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Bra–ket Notation
Bra–ket notation, also called Dirac notation, is a notation for linear algebra and linear operators on complex vector spaces together with their dual space both in the finite-dimensional and infinite-dimensional case. It is specifically designed to ease the types of calculations that frequently come up in quantum mechanics. Its use in quantum mechanics is quite widespread. Bra–ket notation was created by Paul Dirac in his 1939 publication ''A New Notation for Quantum Mechanics''. The notation was introduced as an easier way to write quantum mechanical expressions. The name comes from the English word "bracket". Quantum mechanics In quantum mechanics and quantum computing, bra–ket 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 physicall ... [...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 BC. It was derived from the earlier Phoenician alphabet, and is the earliest known alphabetic script to systematically write 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 BC, the Ionia, Ionic-based Euclidean alphabet, with 24 letters, ordered from alpha to omega, had become standard throughout the Greek-speaking world and is the version that is still used for Greek writing today. The letter case, uppercase and lowercase forms of the 24 letters are: : , , , , , , , , , , , , , , , , , , , , , , , The Greek alphabet is the ancestor of several scripts, such as the Latin script, Latin, Gothic alphabet, Gothic, Coptic script, Coptic, and Cyrillic scripts. Throughout antiquity, Greek had only a single uppercas ... [...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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wave Function
In quantum physics, a wave function (or wavefunction) is a mathematical description of the quantum state of an isolated quantum system. The most common symbols for a wave function are the Greek letters and (lower-case and capital psi (letter), psi, respectively). Wave functions are complex number, complex-valued. For example, a wave function might assign a complex number to each point in a region of space. The Born rule provides the means to turn these complex probability amplitudes into actual probabilities. In one common form, it says that the squared modulus of a wave function that depends upon position is the probability density function, probability density of measurement in quantum mechanics, measuring a particle as being at a given place. The integral of a wavefunction's squared modulus over all the system's degrees of freedom must be equal to 1, a condition called ''normalization''. Since the wave function is complex-valued, only its relative phase and relative magnitud ... [...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 of the same name (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 Eastern Orthodoxy, otherwise known as Eastern Orthodox Christianity or Byzantine Christianity, is one of the three main Branches of Christianity, branches of Chalcedonian Christianity, alongside Catholic Church, Catholicism and Protestantism ... 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' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Greek Numerals
Greek numerals, also known as Ionic, Ionian, Milesian, or Alexandrian numerals, is a numeral system, system of writing numbers using the letters of the Greek alphabet. In modern Greece, they are still used for ordinal number (linguistics), ordinal numbers and in contexts similar to those in which Roman numerals are still used in the Western world. For ordinary cardinal number (linguistics), cardinal numbers, however, modern Greece uses Arabic numerals. History The Minoans, Minoan and Mycenaean civilizations' 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 composed another system that came into use perhaps in the 7th century BC. They were acrophonic, derived (after the initial one) from the first letters of the names of the numbers represented. They ran = 1, = ... [...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 or Homeric 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, which are the best-attested periods and considered most typical of Ancient Greek. From the Hellenistic period (), Ancient Greek was followed by Koine Greek, which is regarded as a separate historical stage, though its earliest form closely resembles Attic Greek, and its latest form approaches Medieval Greek, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parapsychology
Parapsychology is the study of alleged psychic phenomena (extrasensory perception, telepathy, teleportation, precognition, clairvoyance, psychokinesis (also called telekinesis), and psychometry (paranormal), 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 been criticised for continuing investigation despite being unable to provide reproducible evidence for the existence of any psychic phenomena after more than a century of research. Parapsychology research rarely appears in mainstream scientific journals; a few niche journals publish most papers about parapsychology. Terminology The term ''parapsychology'' was coined in 1889 by philosopher Max Dessoir as the German . It was adopted by Joseph Banks Rhine, J. B. Rhine in the 1930s as a replacement for the term ''psychical research'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Early Cyrillic Alphabet
The Early Cyrillic alphabet, also called classical Cyrillic or paleo-Cyrillic, is an alphabetic writing system that was developed in Medieval Bulgaria in the Preslav Literary School during the late 9th century. It is used to write the Church Slavonic language, and was historically used for its ancestor, Old Church Slavonic. It was also used for other languages, but between the 18th and 20th centuries was mostly replaced by the modern Cyrillic script, which is used for some Slavic languages (such as Russian language, Russian), and for East European and Asian languages that have experienced a great amount of Russian cultural influence. History The earliest form of manuscript Cyrillic, known as ''Ustav (script), ustav'', was based on Uncial script, Greek uncial script, augmented by typographic ligature, ligatures and by letters from the Glagolitic alphabet for phonemes not found in Greek. The Glagolitic script was created by the Byzantine monk Saints Cyril and Methodius, Sa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Psychiatry
Psychiatry is the medical specialty devoted to the diagnosis, treatment, and prevention of deleterious mental disorder, mental conditions. These include matters related to cognition, perceptions, Mood (psychology), mood, emotion, and behavior. Initial psychiatric assessment of a person begins with creating a Medical history, case history and conducting a mental status examination. Laboratory tests, physical examinations, and psychological tests may be conducted. On occasion, neuroimaging or neurophysiological studies are performed. Mental disorders are diagnosed in accordance with diagnostic manuals such as the ''International Classification of Diseases'' (ICD), edited by the World Health Organization (WHO), and the ''Diagnostic and Statistical Manual of Mental Disorders'' (DSM), published by the American Psychiatric Association (APA). The fifth edition of the DSM (DSM-5) was published in May 2013. Treatment may include psychotropics (psychiatric medicines), psychotherapy, su ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Modern Greek
Modern Greek (, or , ), 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 language sometimes referred to as Varieties of Modern Greek#Standard Modern Greek, 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, having begun 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. Variet ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gamma Function
In mathematics, the gamma function (represented by Γ, capital Greek alphabet, Greek letter gamma) is the most common extension of the factorial function to complex numbers. Derived by Daniel Bernoulli, the gamma function \Gamma(z) is defined for all complex numbers z except non-positive integers, and for every positive integer z=n, \Gamma(n) = (n-1)!\,.The gamma function can be defined via a convergent improper integral for complex numbers with positive real part: \Gamma(z) = \int_0^\infty t^ e^\textt, \ \qquad \Re(z) > 0\,.The gamma function then is defined in the complex plane as the analytic continuation of this integral function: it is a meromorphic function which is holomorphic function, holomorphic except at zero and the negative integers, where it has simple Zeros and poles, poles. The gamma function has no zeros, so the reciprocal gamma function is an entire function. In fact, the gamma function corresponds to the Mellin transform of the negative exponential functi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
