Branch of physics describin nature on a atomic scale

For a mo' accessible n' less technical introduction ta dis topic, peep Introduction ta quantum mechanics

[1] Da brighta areas represent a higher probabilitizzle of findin tha electron. Wavefunctions of tha electron up in a hydrogen atom at different juice levels. Quantum mechanics cannot predict tha exact location of a particle up in space, only tha probabilitizzle of findin it at different locations.Da brighta areas represent a higher probabilitizzle of findin tha electron.

Quantum mechanics be a gangbangin' fundamenstrual theory up in physics dat serves up a thugged-out description of tha physical propertizzlez of nature all up in tha scale of atoms n' subatomic particlez.[2] It be tha foundation of all quantum physics includin quantum chemistry, quantum field theory, quantum technology, n' quantum shiznit science.

Classical physics, tha description of physics dat existed before tha theory of relativity n' quantum mechanics, raps bout nuff aspectz of nature at a ordinary (macroscopic) scale, while quantum mechanics explains tha aspectz of nature at lil' small-ass (atomic n' subatomic) scales, fo' which old-ass mechanics is insufficient. Most theories up in old-ass physics can be derived from quantum mechanics as a approximation valid at big-ass (macroscopic) scale.[3]

Quantum mechanics differs from old-ass physics up in dat energy, momentum, angular momentum, n' other quantitizzlez of a funky-ass bound system is restricted ta discrete joints (quantization), objects have characteristics of both particlez n' waves (wave-particle duality), n' there be limits ta how tha fuck accurately tha value of a physical quantitizzle can be predicted prior ta its measurement, given a cold-ass lil complete set of initial conditions (the uncertainty principle).[note 1]

Quantum mechanics arose gradually, from theories ta explain observations which could not be reconciled wit old-ass physics, like fuckin Max Planckz solution up in 1900 ta tha black-body radiation problem, n' tha correspondence between juice n' frequency up in Albert Einsteinz 1905 paper which explained tha photoelectric effect. Early quantum theory was profoundly re-conceived up in tha mid-1920s by Niels Bohr, Erwin Schrödinger, Werner Heisenberg, Max Born n' others. Da original gangsta interpretation of quantum mechanics is tha Copenhagen interpretation, pimped by Niels Bohr n' Werner Heisenberg up in Copenhagen durin tha 1920s. Da modern theory is formulated up in various specially pimped mathematical formalisms. In one of them, a mathematical function, tha wave function, serves up shiznit bout tha probabilitizzle amplitude of juice, momentum, n' other physical propertizzlez of a particle.

History [ edit ]

Yo, scientific inquiry tha fuck into tha wave nature of light fuckin started up in tha 17th n' 18th centuries, when scientists like fuckin Robert Hooke, Christiaan Huygens n' Leonhard Euler proposed a wave theory of light based on experimenstrual observations.[5] In 1803 Gangsta polymath Thomas Young busted lyrics bout tha hyped double-slit experiment.[6] This experiment played a major role up in tha general acceptizzle of tha wave theory of light.

In 1838 Mike Faraday discovered cathode rays. These studies was followed by tha 1859 statement of tha black-body radiation problem by Gustav Kirchhoff, tha 1877 suggestion by Ludwig Boltzmann dat tha juice statez of a physical system can be discrete, n' tha 1900 quantum hypothesiz of Max Planck.[7] Planckz hypothesis dat juice is radiated n' absorbed up in discrete "quanta" (or juice packets) precisely matched tha observed patternz of black-body radiation.

In 1896 Wilhelm Wien empirically determined a gangbangin' finger-lickin' distribution law of black-body radiation,[8] called Wienz law. Ludwig Boltzmann independently arrived at dis result by considerationz of Maxwellz equations. But fuck dat shiznit yo, tha word on tha street is dat dat shiznit was valid only at high frequencies n' underestimated tha radiizzle at low frequencies.

Da foundationz of quantum mechanics was established durin tha straight-up original gangsta half of tha 20th century by Max Planck, Niels Bohr, Werner Heisenberg, Louis de Broglie, Arthur Compton, Albert Einstein, Erwin Schrödinger, Max Born, Jizzy von Neumann, Pizzle Dirac, Enrico Fermi, Wolfgang Pauli, Max von Laue, Freeman Dyson, Dizzy Hilbert, Wilhelm Wien, Satyendra Nath Bose, Arnold Sommerfeld, n' others. Da Copenhagen interpretation of Niels Bohr became widely accepted.

Max Planck erected dis model rockin Boltzmannz statistical interpretation of thermodynamics n' proposed what tha fuck is now called Planckz law, which hustled ta tha pimpment of quantum mechanics fo' realz. Afta Planckz solution up in 1900 ta tha black-body radiation problem (reported 1859), Albert Einstein offered a quantum-based explanation of tha photoelectric effect (1905, reported 1887). Around 1900�"1910, tha atomic theory but not tha corpuscular theory of light[9] first came ta be widely accepted as scientistical fact; these latta theories can be considered quantum theoriez of matter n' electromagnetic radiation, respectively. But fuck dat shiznit yo, tha word on tha street is dat tha photon theory was not widely accepted until bout 1915. Even until Einsteinz Nobel Prize, Niels Bohr did not believe up in tha photon.[10]

Among tha straight-up original gangsta ta study quantum phenomena was Arthur Compton, C. V. Raman, n' Pieta Zeeman, each of whom has a quantum effect named afta his muthafuckin ass. Robert Andrews Millikan studied tha photoelectric effect experimentally, n' Albert Einstein pimped a theory fo' it fo' realz. At tha same time, Ernest Rutherford experimentally discovered tha nuclear model of tha atom, n' Niels Bohr pimped a theory of atomic structure, confirmed by tha experimentz of Henry Moseley. In 1913 Peta Debye extended Bohrz theory by introducin elliptical orbits, a cold-ass lil concept also introduced by Arnold Sommerfeld.[11] This phase is known as old quantum theory.

Accordin ta Planck, each juice element (E) is proportionizzle ta its frequency (ν):

E = h ν {\displaystyle E=h

u \ }

where h is Planckz constant.

Planck cautiously insisted dat dis was only a aspect of tha processez of absorption n' emission of radiation n' was not tha physical reality of tha radiation.[12] In fact, his schmoooove ass considered his quantum hypothesis a mathematical trick ta git tha right answer rather than a sizable discovery.[13] But fuck dat shiznit yo, tha word on tha street is dat up in 1905 Albert Einstein interpreted Planckz quantum hypothesis realistically n' used it ta explain tha photoelectric effect, up in which shinin light on certain shiznit can eject electrons from tha material. It aint nuthin but tha nick nack patty wack, I still gots tha bigger sack. Einstein won tha 1921 Nobel Prize up in Physics fo' dis work.

Einstein further pimped dis scam ta show dat a electromagnetic wave like fuckin light could also be busted lyrics bout as a particle (lata called tha photon), wit a gangbangin' finger-lickin' discrete amount of juice dat dependz on its frequency.[14] In his thugged-out lil' paper “On tha Quantum Theory of Radiation,” Einstein expanded on tha interaction between juice n' matta ta explain tha absorption n' emission of juice by atoms fo' realz. Although overshadowed all up in tha time by his wild lil' freakadelic general theory of relativity, dis paper articulated tha mechanizzle underlyin tha stimulated emission of radiation,[15] which became tha basiz of tha laser.

