115-501: [REDACTED] Look up orbital in Wiktionary, the free dictionary. Orbital may refer to: Sciences [ edit ] Chemistry and physics [ edit ] Atomic orbital Molecular orbital Hybrid orbital Astronomy and space flight [ edit ] Orbit Earth orbit Medicine and physiology [ edit ] Orbit (anatomy) , also known as
230-475: A {\displaystyle a} larger we make the spread in momentum smaller, but the spread in position gets larger. This illustrates the uncertainty principle. As we let the Gaussian wave packet evolve in time, we see that its center moves through space at a constant velocity (like a classical particle with no forces acting on it). However, the wave packet will also spread out as time progresses, which means that
345-441: A " subshell ". Because of the quantum mechanical nature of the electrons around a nucleus, atomic orbitals can be uniquely defined by a set of integers known as quantum numbers. These quantum numbers occur only in certain combinations of values, and their physical interpretation changes depending on whether real or complex versions of the atomic orbitals are employed. In physics, the most common orbital descriptions are based on
460-472: A 3d subshell but this is at higher energy than the 3s and 3p in argon (contrary to the situation for hydrogen) and remains empty. Immediately after Heisenberg discovered his uncertainty principle , Bohr noted that the existence of any sort of wave packet implies uncertainty in the wave frequency and wavelength, since a spread of frequencies is needed to create the packet itself. In quantum mechanics, where all particle momenta are associated with waves, it
575-451: A Bohr electron "wavelength" could be seen to be a function of its momentum; so a Bohr orbiting electron was seen to orbit in a circle at a multiple of its half-wavelength. The Bohr model for a short time could be seen as a classical model with an additional constraint provided by the 'wavelength' argument. However, this period was immediately superseded by the full three-dimensional wave mechanics of 1926. In our current understanding of physics,
690-565: A Franco-Belgian science fiction comics series Orbital piercing See also [ edit ] Orbit (disambiguation) Topics referred to by the same term [REDACTED] This disambiguation page lists articles associated with the title Orbital . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=Orbital&oldid=1257027600 " Category : Disambiguation pages Hidden categories: Short description
805-447: A complete set of s, p, d, and f orbitals, respectively, though for higher values of quantum number n , particularly when the atom bears a positive charge, energies of certain sub-shells become very similar and so, order in which they are said to be populated by electrons (e.g., Cr = [Ar]4s 3d and Cr = [Ar]3d ) can be rationalized only somewhat arbitrarily. With the development of quantum mechanics and experimental findings (such as
920-460: A definite prediction of what the quantum state ψ ( t ) {\displaystyle \psi (t)} will be at any later time. Some wave functions produce probability distributions that are independent of time, such as eigenstates of the Hamiltonian . Many systems that are treated dynamically in classical mechanics are described by such "static" wave functions. For example,
1035-510: A family of unitary operators parameterized by a variable t {\displaystyle t} . Under the evolution generated by A {\displaystyle A} , any observable B {\displaystyle B} that commutes with A {\displaystyle A} will be conserved. Moreover, if B {\displaystyle B} is conserved by evolution under A {\displaystyle A} , then A {\displaystyle A}
1150-471: A loss of information, though: knowing the reduced density matrices of the individual systems is not enough to reconstruct the state of the composite system. Just as density matrices specify the state of a subsystem of a larger system, analogously, positive operator-valued measures (POVMs) describe the effect on a subsystem of a measurement performed on a larger system. POVMs are extensively used in quantum information theory. As described above, entanglement
1265-426: A mathematical formulation of quantum mechanics and survey its application to some useful and oft-studied examples. In the mathematically rigorous formulation of quantum mechanics, the state of a quantum mechanical system is a vector ψ {\displaystyle \psi } belonging to a ( separable ) complex Hilbert space H {\displaystyle {\mathcal {H}}} . This vector
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#17328510960711380-779: A maximum of two electrons, each with its own projection of spin m s {\displaystyle m_{s}} . The simple names s orbital , p orbital , d orbital , and f orbital refer to orbitals with angular momentum quantum number ℓ = 0, 1, 2, and 3 respectively. These names, together with their n values, are used to describe electron configurations of atoms. They are derived from description by early spectroscopists of certain series of alkali metal spectroscopic lines as sharp , principal , diffuse , and fundamental . Orbitals for ℓ > 3 continue alphabetically (g, h, i, k, ...), omitting j because some languages do not distinguish between letters "i" and "j". Atomic orbitals are basic building blocks of
1495-417: A measurement of its position and also at the same time for a measurement of its momentum . Another consequence of the mathematical rules of quantum mechanics is the phenomenon of quantum interference , which is often illustrated with the double-slit experiment . In the basic version of this experiment, a coherent light source , such as a laser beam, illuminates a plate pierced by two parallel slits, and
1610-468: A positively charged jelly-like substance, and between the electron's discovery and 1909, this " plum pudding model " was the most widely accepted explanation of atomic structure. Shortly after Thomson's discovery, Hantaro Nagaoka predicted a different model for electronic structure. Unlike the plum pudding model, the positive charge in Nagaoka's "Saturnian Model" was concentrated into a central core, pulling
