momentum, and other physical properties may yield. | 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 problems, such as Wigner's friend, and its application to the universe as a whole remains speculative. Predictions of quantum mechanics have been verified experimentally to an extremely high degree of accuracy. | 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 physicist Max Born. For example, a quantum particle like an electron can be described by a wave function, which associates to each point in space a probability amplitude. Applying the Born rule to these amplitudes gives a probability density function for the position that the electron will be found to have when an experiment is performed to measure it. This is the best the theory can do; it cannot say for certain where the electron will be found. The Schrödinger equation relates the collection of probability amplitudes that pertain to one moment of time to the collection of probability amplitudes that pertain to another. | 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 precise prediction for a measurement of its position and also at the same time for a measurement of its momentum. | 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 light passing through the slits is observed on a screen behind the plate. The wave nature of light causes the light waves passing through the two slits to interfere, producing bright and dark bands on the screen – a result that would not be expected if light consisted of classical particles. However, the light is always found to be absorbed at the screen at discrete points, as individual particles rather than waves; the interference pattern appears via the varying density of these particle hits on the screen. Furthermore, versions of the experiment that include detectors at the slits find that each detected photon passes through one slit (as would a classical particle), and not through both slits (as would a wave). However, such experiments demonstrate that particles do not form the interference pattern if one detects which slit they pass through. Other atomic-scale entities, such as electrons, are found to exhibit the same behavior when fired towards a double slit. This behavior is known as wave–particle duality. | 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 important consequences, enabling radioactive decay, nuclear fusion in stars, and applications such as scanning tunnelling microscopy and the tunnel diode. | 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 that enforces its entire departure from classical lines of thought". Quantum entanglement enables the counter-intuitive properties of quantum pseudo-telepathy, and can be a valuable resource in communication protocols, such as quantum key distribution and superdense coding. Contrary to popular misconception, entanglement does not allow sending signals faster than light, as demonstrated by the no-communication theorem. | 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, have demonstrated that broad classes of such hidden-variable theories are in fact incompatible with quantum physics. According to Bell's theorem, if nature actually operates in accord with any theory of "local" hidden variables, then the results of a Bell test will be constrained in a particular, quantifiable way. Many Bell tests have been performed, using entangled particles, and they have shown results incompatible with the constraints imposed by local hidden variables. | 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, this article will present a mathematical formulation of quantum mechanics and survey its application to some useful and oft-studied examples. | 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-defined up to a complex number of modulus 1 (the global phase), that is, formula_1 and formula_5 represent the same physical system. In other words, the possible states are points in the projective space of a Hilbert space, usually called the complex projective space. The exact nature of this Hilbert space is dependent on the system – for example, for describing position and momentum the Hilbert space is the space of complex square-integrable functions formula_6, while the Hilbert space for the spin of a single proton is simply the space of two-dimensional complex vectors formula_7 with the usual inner product. | 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 corresponds to the value of the observable in that eigenstate. More generally, a quantum state will be a linear combination of the eigenstates, known as a quantum superposition. When an observable is measured, the result will be one of its eigenvalues with probability given by the Born rule: in the simplest case the eigenvalue formula_8 is non-degenerate and the probability is given by formula_9, where formula_10 is its associated eigenvector. More generally, the eigenvalue is degenerate and the probability is given by formula_11, where formula_12 is the projector onto its associated eigenspace. In the continuous case, these formulas give instead the probability density. | 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 quantum systems to understand. It was the central topic in the famous Bohr–Einstein debates, in which the two scientists attempted to clarify these fundamental principles by way of thought experiments. In the decades after the formulation of quantum mechanics, the question of what constitutes a "measurement" has been extensively studied. Newer interpretations of quantum mechanics have been formulated that do away with the concept of "wave function collapse" (see, for example, the many-worlds interpretation). The basic idea is that when a quantum system interacts with a measuring apparatus, their respective wave functions become entangled so that the original quantum system ceases to exist as an independent entity. For details, see the article on measurement in quantum mechanics. | 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 classical system; the ability to make such an approximation in certain limits is called the correspondence principle. | 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 particle moving in a circular trajectory around the atomic nucleus, whereas in quantum mechanics, it is described by a static wave function surrounding the nucleus. For example, the electron wave function for an unexcited hydrogen atom is a spherically symmetric function known as an "s" orbital (). | 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 analytic treatment. | 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-classical equation of motion", which applies to systems for which quantum mechanics produces only small deviations from classical behavior. These deviations can then be computed based on the classical motion. This approach is particularly important in the field of quantum chaos. | 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 observables, meaning that they are represented by Hermitian operators. The position operator formula_24 and momentum operator formula_25 do not commute, but rather satisfy the canonical commutation relation: | 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 Fourier transform of the dependence in position means that the momentum operator is equivalent (up to an formula_36 factor) to taking the derivative according to the position, since in Fourier analysis differentiation corresponds to multiplication in the dual space. This is why in quantum equations in position space, the momentum formula_37 is replaced by formula_38, and in particular in the non-relativistic Schrödinger equation in position space the momentum-squared term is replaced with a Laplacian times formula_39. | 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 then | 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 formula_44 and formula_51 are both possible states for system formula_33, then | 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 information, though: knowing the reduced density matrices of the individual systems is not enough to reconstruct the state of the composite system. Just as density matrices specify the state of a subsystem of a larger system, analogously, positive operator-valued measures (POVMs) describe the effect on a subsystem of a measurement performed on a larger system. POVMs are extensively used in quantum information theory. | 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 explain why, in practice, quantum effects are difficult to observe in systems larger than microscopic. | 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 (invented by Erwin Schrödinger). An alternative formulation of quantum mechanics is Feynman's path integral formulation, in which a quantum-mechanical amplitude is considered as a sum over all possible classical and non-classical paths between the initial and final states. This is the quantum-mechanical counterpart of the action principle in classical mechanics. | 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 expectation value will not change over time. This statement generalizes, as mathematically, any Hermitian operator formula_32 can generate a family of unitary operators parameterized by a variable formula_56. Under the evolution generated by formula_32, any observable formula_33 that commutes with formula_32 will be conserved. Moreover, if formula_33 is conserved by evolution under formula_32, then formula_32 is conserved under the evolution generated by formula_33. This implies a quantum version of the result proven by Emmy Noether in classical (Lagrangian) mechanics: for every differentiable symmetry of a Hamiltonian, there exists a corresponding conservation law. | 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 in momentum, however, stays constant. | 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. For the one-dimensional case in the formula_80 direction, the time-independent Schrödinger equation may be written | 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 wave function must satisfy more complicated mathematical boundary conditions as it is nonzero in regions outside the well. Another related problem is that of the rectangular potential barrier, which furnishes a model for the quantum tunneling effect that plays an important role in the performance of modern technologies such as flash memory and scanning tunneling microscopy. | 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 eraser, the Elitzur–Vaidman bomb tester, and in studies of quantum entanglement. | 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 both beam splitters, and ends at the right. The quantum state of the photon is therefore a vector formula_111 that is a superposition of the "lower" path formula_112 and the "upper" path formula_113, that is, formula_114 for complex formula_115. In order to respect the postulate that formula_116 we require that formula_117. | 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 is modelled as the unitary matrix formula_121, which means that if the photon is on the "upper" path it will gain a relative phase of formula_122, and it will stay unchanged if it is in the lower path. | https://en.wikipedia.org/wiki?curid=25202 |
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