In tha mid-1920s quantum mechanics was pimped ta become tha standard formulation fo' atomic physics. In tha summer of 1925, Bohr n' Heisenberg published thangs up in dis biatch dat closed tha oldschool quantum theory. Cuz of they particle-like behavior up in certain processes n' measurements, light quanta came ta be called photons (1926). In 1926 Erwin Schrödinger suggested a partial differential equation fo' tha wave functionz of particlez like electrons fo' realz. And when effectively restricted ta a gangbangin' finite region, dis equation allowed only certain modes, correspondin ta discrete quantum states �" whose propertizzles turned up ta be exactly tha same as implied by matrix mechanics.[16] Einsteinz simple postulation spurred a gangbangin' flurry of debate, theorizing, n' testing. Thus, tha entire field of quantum physics emerged, leadin ta its wider acceptizzle all up in tha Fifth Solvay Conference up in 1927.[17]

Dat shiznit was found dat subatomic particlez n' electromagnetic waves is neither simply particle nor wave but have certain propertizzlez of each. This originated tha concept of wave�"particle duality.[18]

By 1930 quantum mechanics had been further unified n' formalized by Dizzy Hilbert, Pizzle Dirac n' Jizzy von Neumann[19] wit pimped outa emphasis on measurement, tha statistical nature of our knowledge of reality, n' philosophical speculation bout tha 'observer'.[20] It has since permeated nuff disciplines, includin quantum chemistry, quantum electronics, quantum optics, n' quantum shiznit science. Well shiiiit, it also serves up a useful framework fo' nuff featurez of tha modern periodic table of elements, n' raps bout tha behaviorz of atoms durin chemical bonding n' tha flow of electrons up in computa semiconductors, n' therefore skits a cold-ass lil crucial role up in nuff modern technologies.[18] Its speculatizzle modern pimpments include strin theory n' quantum gravity theory.

While quantum mechanics was constructed ta describe tha ghetto of tha straight-up small, it be also needed ta explain some macroscopic phenomena like fuckin superconductors[21] n' superfluids.[22]

Da word quantum derives from tha Latin, meanin "how pimped out" or "how much".[23] In quantum mechanics, it refers ta a gangbangin' finger-lickin' discrete unit assigned ta certain physical quantities like fuckin tha energy of a atom at rest (see Figure 1). Da discovery dat particlez is discrete packetz of juice wit wave-like propertizzles hustled ta tha branch of physics dealin wit atomic n' subatomic systems which is todizzle called quantum mechanics. Well shiiiit, it underlies tha mathematical framework of nuff fieldz of physics n' chemistry, includin condensed matta physics, solid-state physics, atomic physics, molecular physics, computationizzle physics, computationizzle chemistry, quantum chemistry, particle physics, nuclear chemistry, n' nuclear physics.[24][better source needed] Some fundamenstrual aspectz of tha theory is still actively studied.[25]

Quantum mechanics is essential fo' understandin tha behavior of systems at atomic length scalez n' smalla n' shit. If tha physical nature of a atom was solely busted lyrics bout by classical mechanics, electrons would not orbit tha nucleus, since orbitin electrons emit radiation (due ta circular motion) n' so would quickly lose juice n' collide wit tha nucleus. This framework was unable ta explain tha stabilitizzle of atoms. Boy it's gettin hot, yes indeed it is. Instead, electrons remain up in a uncertain, non-deterministic, smeared, probabilistic wave�"particle orbital bout tha nucleus, defyin tha traditionizzle assumptionz of old-ass mechanics n' electromagnetism.[26]

Quantum mechanics was initially pimped ta provide a funky-ass betta explanation n' description of tha atom, especially tha differences up in tha spectra of light emitted by different isotopes of tha same chemical element, as well as subatomic particles. In short, tha quantum-mechanical atomic model has succeeded spectacularly up in tha realm where old-ass mechanics n' electromagnetizzle falter.

Broadly bustin lyrics, quantum mechanics incorporates four classez of phenomena fo' which old-ass physics cannot account[20]:

Mathematical formulations [ edit ]

In tha mathematically rigorous formulation of quantum mechanics pimped by Pizzle Dirac,[27] Dizzy Hilbert,[28] Jizzy von Neumann,[29] n' Hermann Weyl,[30] tha possible statez of a quantum mechanical system is symbolized[31] as unit vectors (called state vectors). Formally, these vectors is elementz of a complex separable Hilbert space �" variously called tha state space or tha associated Hilbert space of tha system �" dat is well defined up ta a cold-ass lil complex number of norm 1 (the phase factor). In other lyrics, tha possible states is points up in tha projectizzle space of a Hilbert space, probably called tha complex projectizzle space. Da exact nature of dis Hilbert space is dependent on tha system �" fo' example, tha state space fo' posizzle n' momentum states is tha space of square-integrable functions, while tha state space fo' tha spin of a single proton is just tha thang of two complex planes. Each observable is represented by a maximally Hermitian (precisely: by a self-adjoint) linear operator actin on tha state space. Each eigenstate of a observable correspondz ta a eigenvector of tha operator, n' tha associated eigenvalue correspondz ta tha value of tha observable up in dat eigenstate. If tha operatorz spectrum is discrete, tha observable can attain only dem discrete eigenvalues.

In tha formalizzle of quantum mechanics, tha state of a system at a given time is busted lyrics bout by a complex wave function, also referred ta as state vector up in a cold-ass lil complex vector space.[32] This abstract mathematical object allows fo' tha calculation of probabilities of outcomez of concrete experiments, n' you can put dat on yo' toast. For example, it allows one ta compute tha probabilitizzle of findin a electron up in a particular region round tha nucleus at a particular time. Contrary ta old-ass mechanics, one can never make simultaneous predictionz of conjugate variablez, like fuckin posizzle n' momentum, ta arbitrary precision. I aint talkin' bout chicken n' gravy biatch. For instance, electrons may be considered (to a cold-ass lil certain probability) ta be located somewhere within a given region of space yo, but wit they exact positions unknown. I aint talkin' bout chicken n' gravy biatch. Contourz of constant probabilitizzle density, often referred ta as "clouds", may be drawn round tha nucleuz of a atom ta conceptualize where tha electron might be located wit da most thugged-out probabilitizzle yo. Heisenbergz uncertainty principle quantifies tha inabilitizzle ta precisely locate tha particle given its conjugate momentum.[33]

Accordin ta one interpretation, as tha result of a measurement, tha wave function containin tha probabilitizzle shiznit fo' a system collapses from a given initial state ta a particular eigenstate. Da possible thangs up in dis biatch of a measurement is tha eigenvaluez of tha operator representin tha observable �" which explains tha chizzle of Hermitian operators, fo' which all tha eigenvalues is real. It aint nuthin but tha nick nack patty wack, I still gots tha bigger sack. Da probabilitizzle distribution of a observable up in a given state can be found by computin tha spectral decomposition of tha correspondin operator yo. Heisenbergz uncertainty principle is represented by tha statement dat tha operators correspondin ta certain observablez do not commute.

Da probabilistic nature of quantum mechanics thus stems from tha act of measurement. This is one of da most thugged-out hard as fuck aspectz of quantum systems ta understand. Y'all KNOW dat shit, muthafucka! Dat shiznit was tha central topic up in tha hyped Bohr�"Einstein debates, up in which tha two scientists attempted ta clarify these fundamenstrual principlez by way of thought experiments. In tha decades afta tha formulation of quantum mechanics, tha question of what tha fuck constitutes a "measurement" has been extensively studied. Y'all KNOW dat shit, muthafucka! Newer interpretationz of quantum mechanics done been formulated dat do away wit tha concept of "wave function collapse" (see, fo' example, tha relatizzle state interpretation). Da basic scam is dat when a quantum system interacts wit a measurin apparatus, they respectizzle wave functions become entangled, so dat tha original gangsta quantum system ceases ta exist as a independent entity. For details, peep tha article on measurement up in quantum mechanics.[34]

Generally, quantum mechanics do not assign definite joints, n' you can put dat on yo' toast. Instead, it cook up a prediction rockin a probabilitizzle distribution; dat is, it raps bout tha probabilitizzle of obtainin tha possible outbe reppin measurin a observable. Often these thangs up in dis biatch is skewed by nuff causes, like fuckin dense probabilitizzle clouds. Probabilitizzle cloudz is approximate (but betta than the Bohr model) whereby electron location is given by a probabilitizzle function, tha wave function eigenvalue, such dat tha probabilitizzle is tha squared moduluz of tha complex amplitude, or quantum state nuclear attraction.[35][36] Naturally, these probabilitizzles will depend on tha quantum state all up in tha "instant" of tha measurement yo. Hence, uncertainty is involved up in tha value. There are, however, certain states dat is associated wit a thugged-out definite value of a particular observable. These is known as eigenstates of tha observable ("eigen" can be translated from German as meanin "inherent" or "characteristic").[37]

In tha everydizzle ghetto, it is natural n' intuitizzle ta be thinkin of every last muthafuckin thang (every observable) as bein up in a eigenstate. Everythang appears ta git a thugged-out definite position, a thugged-out definite momentum, a thugged-out definite juice, n' a thugged-out definite time of occurrence. But fuck dat shiznit yo, tha word on tha street is dat quantum mechanics do not pinpoint tha exact jointz of a particlez posizzle n' momentum (since they is conjugate pairs) or its juice n' time (since they too is conjugate pairs). Rather, it serves up only a range of probabilitizzles up in which dat particle might be given its momentum n' momentum probability. Therefore, it is helpful ta use different lyrics ta describe states havin uncertain joints n' states havin definite joints (eigenstates).

