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Bra–ket Notation
In quantum mechanics, bra–ket notation, or Dirac notation, is used ubiquitously to denote quantum states. The notation uses angle brackets, and , and a vertical bar , to construct "bras" and "kets". A ket is of the form , v \rangle. Mathematically it denotes a vector, \boldsymbol v, in an abstract (complex) vector space V, and physically it represents a state of some quantum system. A bra is of the form \langle f, . Mathematically it denotes a linear form f:V \to \Complex, i.e. a linear map that maps each vector in V to a number in the complex plane \Complex. Letting the linear functional \langle f, act on a vector , v\rangle is written as \langle f , v\rangle \in \Complex. Assume that on V there exists an inner product (\cdot,\cdot) with antilinear first argument, which makes V an inner product space. Then with this inner product each vector \boldsymbol \phi \equiv , \phi\rangle can be identified with a corresponding linear form, by placing the vector in the anti-line ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Greek Alphabet
The Greek alphabet has been used to write the Greek language since the late 9th or early 8th century BCE. It is derived from the earlier Phoenician alphabet, and was the earliest known alphabetic script to have distinct letters for vowels as well as consonants. In Archaic Greece, Archaic and early Classical Greece, Classical times, the Greek alphabet existed in Archaic Greek alphabets, many local variants, but, by the end of the 4th century BCE, the Euclidean alphabet, with 24 letters, ordered from alpha to omega, had become standard and it is this version that is still used for Greek writing today. The letter case, uppercase and lowercase forms of the 24 letters are: : , , , , , , , , , , , , , , , , , /ς, , , , , , . The Greek alphabet is the ancestor of the Latin script, Latin and Cyrillic scripts. Like Latin and Cyrillic, Greek originally had only a single form of each letter; it developed the letter case distinction between uppercase and lowercase in parallel with Latin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polygamma Function
In mathematics, the polygamma function of order is a meromorphic function on the complex numbers \mathbb defined as the th derivative of the logarithm of the gamma function: :\psi^(z) := \frac \psi(z) = \frac \ln\Gamma(z). Thus :\psi^(z) = \psi(z) = \frac holds where is the digamma function and is the gamma function. They are holomorphic on \mathbb \backslash\mathbb_. At all the nonpositive integers these polygamma functions have a pole of order . The function is sometimes called the trigamma function. Integral representation When and , the polygamma function equals :\begin \psi^(z) &= (-1)^\int_0^\infty \frac\,\mathrmt \\ &= -\int_0^1 \frac(\ln t)^m\,\mathrmt\\ &= (-1)^m!\zeta(m+1,z) \end where \zeta(s,q) is the Hurwitz zeta function. This expresses the polygamma function as the Laplace transform of . It follows from Bernstein's theorem on monotone functions that, for and real and non-negative, is a completely monotone function. Setting in the above formula ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wave Function
A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements made on the system can be derived from it. The most common symbols for a wave function are the Greek letters and (lower-case and capital psi, respectively). The wave function is a function of the degrees of freedom corresponding to some maximal set of commuting observables. Once such a representation is chosen, the wave function can be derived from the quantum state. For a given system, the choice of which commuting degrees of freedom to use is not unique, and correspondingly the domain of the wave function is also not unique. For instance, it may be taken to be a function of all the position coordinates of the particles over position space, or the momenta of all the particles over momentum space; the two are related by a Fourier tran ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Psi (Cyrillic)
Psi (Ѱ, ѱ; italics: ) is a letter in the early Cyrillic alphabet, derived from the Greek letter psi (Ψ, ψ). It represents the sound /ps/, as in English ''naps''. According to the school rules developed in the 16th and the 17th centuries, such as Meletius Smotrytsky's grammar book, it was intended for use in words of Greek origin, but it was occasionally used for writing native words as well like Ukrainian (''psy'', “dogs”).Simovyč 1963, p 515a. It was used especially in words relating to the Eastern Orthodox Church, as can be seen in its continuing use in Church Slavonic. Psi was eliminated from the Russian orthography, along with ksi, omega, and the yuses, in the Civil Script of 1708 (Peter the Great's ''Grazhdansky Shrift''), and it has also been dropped from other secular languages. It continues to be used in Church Slavonic. Computing codes See also * Psi (other) Psi, PSI or Ψ may refer to: Alphabetic letters * Psi (Greek) (Ψ, ψ), the 23rd l ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Greek Numerals
Greek numerals, also known as Ionic, Ionian, Milesian, or Alexandrian numerals, are a system of writing numbers using the letters of the Greek alphabet. In modern Greece, they are still used for ordinal numbers and in contexts similar to those in which Roman numerals are still used in the Western world. For ordinary cardinal numbers, however, modern Greece uses Arabic numerals. History The Minoan and Mycenaean civilization Mycenaean Greece (or the Mycenaean civilization) was the last phase of the Bronze Age in Ancient Greece, spanning the period from approximately 1750 to 1050 BC.. It represents the first advanced and distinctively Greek civilization in mainland ...s' Linear A and Linear B alphabets used a different system, called Aegean numerals, which included number-only symbols for powers of ten: = 1, = 10, = 100, = 1000, and = 10000. Attic numerals comprised another system that came into use perhaps in th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Classical Greek