Usually, a system aint gonna be up in a eigenstate of tha observable (particle) we is interested in. I aint talkin' bout chicken n' gravy biatch. But fuck dat shiznit yo, tha word on tha street is dat if one measures tha observable, tha wave function will instantaneously be a eigenstate (or "generalized" eigenstate) of dat observable. This process is known as wave function collapse, a cold-ass lil controversial n' much-debated process[38] dat involves expandin tha system under study ta include tha measurement device. If one knows tha correspondin wave function all up in tha instant before tha measurement, one is ghon be able ta compute tha probabilitizzle of tha wave function collapsin tha fuck into each of tha possible eigenstates.

For example, tha free particle up in tha previous example will probably gotz a wave function dat be a wave packet centered round some mean posizzle x 0 (neither a eigenstate of posizzle nor of momentum). When one measures tha posizzle of tha particle, it is impossible ta predict wit certainty tha result.[34] It be probable yo, but not certain, dat it is ghon be near x 0 , where tha amplitude of tha wave function is pimpin' fo' realz. Afta tha measurement is performed, havin obtained some result x, tha wave function collapses tha fuck into a posizzle eigenstate centered at x.[39]

Da time evolution of a quantum state is busted lyrics bout by tha Schrödinger equation, up in which tha Hamiltonian (the operator correspondin ta tha total juice of tha system) generates tha time evolution. I aint talkin' bout chicken n' gravy biatch. Da time evolution of wave functions is deterministic up in tha sense that �" given a wave function at a initial time �" it cook up a thugged-out definite prediction of what tha fuck tha wave function is ghon be at any later time.[40]

Durin a measurement, on tha other hand, tha chizzle of tha initial wave function tha fuck into another, lata wave function aint deterministic, it is unpredictable (i.e., random) fo' realz. A time-evolution simulation can be peeped here.[41][42]

Wave functions chizzle as time progresses. Da Schrödinger equation raps bout how tha fuck wave functions chizzle up in time, playin a role similar ta Newtonz second law up in classical mechanics. Da Schrödinger equation, applied ta tha aforementioned example of tha free particle, predicts dat tha centa of a wave packet will move all up in space at a cold-ass lil constant velocitizzle (like a old-ass particle wit no forces actin on it). But fuck dat shiznit yo, tha word on tha street is dat tha wave packet will also spread up as time progresses, which means dat tha posizzle becomes mo' uncertain wit time. This also has tha effect of turnin a posizzle eigenstate (which can be thought of as a infinitely sharp wave packet) tha fuck into a funky-ass broadened wave packet dat no longer represents a (definite, certain) posizzle eigenstate.[43]

n = 1, 2, 3, ...) n' angular momenta (increasin across from left ta right: s, p, d, ...). Denser areas correspond ta higher probabilitizzle densitizzle up in a posizzle measurement. Right back up in yo muthafuckin ass. Such wave functions is directly comparable ta only discrete joints like dem shown (as is tha case fo' Fig. 1: Probabilitizzle densities correspondin ta tha wave functionz of a electron up in a hydrogen atom possessin definite juice levels (increasin from tha top of tha image ta tha bottom:= 1, 2, 3, ...) n' angular momenta (increasin across from left ta right:, ...). Denser areas correspond ta higher probabilitizzle densitizzle up in a posizzle measurement. Right back up in yo muthafuckin ass. Such wave functions is directly comparable ta Chladniz figures of acoustic modez of vibration up in classical physics n' is modez of oscillation as well, possessin a gangbangin' finger-lickin' dirty-ass sharp energy n' thus, a thugged-out definite frequency . Da angular momentum n' juice is quantized n' takediscrete joints like dem shown (as is tha case fo' resonant frequencies up in acoustics)

Yo, some wave functions produce probabilitizzle distributions dat is constant, or independent of time �" like fuckin when up in a stationary state of definite juice, time vanishes up in tha absolute square of tha wave function (this is tha basis fo' tha juice-time uncertainty principle). Many systems dat is treated dynamically up in old-ass mechanics is busted lyrics bout by such "static" wave functions. For example, a single electron up in a unexcited atom is pictured classically as a particle movin up in a cold-ass lil circular trajectory round tha atomic nucleus, whereas up in quantum mechanics, it is busted lyrics bout by a static, spherically symmetric wave function surroundin tha nucleus (Fig. 1) (however, only tha lowest angular momentum states, labeled s, is spherically symmetric.)[44]

Da Schrödinger equation acts on tha entire probabilitizzle amplitude, not merely its absolute value. Whereas tha absolute value of tha probabilitizzle amplitude encodes shiznit bout probabilities, its phase encodes shiznit bout tha interference between quantum states. This gives rise ta tha "wave-like" behavior of quantum states.

Analytic solutionz of tha Schrödinger equation is known fo' very few relatively simple model Hamiltonians includin tha quantum harmonic oscillator, tha particle up in a funky-ass box, tha dihydrogen cation, n' tha hydrogen atom. Even tha helium atom �" which gotz nuff just two electrons �" has defied all attempts at a gangbangin' straight-up analytic treatment.

But fuck dat shiznit yo, tha word on tha street is dat there be steez fo' findin approximate solutions. One method, called perturbation theory, uses tha analytic result fo' a simple quantum mechanical model ta create a result fo' a related but mo' fucked up model by (for example) tha addizzle of a weak potential juice fo' realz. Another method is called "semi-classical equation of motion", which applies ta systems fo' which quantum mechanics produces only lil' small-ass deviations from old-ass behavior. Shiiit, dis aint no joke. These deviations can then be computed based on tha old-ass motion. I aint talkin' bout chicken n' gravy biatch. This approach is particularly blingin up in tha field of quantum chaos.

Mathematically equivalent formulations [ edit ]

There is nuff mathematically equivalent formulationz of quantum mechanics. One of tha crazy oldschool n' most common is tha "transformation theory" proposed by Pizzle Dirac, which unifies n' generalizes tha two earliest formulationz of quantum mechanics �" matrix mechanics (invented by Werner Heisenberg) n' wave mechanics (invented by Erwin Schrödinger).[45]

Especially since Heisenberg was awarded tha Nobel Prize up in Physics up in 1932 fo' tha creation of quantum mechanics, tha role of Max Born up in tha pimpment of QM was overlooked until tha 1954 Nobel award. Y'all KNOW dat shit, muthafucka! Da role is noted up in a 2005 bibliography of Born, which recounts his bangin role up in tha matrix formulation n' tha use of probabilitizzle amplitudes yo. Heisenberg acknowledges havin hustled matrices from Born, as published up in a 1940 festschrift honorin Max Planck.[46] In tha matrix formulation, tha instantaneous state of a quantum system encodes tha probabilitizzlez of its measurable properties, or "observablez". Examplez of observablez include energy, position, momentum, n' angular momentum. Observablez can be either continuous (e.g., tha posizzle of a particle) or discrete (e.g., tha juice of a electron bound ta a hydrogen atom).[47] An alternatizzle formulation of quantum mechanics is Feynmanz path integral formulation, up in which a quantum-mechanical amplitude is considered as a sum over all possible old-ass n' non-classical paths between tha initial n' final states. This is tha quantum-mechanical counterpart of tha action principle up in old-ass mechanics.

Relation ta other scientistical theories [ edit ]

Da rulez of quantum mechanics is fundamental. It aint nuthin but tha nick nack patty wack, I still gots tha bigger sack. They assert dat tha state space of a system be a Hilbert space (crucially, dat tha space has a inner product) n' dat observablez of tha system is Hermitian operators actin on vectors up in dat space �" although they do not tell our asses which Hilbert space or which operators. These can be chosen appropriately up in order ta obtain a quantitatizzle description of a quantum system fo' realz. An blingin guide fo' makin these chizzlez is tha correspondence principle, which states dat tha predictionz of quantum mechanics reduce ta dem of old-ass mechanics when a system moves ta higher energies or, equivalently, larger quantum numbers, i.e. whereas a single particle exhibits a thugged-out degree of randomness, up in systems incorporatin millionz of particlez averagin takes over and, all up in tha high juice limit, tha statistical probabilitizzle of random behaviour approaches zero. In other lyrics, old-ass mechanics is simply a quantum mechanics of big-ass systems. Boy it's gettin hot, yes indeed it is. This "high juice" limit is known as tha classical or correspondence limit. One can even start from a established old-ass model of a particular system, then try ta guess tha underlyin quantum model dat would give rise ta tha old-ass model up in tha correspondence limit.

Unsolved problem up in physics:

quantum mechanics: Is there a preferred interpretation of quantum mechanics, biatch? How tha fuck do tha quantum description of reality, which includes elements like fuckin tha " In tha correspondence limit of: Is there a preferred interpretation of quantum mechanics, biatch? How tha fuck do tha quantum description of reality, which includes elements like fuckin tha " superposition of states" n' " wave function collapse ", give rise ta tha realitizzle we perceive? (more unsolved problems up in physics)

When quantum mechanics was originally formulated, dat shiznit was applied ta models whose correspondence limit was non-relativistic classical mechanics. For instance, tha well-known model of tha quantum harmonic oscillator uses a explicitly non-relativistic expression fo' tha kinetic juice of tha oscillator, n' is thus a quantum version of tha classical harmonic oscillator.

Early attempts ta merge quantum mechanics wit special relativity involved tha replacement of tha Schrödinger equation wit a cold-ass lil covariant equation like fuckin tha Klein�"Gordon equation or tha Dirac equation. While these theories was successful up in explainin nuff experimenstrual thangs up in dis biatch, they had certain unsatisfactory qualitizzles stemmin from they neglect of tha relativistic creation n' annihilation of particlez fo' realz. A straight-up relativistic quantum theory required tha pimpment of quantum field theory, which applies quantization ta a gangbangin' field (rather than a gangbangin' fixed set of particles). Da first complete quantum field theory, quantum electrodynamics, serves up a gangbangin' straight-up quantum description of tha electromagnetic interaction. Da full apparatuz of quantum field theory is often unnecessary fo' describin electrodynamic systems fo' realz. A simpla approach, one dat has been used since tha inception of quantum mechanics, is ta treat charged particlez as quantum mechanical objects bein acted on by a old-ass electromagnetic field. For example, tha elementary quantum model of tha hydrogen atom raps bout tha electric field of tha hydrogen atom rockin a old-ass − e 2 / ( 4 π ϵ 0 r ) {\displaystyle \textstyle -e^{2}/(4\pi \epsilon _{_{0}}r)} Coulomb potential. This "semi-classical" approach fails if quantum fluctuations up in tha electromagnetic field play a blingin role, like fuckin up in tha emission of photons by charged particlez.

Quantum field theories fo' tha strong nuclear force n' tha weak nuclear force have also been pimped. Y'all KNOW dat shit, muthafucka! Da quantum field theory of tha phat nuclear force is called quantum chromodynamics, n' raps bout tha interactionz of subnuclear particlez like fuckin quarks n' gluons. Da weak nuclear force n' tha electromagnetic force was unified, up in they quantized forms, tha fuck into a single quantum field theory (known as electroweak theory), by tha physicists Abdus Salam, Sheldon Glashow n' Steven Weinberg. These three pimps shared tha Nobel Prize up in Physics up in 1979 fo' dis work.[48]

It has proven hard as fuck ta construct quantum modelz of gravity, tha remainin fundamenstrual force. Right back up in yo muthafuckin ass. Semi-classical approximations is workable, n' have hustled ta predictions like fuckin Hawkin radiation. But fuck dat shiznit yo, tha word on tha street is dat tha formulation of a cold-ass lil complete theory of quantum gravity is hindered by apparent incompatibilitizzles between general relativity (da most thugged-out accurate theory of gravitizzle currently known) n' a shitload of tha fundamenstrual assumptionz of quantum theory. Da resolution of these incompatibilitizzles be a area of actizzle research. Candidates fo' a gangbangin' future theory of quantum gravitizzle include strin theory.

Classical mechanics has also been extended tha fuck into tha complex domain, wit complex old-ass mechanics exhibitin behaviors similar ta quantum mechanics.[49]

Relation ta old-ass physics [ edit ]

Predictionz of quantum mechanics done been verified experimentally ta a mad high degree of accuracy.[50] Accordin ta tha correspondence principle between old-ass n' quantum mechanics, all objects obey tha lawz of quantum mechanics, n' old-ass mechanics is just a approximation fo' big-ass systemz of objects (or a statistical quantum mechanics of a big-ass collection of particles).[51] Da lawz of old-ass mechanics thus follow from tha lawz of quantum mechanics as a statistical average all up in tha limit of big-ass systems or big-ass quantum numbers (Ehrenfest theorem).[52][53] But fuck dat shiznit yo, tha word on tha street is dat chaotic systems aint gots phat quantum numbers, n' quantum chaos studies tha relationshizzle between old-ass n' quantum descriptions up in these systems.

Quantum coherence be a essential difference between old-ass n' quantum theories as illustrated by tha Einstein�"Podolsky�"Rosen (EPR) paradox �" a whoopin' on a cold-ass lil certain philosophical interpretation of quantum mechanics by a appeal ta local realism.[54] Quantum interference involves addin together probabilitizzle amplitudes, whereas old-ass "waves" infer dat there be a addin together of intensities. For microscopic bodies, tha extension of tha system is much smalla than tha coherence length, which gives rise ta long-range entanglement n' other nonlocal phenomena characteristic of quantum systems.[55] Quantum coherence aint typically evident at macroscopic scales, except maybe at temperatures approachin absolute zero at which quantum behavior may manifest macroscopically.[56] This is up in accordizzle wit tha followin observations:

Many macroscopic propertizzlez of a old-ass system is a gangbangin' finger-lickin' direct consequence of tha quantum behavior of its parts, n' you can put dat on yo' toast. For example, tha stabilitizzle of bulk matta (consistin of atoms n' moleculez which would quickly collapse under electric forces alone), tha rigiditizzle of solids, n' tha mechanical, thermal, chemical, optical n' magnetic propertizzlez of matta is all thangs up in dis biatch of tha interaction of electric charges under tha rulez of quantum mechanics. [57]

While tha seemingly "exotic" behavior of matta posited by quantum mechanics n' relativitizzle theory become mo' apparent fo' mad lil' small-ass particlez or fo' velocitizzles approachin tha speed of light, tha lawz of classical, often considered "Newtonian", physics remain accurate up in predictin tha behavior of tha vast majoritizzle of "large" objects (on tha order of tha size of big-ass moleculez or bigger) at velocitizzles much smalla than tha velocitizzle of light.[58]

Copenhagen interpretation of quantum versus old-ass kinematics [ edit ]

A big-ass difference between old-ass n' quantum mechanics is dat they use straight-up different kinematic descriptions.[59]