Ancient Greek includes the forms of the Greek language used in ancient Greece and the ancient world from around 1500 BC to 300 BC. It is often roughly divided into the following periods: Mycenaean Greek (), Dark Ages (), the Archaic period (), and the Classical period (). Ancient Greek was the language of Homer and of fifth-century Athenian historians, playwrights, and philosophers. It has contributed many words to English vocabulary and has been a standard subject of study in educational institutions of the Western world since the Renaissance. This article primarily contains information about the Epic and Classical periods of the language. From the Hellenistic period (), Ancient Greek was followed by Koine Greek, which is regarded as a separate historical stage, although its earliest form closely resembles Attic Greek and its latest form approaches Medieval Greek. There were several regional dialects of Ancient Greek, of which Attic Greek developed into Koine. Dia ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parapsychology
Parapsychology is the study of alleged psychic phenomena (extrasensory perception, telepathy, precognition, clairvoyance, psychokinesis (also called telekinesis), and psychometry) and other paranormal claims, for example, those related to near-death experiences, synchronicity, apparitional experiences, etc. Criticized as being a pseudoscience, the majority of mainstream scientists reject it. Parapsychology has also been criticised by mainstream critics for many of its practitioners claiming that their studies are plausible in spite of there being no convincing evidence for the existence of any psychic phenomena after more than a century of research. Parapsychology research rarely appears in mainstream scientific journals; instead, most papers about parapsychology are published in a small number of niche journals. Terminology The term ''parapsychology'' was coined in 1889 by philosopher Max Dessoir as the German . It was adopted by J. B. Rhine in the 1930s as a replacement fo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Early Cyrillic Alphabet
The Early Cyrillic alphabet, also called classical Cyrillic or paleo-Cyrillic, is a writing system that was developed in the First Bulgarian Empire during the late 9th century on the basis of the Greek alphabet for the Slavic people living near the Byzantine Empire in South East and Central Europe. It was used by Slavic peoples in South East, Central and Eastern Europe. It was developed in the Preslav Literary School in the capital city of the First Bulgarian Empire in order to write the Old Church Slavonic language. The modern Cyrillic script is still used primarily for some Slavic languages (such as Bulgarian, Macedonian, Serbian, Russian and Ukrainian), Kazakhstanand for East European and Asian languages that have experienced a great amount of Russian cultural influence. Among some of the traditionally culturally influential countries using Cyrillic script are Bulgaria, Russia, Serbia and Ukraine. Set А Б В Г Д Є Ж З И І К Л М Н О П Р С Т Ꙋ Ф Х ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gamma Function
In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except the non-positive integers. For every positive integer , \Gamma(n) = (n-1)!\,. Derived by Daniel Bernoulli, for complex numbers with a positive real part, the gamma function is defined via a convergent improper integral: \Gamma(z) = \int_0^\infty t^ e^\,dt, \ \qquad \Re(z) > 0\,. The gamma function then is defined as the analytic continuation of this integral function to a meromorphic function that is holomorphic in the whole complex plane except zero and the negative integers, where the function has simple poles. The gamma function has no zeroes, so the reciprocal gamma function is an entire function. In fact, the gamma function corresponds to the Mellin transform of the negative exponential function: \Gamma(z) = \mathcal M \ (z ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear B
Linear B was a syllabic script used for writing in Mycenaean Greek, the earliest attested form of Greek. The script predates the Greek alphabet by several centuries. The oldest Mycenaean writing dates to about 1400 BC. It is descended from the older Linear A, an undeciphered earlier script used for writing the Minoan language, as is the later Cypriot syllabary, which also recorded Greek. Linear B, found mainly in the palace archives at Knossos, Kydonia, Pylos, Thebes and Mycenae, disappeared with the fall of Mycenaean civilization during the Late Bronze Age collapse. The succeeding period, known as the Greek Dark Ages, provides no evidence of the use of writing. Linear B, deciphered by English architect and self-taught linguist Michael Ventris—based on the research of American classicist Alice Kober—is the only Bronze Age Aegean script to have been deciphered. Linear B consists of around 87 syllabic signs and over 100 ideographic signs. These ideograms or "signifyin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Modern Greek
Modern Greek (, , or , ''Kiní Neoellinikí Glóssa''), generally referred to by speakers simply as Greek (, ), refers collectively to the dialects of the Greek language spoken in the modern era, including the official standardized form of the languages sometimes referred to as Standard Modern Greek. The end of the Medieval Greek period and the beginning of Modern Greek is often symbolically assigned to the fall of the Byzantine Empire in 1453, even though that date marks no clear linguistic boundary and many characteristic features of the modern language arose centuries earlier, beginning around the fourth century AD. During most of the Modern Greek period, the language existed in a situation of diglossia, with regional spoken dialects existing side by side with learned, more archaic written forms, as with the vernacular and learned varieties (''Dimotiki'' and ''Katharevousa'') that co-existed in Greece throughout much of the 19th and 20th centuries. Varieties Varieties of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