In Niels Bohrz mature view, quantum mechanical phenomena is required ta be experiments, wit complete descriptionz of all tha devices fo' tha system, preparative, intermediary, n' finally measuring. Da descriptions is up in macroscopic terms, expressed up in ordinary language, supplemented wit tha conceptz of old-ass mechanics.[60][61][62][63] Da initial condizzle n' tha final condizzle of tha system is respectively busted lyrics bout by joints up in a cold-ass lil configuration space, fo' example a posizzle space, or some equivalent space like fuckin a momentum space. Quantum mechanics do not admit a cold-ass lil straight-up precise description, up in termz of both posizzle n' momentum, of a initial condizzle or "state" (in tha old-ass sense of tha word) dat would support a precisely deterministic n' causal prediction of a gangbangin' final condition.[64][65] In dis sense, a quantum phenomenon be a process, a passage from initial ta final condition, not a instantaneous "state" up in tha old-ass sense of dat word.[66][67] Thus there be two kindz of processes up in quantum mechanics: stationary n' transitional. It aint nuthin but tha nick nack patty wack, I still gots tha bigger sack. For a stationary process, tha initial n' final condizzle is tha same. For a transition, they is different. Obviously by definition, if only tha initial condizzle is given, tha process aint determined.[64] Given its initial condition, prediction of its final condizzle is possible, causally but only probabilistically, cuz tha Schrödinger equation is deterministic fo' wave function evolution yo, but tha wave function raps bout tha system only probabilistically.[68][69]

For nuff experiments, it is possible ta be thinkin of tha initial n' final conditionz of tha system as bein a particle. In some cases it appears dat there be potentially nuff muthafuckin spatially distinct pathways or trajectories by which a particle might pass from initial ta final condition. I aint talkin' bout chicken n' gravy biatch. Well shiiiit, it be a blingin feature of tha quantum kinematic description dat it do not permit a unique definite statement of which of dem pathways is straight-up followed. Y'all KNOW dat shit, muthafucka! Only tha initial n' final conditions is definite, and, as stated up in tha foregoin paragraph, they is defined only as precisely as allowed by tha configuration space description or its equivalent. In every last muthafuckin case fo' which a quantum kinematic description is needed, there be always a cold-ass lil compellin reason fo' dis restriction of kinematic precision. I aint talkin' bout chicken n' gravy biatch fo' realz. An example of such a reason is dat fo' a particle ta be experimentally found up in a thugged-out definite position, it must be held motionless; fo' it ta be experimentally found ta git a thugged-out definite momentum, it must have free motion; these two is logically incompatible.[70][71]

Classical kinematics do not primarily demand experimenstrual description of its phenomena. Well shiiiit, it allows straight-up precise description of a instantaneous state by a value up in phase space, tha Cartesian thang of configuration n' momentum spaces. This description simply assumes or imagines a state as a physically existin entitizzle without concern bout its experimenstrual measurability. Right back up in yo muthafuckin ass. Such a thugged-out description of a initial condition, together wit Newtonz lawz of motion, allows a precise deterministic n' causal prediction of a gangbangin' final condition, wit a thugged-out definite trajectory of passage. Hamiltonian dynamics can be used fo' all dis bullshit. Classical kinematics also allows tha description of a process analogous ta tha initial n' final condizzle description used by quantum mechanics. Lagrangian mechanics applies ta all dis bullshit.[72] For processes dat need account ta be taken of actionz of a lil' small-ass number of Planck constants, old-ass kinematics aint adequate; quantum mechanics is needed.

Relation ta general relativity [ edit ]

Even wit tha definin postulatez of both Einsteinz theory of general relativitizzle n' quantum theory bein indisputably supported by rigorous n' repeated empirical evidence, n' while they do not directly contradict each other theoretically (at least wit regard ta they primary fronts), they have proven mad hard as fuck ta incorporate tha fuck into one consistent, cohesive model.[73]

Gravitizzle is negligible up in nuff areaz of particle physics, so dat unification between general relativitizzle n' quantum mechanics aint a urgent issue up in dem particular applications. But fuck dat shiznit yo, tha word on tha street is dat tha lack of a cold-ass lil erect theory of quantum gravity be a blingin issue up in physical cosmology n' tha search by physicists fo' a elegant "Theory of Everything" (TOE). Consequently, resolvin tha inconsistencies between both theories has been a major goal of 20th- n' 21st-century physics. Many prominent physicists, includin Stephen Hawking, hit dat shiznit fo' nuff muthafuckin years ta create a theory underlyin everything. This TOE would combine not only tha modelz of subatomic physics yo, but also derive tha four fundamenstrual forcez of nature �" tha strong force, electromagnetism, tha weak force, n' gravity �" from a single force or phenomenon. I aint talkin' bout chicken n' gravy biatch. But fuck dat shiznit yo, tha word on tha street is dat afta thankin bout Gödelz Incompletenizz Theorem, Hawkin concluded dat a theory of every last muthafuckin thang aint possible, n' stated so publicly up in his fuckin lecture "Gödel n' tha End of Physics" (2002).[74]

Attempts at a unified field theory [ edit ]

Da quest ta unify tha fundamenstrual forces all up in quantum mechanics is ongoing. Quantum electrodynamics (or "quantum electromagnetism"), which is (at least up in tha perturbatizzle regime) da most thugged-out accurately tested physical theory up in competizzle wit general relativity,[75][76] has been merged wit tha weak nuclear force tha fuck into tha electroweak force; work continues, ta merge it wit tha phat force tha fuck into tha electrostrong force. Current predictions state dat at round 1014 GeV these three forces fuse tha fuck into a single field.[77] Beyond dis "grand unification", it is speculated dat it may be possible ta merge gravitizzle wit tha other three gauge symmetries, sposed ta fuckin occur at roughly 1019 GeV yo. However �" n' while special relativitizzle is parsimoniously incorporated tha fuck into quantum electrodynamics �" tha expanded general relativity, currently tha dopest theory describin tha gravitation force, has not been straight-up incorporated tha fuck into quantum theory. One of dem searchin fo' a cold-ass lil coherent TOE is Edward Witten, a theoretical physicist whoz ass formulated tha M-theory, which be a attempt at describin tha supersymmetrical based strin theory. M-theory posits dat our apparent 4-dimensionizzle spacetime is, up in reality, straight-up a 11-dimensionizzle spacetime containin 10 spatial dimensions n' 1 time dimension, although 7 of tha spatial dimensions is �" at lower energies �" straight-up "compactified" (or infinitely curved) n' not readily amenable ta measurement or probing.

Another ghettofab theory is loop quantum gravity (LQG) proposed by Carlo Rovelli, dat raps bout quantum propertizzlez of gravity. Well shiiiit, it be also a theory of quantum spacetime n' quantum time, cuz up in general relativitizzle tha geometry of spacetime be a manifestation of gravity. LQG be a attempt ta merge n' adapt standard quantum mechanics n' standard general relativity. This theory raps bout space as granular analogous ta tha granularitizzle of photons up in tha quantum theory of electromagnetizzle n' tha discrete juice levelz of atoms. Boy it's gettin hot, yes indeed it is. Mo' precisely, space be a mad fine fabric or networks "woven" of finite loops called spin networks. Da evolution of a spin network over time is called a spin foam. Da predicted size of dis structure is tha Planck length, which be approximately 1.616�-10−35 m fo' realz. Accordin ta dis theory, there is no meanin ta length shorta than dis (cf. Planck scale juice).

Philosophical implications [ edit ]

Yo, since its inception, tha nuff counter-intuitive aspects n' thangs up in dis biatch of quantum mechanics have provoked phat philosophical debates n' nuff interpretations. Even fundamenstrual issues, like fuckin Max Bornz basic rulez bout probabilitizzle amplitudes n' probabilitizzle distributions, took decades ta be appreciated by society n' nuff leadin scientists, n' you can put dat on yo' toast. Slick Rick Feynman once holla'd, "I be thinkin I can safely say dat no muthafucka understandz quantum mechanics."[78] Accordin ta Steven Weinberg, "There is now up in mah opinion no entirely satisfactory interpretation of quantum mechanics."[79]

Da Copenhagen interpretation �" due largely ta Niels Bohr n' Werner Heisenberg �" remains most widely accepted some 75 muthafuckin years afta its enunciation. I aint talkin' bout chicken n' gravy biatch fo' realz. Accordin ta dis interpretation, tha probabilistic nature of quantum mechanics aint a temporary feature which will eventually be replaced by a thugged-out deterministic theory yo, but is instead a final renunciation of tha old-ass scam of "causality". Well shiiiit, it also states dat any well-defined application of tha quantum mechanical formalizzle must always make reference ta tha experimenstrual arrangement, cuz of tha conjugate nature of evidence obtained under different experimenstrual thangs.

Albert Einstein, his dirty ass one of tha foundaz of quantum theory, did not accept a shitload of tha mo' philosophical or metaphysical interpretationz of quantum mechanics, like fuckin rejection of determinism n' of causality yo. Dude famously holla'd bout this, "Dogg do not fuck wit dice".[80] Dude rejected tha concept dat tha state of a physical system dependz on tha experimenstrual arrangement fo' its measurement yo. Dude held dat a state of nature occurs up in its own right, regardless of whether or how tha fuck it might be observed. Y'all KNOW dat shit, muthafucka! That view is supported by tha currently accepted definizzle of a quantum state, which do not depend on tha configuration space fo' its representation, dat is ta say, manner of observation. I aint talkin' bout chicken n' gravy biatch. Einstein also believed dat underlyin quantum mechanics must be a theory dat thoroughly n' directly expresses tha rule against action at a gangbangin' finger-lickin' distance; up in other lyrics, he insisted on tha principle of locality yo. Dude considered yo, but rejected on theoretical grounds, a particular proposal fo' hidden variablez ta obviate tha indeterminizzle or acausalitizzle of quantum mechanical measurement yo. Dude believed dat quantum mechanics was a cold-ass lil currently valid but not a permanently definitizzle theory fo' quantum phenomena yo. Dude thought its future replacement would require profound conceptual advances, n' would not come quickly or doggystyle. Da Bohr-Einstein debates provide a vibrant critique of tha Copenhagen interpretation from a epistemological point of view. In jumpin off bout some shiznit fo' his views, he produced a seriez of objections, of which da most thugged-out hyped has become known as tha Einstein�"Podolsky�"Rosen paradox.

Jizzy Bell flossed dat dis EPR paradox hustled ta experimentally testable differences between quantum mechanics n' theories dat rely on local hidden variables. Experiments confirmed tha accuracy of quantum mechanics, thereby showin dat quantum mechanics cannot be improved upon by addizzle of local hidden variables.[81] Alain Aspectz experiments up in 1982 n' nuff lata experiments definitively verified quantum entanglement. Entanglement, as demonstrated up in Bell-type experiments, do not violate causality, since it do not involve transfer of shiznit. I aint talkin' bout chicken n' gravy biatch. By tha early 1980s, experiments had shown dat such inequalitizzles was indeed violated up in practice �" so dat there was up in fact correlationz of tha kind suggested by quantum mechanics fo' realz. At first these just seemed like isolated esoteric effects yo, but by tha mid-1990s, they was bein codified up in tha field of quantum shiznit theory, n' hustled ta constructions wit names like quantum cryptography n' quantum teleportation.[82] Quantum cryptography is proposed fo' use up in high-securitizzle applications up in bankin n' posse.

Da Everett many-worldz interpretation, formulated up in 1956, holdz dat all tha possibilitizzles busted lyrics bout by quantum theory simultaneously occur up in a multiverse composed of mostly independent parallel universes.[83] This aint accomplished by introducin a "new axiom" ta quantum mechanics yo, but by removing tha axiom of tha collapse of tha wave packet. All possible consistent statez of tha measured system n' tha measurin apparatus (includin tha observer) is present up in a real physical �" not just formally mathematical, as up in other interpretations �" quantum superposition. Right back up in yo muthafuckin ass. Such a superposizzle of consistent state combinationz of different systems is called a entangled state. While tha multiverse is deterministic, we perceive non-deterministic behavior governed by probabilities, cuz we can only observe tha universe (i.e., tha consistent state contribution ta tha aforementioned superposition) dat we, as observers, inhabit. Everettz interpretation is perfectly consistent wit Jizzy Bellz experiments n' make dem intuitively understandable. But fuck dat shiznit yo, tha word on tha street is dat accordin ta tha theory of quantum decoherence, these "parallel universes" aint NEVER gonna be accessible ta us. Da inaccessibilitizzle can be understood as bigs up: once a measurement is done, tha measured system becomes entangled wit both tha physicist whoz ass measured it and a big-ass number of other particles, a shitload of which is photons flyin away all up in tha speed of light towardz tha other end of tha universe. In order ta prove dat tha wave function did not collapse, one would gotta brang all these particlez back n' measure dem again, together wit tha system dat was originally measured. Y'all KNOW dat shit, muthafucka! Not only is dis straight-up impractical yo, but even if one could theoretically do this, it would gotta fuck wit any evidence dat tha original gangsta measurement took place (includin tha physicistz memory).

In light of tha Bell tests, Cramer up in 1986 formulated his transactionizzle interpretation[84] which is unique up in providin a physical explanation fo' tha Born rule.[85] Relationizzle quantum mechanics rocked up in tha late 1990s as tha modern derivatizzle of tha Copenhagen interpretation.

Applications [ edit ]

Quantum mechanics has had enormous[18] success up in explainin nuff of tha featurez of our universe. Quantum mechanics is often tha only theory dat can reveal tha individual behaviorz of tha subatomic particlez dat make up all formz of matta (electrons, protons, neutrons, photons, n' others). Quantum mechanics has straight fuckin hyped up strin theories, muthafuckas fo' a Theory of Everything (see reductionism).

Quantum mechanics be also critically blingin fo' understandin how tha fuck individual atoms is joined by covalent bondz ta form moleculez. Da application of quantum mechanics ta chemistry is known as quantum chemistry. Quantum mechanics can also provide quantitatizzle insight tha fuck into ionic n' covalent bonding processes by explicitly showin which moleculez is energetically favorable ta which others n' tha magnitudez of tha energies involved.[86] Furthermore, most of tha calculations performed up in modern computationizzle chemistry rely on quantum mechanics.

In nuff aspects modern technologizzle operates at a scale where quantum effects is significant. Important applicationz of quantum theory include quantum chemistry, quantum optics, quantum computing, superconductin magnets, light-emittin diodes, tha optical amplifier n' tha laser, tha transistor n' semiconductors like fuckin tha microprocessor, medical n' research imaging like fuckin magnetic resonizzle imaging n' electron microscopy.[87] Explanations fo' nuff astrological n' physical phenomena is rooted up in tha nature of tha chemical bond, most notably tha macro-molecule DNA.[88]

Electronics [ edit ]

Many modern electronic devices is designed rockin quantum mechanics. Examplez include tha laser, tha transistor (and thus tha microchip), tha electron microscope, n' magnetic resonizzle imaging (MRI). Da study of semiconductors hustled ta tha invention of tha diode n' tha transistor, which is indispensable partz of modern electronics systems, computer n' telecommunication devices fo' realz. Another application is fo' makin laser diodes n' light emittin diodes, which is a high-efficiency source of light.

Many electronic devices operate under effect of quantum tunneling. Well shiiiit, it even exists up in tha simple light switch. Da switch would not work if electrons could not quantum tunnel all up in tha layer of oxidation on tha metal contact surfaces. Flash memory chips found up in STD drives use quantum tunnelin ta erase they memory cells. Right back up in yo muthafuckin ass. Some wack differential resistizzle devices also utilize quantum tunnelin effect, like fuckin resonant tunnelin diodes. Unlike old-ass diodes, its current is carried by resonant tunneling all up in two or mo' potential barriers (see right figure). Its wack resistizzle behavior can only be understood wit quantum mechanics: As tha confined state moves close ta Fermi level, tunnel current increases fo' realz. As it moves away, current decreases. Quantum mechanics is necessary ta KNOW n' design such electronic devices.

Cryptography [ edit ]

Researchers is currently seekin robust methodz of directly manipulatin quantum states. Efforts is bein made ta mo' straight-up pimp quantum cryptography, which will theoretically allow guaranteed secure transmission of shiznit.

An inherent advantage yielded by quantum cryptography when compared ta old-ass cryptography is tha detection of passive eavesdropping. This be a natural result of tha behavior of quantum bits; cuz of tha observer effect, if a lil' bit up in a superposizzle state was ta be observed, tha superposizzle state would collapse tha fuck into a eigenstate. Because tha intended recipient was expectin ta receive tha bit up in a superposizzle state, tha intended recipient would know there was a attack, cuz tha bitz state would no longer be up in a superposition.[89]

Quantum computing [ edit ]

Another goal is tha pimpment of quantum computers, which is sposed ta fuckin big-ass up certain computationizzle tasks exponentially fasta than old-ass computers. Instead of rockin old-ass bits, quantum computas use qubits, which can be up in superpositions of states. Quantum programmers is able ta manipulate tha superposizzle of qubits up in order ta solve problems dat old-ass computin cannot do effectively, like fuckin searchin unsorted databases or integer factorization. IBM fronts dat tha advent of quantum computin may progress tha fieldz of medicine, logistics, financial skillz, artificial intelligence n' cloud security.[90]

Another actizzle research topic is quantum teleportation, which deals wit steez ta transmit quantum shiznit over arbitrary distances.

Macroscale quantum effects [ edit ]

While quantum mechanics primarily applies ta tha smalla atomic regimez of matta n' juice, some systems exhibit quantum mechanical effects on a big-ass scale. Superfluidity, tha frictionless flow of a liquid at temperatures near absolute zero, is one well-known example. Right back up in yo muthafuckin ass. So is tha closely related phenomenon of superconductivity, tha frictionless flow of a electron gas up in a cold-ass lil conductin material (an electric current) at sufficiently low temperatures. Da fractionizzle quantum Hall effect be a topological ordered state which correspondz ta patternz of long-range quantum entanglement.[91] States wit different topological ordaz (or different patternz of long range entanglements) cannot chizzle tha fuck into each other without a phase transition.

Other phenomena [ edit ]

Quantum theory also serves up accurate descriptions fo' nuff previously unexplained phenomena, like fuckin black-body radiation n' tha stabilitizzle of tha orbitals of electrons up in atoms. Boy it's gettin hot, yes indeed it is. Well shiiiit, it has also given insight tha fuck into tha workingz of nuff different astrological systems, includin smell receptors n' protein structures.[92] Recent work on photosynthesis has provided evidence dat quantum correlations play a essential role up in dis fundamenstrual process of plants n' nuff other organisms.[93] Even so, classical physics can often provide phat approximations ta thangs up in dis biatch otherwise obtained by quantum physics, typically up in circumstances wit big-ass numberz of particlez or big-ass quantum numbers. Right back up in yo muthafuckin ass. Since old-ass formulas is much simpla n' easier ta compute than quantum formulas, old-ass approximations is used n' preferred when tha system is big-ass enough ta render tha effectz of quantum mechanics insignificant.

Examplez [ edit ]

Jacked particle [ edit ]

For example, consider a free particle. In quantum mechanics, a gangbangin' free matta is busted lyrics bout by a wave function. I aint talkin' bout chicken n' gravy biatch. Da particle propertizzlez of tha matta become apparent when we measure its posizzle n' velocity. Da wave propertizzlez of tha matta become apparent when we measure its wave propertizzles like interference. Da wave�"particle duality feature is incorporated up in tha relationz of coordinates n' operators up in tha formulation of quantum mechanics. Since tha matta is free (not subject ta any interactions), its quantum state can be represented as a wave of arbitrary shape n' extendin over space as a wave function. Da posizzle n' momentum of tha particle is observablez. Da Uncertainty Principle states dat both tha posizzle n' tha momentum cannot simultaneously be measured wit complete precision. I aint talkin' bout chicken n' gravy biatch. But fuck dat shiznit yo, tha word on tha street is dat one can measure tha posizzle (alone) of a movin free particle, bustin a eigenstate of posizzle wit a wave function dat is straight-up big-ass (a Dirac delta) at a particular posizzle x, n' zero everywhere else. If one performs a posizzle measurement on such a wave function, tha resultant x is ghon be obtained wit 100% probabilitizzle (i.e., wit full certainty, or complete precision). This is called a eigenstate of position �" or, stated up in mathematical terms, a generalized posizzle eigenstate (eigendistribution). If tha particle is up in a eigenstate of position, then its momentum is straight-up unknown. I aint talkin' bout chicken n' gravy biatch. On tha other hand, if tha particle is up in a eigenstate of momentum, then its posizzle is straight-up unknown.[94] In a eigenstate of momentum havin a plane wave form, it can be shown dat tha wavelength is equal ta h/p, where h is Planckz constant n' p is tha momentum of tha eigenstate.[95]

Particle up in a funky-ass box [ edit ]

1-dimensionizzle potential juice box (or infinite potential well)

Da particle up in a one-dimensionizzle potential juice box is da most thugged-out mathematically simple example where restraints lead ta tha quantization of juice levels. Da box is defined as havin zero potential juice everywhere inside a cold-ass lil certain region, n' therefore infinite potential juice everywhere outside dat region. I aint talkin' bout chicken n' gravy biatch. For tha one-dimensionizzle case up in tha x {\displaystyle x} direction, tha time-independent Schrödinger equation may be written[96]

− ℏ 2 2 m d 2 ψ d x 2 = E ψ . {\displaystyle -{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}\psi }{dx^{2}}}=E\psi .}

With tha differential operator defined by

p ^ x = − i ℏ d d x {\displaystyle {\hat {p}}_{x}=-i\hbar {\frac {d}{dx}}}

the previous equation is evocatizzle of tha funky-ass kinetic juice analogue,

1 2 m p ^ x 2 = E , {\displaystyle {\frac {1}{2m}}{\hat {p}}_{x}^{2}=E,}

with state ψ {\displaystyle \psi } up in dis case havin juice E {\displaystyle E} coincident wit tha kinetic juice of tha particle.

Da general solutionz of tha Schrödinger equation fo' tha particle up in a funky-ass box are

ψ ( x ) = A e i k x + B e − i k x E = ℏ 2 k 2 2 m {\displaystyle \psi (x)=Ae^{ikx}+Be^{-ikx}\qquad \qquad E={\frac {\hbar ^{2}k^{2}}{2m}}}

or, from Eulerz formula,

ψ ( x ) = C sin ⁡ ( k x ) + D cos ⁡ ( k x ) . {\displaystyle \psi (x)=C\sin(kx)+D\cos(kx).\!}

Da infinite potential wallz of tha box determine tha jointz of C , D , {\displaystyle C,D,} n' k {\displaystyle k} at x = 0 {\displaystyle x=0} n' x = L {\displaystyle x=L} where ψ {\displaystyle \psi } must be zero. Thus, at x = 0 {\displaystyle x=0} ,

ψ ( 0 ) = 0 = C sin ⁡ ( 0 ) + D cos ⁡ ( 0 ) = D {\displaystyle \psi (0)=0=C\sin(0)+D\cos(0)=D}

and D = 0 {\displaystyle D=0} fo' realz. At x = L {\displaystyle x=L} ,

ψ ( L ) = 0 = C sin ⁡ ( k L ) , {\displaystyle \psi (L)=0=C\sin(kL),}

in which C {\displaystyle C} cannot be zero as dis would conflict wit tha Born interpretation. I aint talkin' bout chicken n' gravy biatch. Therefore, since sin ⁡ ( k L ) = 0 {\displaystyle \sin(kL)=0} , k L {\displaystyle kL} must be a integer multiple of π {\displaystyle \pi } ,

k = n π L n = 1 , 2 , 3 , … . {\displaystyle k={\frac {n\pi }{L}}\qquad \qquad n=1,2,3,\ldots .}

Da quantization of juice levels bigs up from dis constraint on k , {\displaystyle k,} since

E = ℏ 2 π 2 n 2 2 m L 2 = n 2 h 2 8 m L 2 . {\displaystyle E={\frac {\hbar ^{2}\pi ^{2}n^{2}}{2mL^{2}}}={\frac {n^{2}h^{2}}{8mL^{2}}}.}

Da ground state juice of tha particlez is E 1 {\displaystyle E_{1}} fo' n = 1. {\displaystyle n=1.}

Da juice of tha particle up in tha n {\displaystyle n} th state is E n = n 2 E 1 , n = 2 , 3 , 4 , … {\displaystyle E_{n}=n^{2}E_{1},\;n=2,3,4,\dots }

Particle up in a funky-ass box wit boundary condizzle V ( x ) = 0 , − a / 2 < x < + a / 2 {\displaystyle V(x)=0,\;-a/2<x<+a/2}

A particle up in a funky-ass box wit a lil chizzle up in tha boundary condition.

In dis condizzle tha general solution is ghon be same, there will lil chizzle ta tha final result, since tha boundary conditions is chizzled only slightly:

ψ ( x ) = C sin ⁡ ( k x ) + D cos ⁡ ( k x ) . {\displaystyle \psi (x)=C\sin(kx)+D\cos(kx).}

At x = 0 , {\displaystyle x=0,} tha wave function aint straight-up zero at all jointz of n . {\displaystyle n.}

Clearly, from tha wave function variation graph our crazy asses have, At n = 1 , 3 , 5 , … , {\displaystyle n=1,3,5,\dots ,} tha wave function bigs up a cold-ass lil cosine curve wit x = 0 {\displaystyle x=0} as tha origin.

At n = 2 , 4 , 6 , … , {\displaystyle n=2,4,6,\dots ,} tha wave function bigs up a sine curve wit x = 0 {\displaystyle x=0} as tha origin. I aint talkin' bout chicken n' gravy biatch.

Wave Function Variation wit x n' n.

From dis observation we can conclude dat tha wave function be alternatively sine n' cosine. So up in dis case tha resultant wave equation is

ψ n ( x ) = { A cos ⁡ ( k n x ) , n = 1 , 3 , 5 , … B sin ⁡ ( k n x ) , n = 2 , 4 , 6 , … {\displaystyle \psi _{n}(x)={\begin{cases}A\cos(k_{n}x),&n=1,3,5,\dots \\B\sin(k_{n}x),&n=2,4,6,\dots \end{cases}}}

Finite potential well [ edit ]

A finite potential well is tha generalization of tha infinite potential well problem ta potential wells havin finite depth.

Da finite potential well problem is mathematically mo' fucked up than tha infinite particle-in-a-box problem as tha wave function aint pinned ta zero all up in tha wallz of tha well. Instead, tha wave function must satisfy mo' fucked up mathematical boundary conditions as it is nonzero up in regions outside tha well.

Rectangular potential barrier [ edit ]

This be a model fo' tha quantum tunneling effect which skits a blingin role up in tha performizzle of modern technologies like fuckin flash memory n' scannin tunnelin microscopy. Quantum tunnelin is central ta physical phenomena involved up in superlattices.

Harmonic oscillator [ edit ]

As up in tha old-ass case, tha potential fo' tha quantum harmonic oscillator is given by

V ( x ) = 1 2 m ω 2 x 2 . {\displaystyle V(x)={\frac {1}{2}}m\omega ^{2}x^{2}.}

This problem can either be treated by directly solvin tha Schrödinger equation, which aint trivial, or by rockin tha mo' elegant "ladder method" first proposed by Pizzle Dirac. Da eigenstates is given by

ψ n ( x ) = 1 2 n n ! ⋅ ( m ω π ℏ ) 1 / 4 ⋅ e − m ω x 2 2 ℏ ⋅ H n ( m ω ℏ x ) , {\displaystyle \psi _{n}(x)={\sqrt {\frac {1}{2^{n}\,n!}}}\cdot \left({\frac {m\omega }{\pi \hbar }}\right)^{1/4}\cdot e^{-{\frac {m\omega x^{2}}{2\hbar }}}\cdot H_{n}\left({\sqrt {\frac {m\omega }{\hbar }}}x\right),\qquad }

n = 0 , 1 , 2 , … . {\displaystyle n=0,1,2,\ldots .}

where H n is tha Hermite polynomials

H n ( x ) = ( − 1 ) n e x 2 d n d x n ( e − x 2 ) , {\displaystyle H_{n}(x)=(-1)^{n}e^{x^{2}}{\frac {d^{n}}{dx^{n}}}\left(e^{-x^{2}}\right),}

and tha correspondin juice levels are

E n = ℏ ω ( n + 1 2 ) . {\displaystyle E_{n}=\hbar \omega \left(n+{1 \over 2}\right).}

This be another example illustratin tha quantification of juice fo' bound states.

Step potential [ edit ]

V 0 , shown up in green. I aint talkin' bout chicken n' gravy biatch. Da amplitudes n' direction of left- n' right-movin waves is indicated. Y'all KNOW dat shit, muthafucka! This type'a shiznit happens all tha time. Yellow is tha incident wave, blue is reflected n' transmitted waves, red do not occur. Shiiit, dis aint no joke. E > V 0 fo' dis figure. Scatterin at a gangbangin' finite potential step of height, shown up in green. I aint talkin' bout chicken n' gravy biatch. Da amplitudes n' direction of left- n' right-movin waves is indicated. Y'all KNOW dat shit, muthafucka! This type'a shiznit happens all tha time. Yellow is tha incident wave, blue is reflected n' transmitted waves, red do not occur. Shiiit, dis aint no joke.fo' dis figure.

Da potential up in dis case is given by:

V ( x ) = { 0 , x < 0 , V 0 , x ≥ 0. {\displaystyle V(x)={\begin{cases}0,&x<0,\\V_{0},&x\geq 0.\end{cases}}}

Da solutions is superpositionz of left- n' right-movin waves:

ψ 1 ( x ) = 1 k 1 ( A → e i k 1 x + A ← e − i k 1 x ) x < 0 {\displaystyle \psi _{1}(x)={\frac {1}{\sqrt {k_{1}}}}\left(A_{\rightarrow }e^{ik_{1}x}+A_{\leftarrow }e^{-ik_{1}x}\right)\qquad x<0}

and

ψ 2 ( x ) = 1 k 2 ( B → e i k 2 x + B ← e − i k 2 x ) x > 0 {\displaystyle \psi _{2}(x)={\frac {1}{\sqrt {k_{2}}}}\left(B_{\rightarrow }e^{ik_{2}x}+B_{\leftarrow }e^{-ik_{2}x}\right)\qquad x>0}

with coefficients A n' B determined from tha boundary conditions n' by imposin a cold-ass lil continuous derivative on tha solution, n' where tha wave vectors is related ta tha juice via

k 1 = 2 m E / ℏ 2 {\displaystyle k_{1}={\sqrt {2mE/\hbar ^{2}}}}

and

k 2 = 2 m ( E − V 0 ) / ℏ 2 {\displaystyle k_{2}={\sqrt {2m(E-V_{0})/\hbar ^{2}}}}

Each term of tha solution can be interpreted as a incident, reflected, or transmitted component of tha wave, allowin tha calculation of transmission n' reflection coefficients, n' you can put dat on yo' toast. Notably, up in contrast ta old-ass mechanics, incident particlez wit energies pimped outa than tha potential step is partially reflected.

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^ precision: If δ x {\displaystyle \delta x} δ p {\displaystyle \delta p} individual measurement n' σ x {\displaystyle \sigma _{x}} σ p {\displaystyle \sigma _{p}} ensemble of individual measurements on similarly prepared systems, then "There are, up in principle, no restrictions on tha precisionz of individual measurements δ x {\displaystyle \delta x} δ p {\displaystyle \delta p} σ x σ p ≥ ℏ / 2 {\displaystyle \sigma _{x}\sigma _{p}\geq \hbar /2} ".[4] N.B. on: Ifn'is tha precisionz of posizzle n' momentum obtained up in ameasurement n'they standard deviations up in aof individual measurements on similarly prepared systems, then "".

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