1725-467: A probability amplitude. Applying the Born rule to these amplitudes gives a probability density function for the position that the electron will be found to have when an experiment is performed to measure it. This is the best the theory can do; it cannot say for certain where the electron will be found. The Schrödinger equation relates the collection of probability amplitudes that pertain to one moment of time to
1840-517: A rapid transit line usually encircling a city centre Orbital engine Other uses [ edit ] Orbital (novel) , a 2023 novel by Samantha Harvey Orbital (The Culture) , artificial worlds from Iain M. Banks's series of science fiction novels, the Culture Orbital (band) , an English electronic dance music duo, whose first single was 'Chime' [1989] Orbital (1991 album) Orbital (1993 album) Orbital (comics) ,
1955-414: A relatively tiny planet (the nucleus). Atomic orbitals exactly describe the shape of this "atmosphere" only when one electron is present. When more electrons are added, the additional electrons tend to more evenly fill in a volume of space around the nucleus so that the resulting collection ("electron cloud" ) tends toward a generally spherical zone of probability describing the electron's location, because of
2070-424: A set of quantum numbers summarized in the term symbol and usually associated with particular electron configurations, i.e., by occupation schemes of atomic orbitals (for example, 1s 2s 2p for the ground state of neon -term symbol: S 0 ). This notation means that the corresponding Slater determinants have a clear higher weight in the configuration interaction expansion. The atomic orbital concept
2185-405: A single electron in an unexcited atom is pictured classically as a particle moving in a circular trajectory around the atomic nucleus , whereas in quantum mechanics, it is described by a static wave function surrounding the nucleus. For example, the electron wave function for an unexcited hydrogen atom is a spherically symmetric function known as an s orbital ( Fig. 1 ). Analytic solutions of
2300-545: A single spatial dimension. A free particle is one which is not subject to external influences, so that its Hamiltonian consists only of its kinetic energy: The general solution of the Schrödinger equation is given by which is a superposition of all possible plane waves e i ( k x − ℏ k 2 2 m t ) {\displaystyle e^{i(kx-{\frac {\hbar k^{2}}{2m}}t)}} , which are eigenstates of
2415-468: Is and this provides the lower bound on the product of standard deviations: Another consequence of the canonical commutation relation is that the position and momentum operators are Fourier transforms of each other, so that a description of an object according to its momentum is the Fourier transform of its description according to its position. The fact that dependence in momentum is the Fourier transform of
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#17328510960712530-478: Is a key feature of models of measurement processes in which an apparatus becomes entangled with the system being measured. Systems interacting with the environment in which they reside generally become entangled with that environment, a phenomenon known as quantum decoherence . This can explain why, in practice, quantum effects are difficult to observe in systems larger than microscopic. There are many mathematically equivalent formulations of quantum mechanics. One of
2645-424: Is a valid joint state that is not separable. States that are not separable are called entangled . If the state for a composite system is entangled, it is impossible to describe either component system A or system B by a state vector. One can instead define reduced density matrices that describe the statistics that can be obtained by making measurements on either component system alone. This necessarily causes
2760-481: Is actually a function of the coordinates of all the electrons, so that their motion is correlated, but this is often approximated by this independent-particle model of products of single electron wave functions. (The London dispersion force , for example, depends on the correlations of the motion of the electrons.) In atomic physics , the atomic spectral lines correspond to transitions ( quantum leaps ) between quantum states of an atom. These states are labeled by
2875-405: Is conserved under the evolution generated by B {\displaystyle B} . This implies a quantum version of the result proven by Emmy Noether in classical ( Lagrangian ) mechanics: for every differentiable symmetry of a Hamiltonian, there exists a corresponding conservation law . The simplest example of a quantum system with a position degree of freedom is a free particle in
2990-1066: Is considered as a sum over all possible classical and non-classical paths between the initial and final states. This is the quantum-mechanical counterpart of the action principle in classical mechanics. The Hamiltonian H {\displaystyle H} is known as the generator of time evolution, since it defines a unitary time-evolution operator U ( t ) = e − i H t / ℏ {\displaystyle U(t)=e^{-iHt/\hbar }} for each value of t {\displaystyle t} . From this relation between U ( t ) {\displaystyle U(t)} and H {\displaystyle H} , it follows that any observable A {\displaystyle A} that commutes with H {\displaystyle H} will be conserved : its expectation value will not change over time. This statement generalizes, as mathematically, any Hermitian operator A {\displaystyle A} can generate
3105-418: Is different from Wikidata All article disambiguation pages All disambiguation pages Atomic orbital In quantum mechanics , an atomic orbital ( / ˈ ɔːr b ɪ t ə l / ) is a function describing the location and wave-like behavior of an electron in an atom . This function describes an electron's charge distribution around the atom's nucleus , and can be used to calculate
3220-448: Is given by The operator U ( t ) = e − i H t / ℏ {\displaystyle U(t)=e^{-iHt/\hbar }} is known as the time-evolution operator, and has the crucial property that it is unitary . This time evolution is deterministic in the sense that – given an initial quantum state ψ ( 0 ) {\displaystyle \psi (0)} – it makes
3335-406: Is its associated eigenvector. More generally, the eigenvalue is degenerate and the probability is given by ⟨ ψ , P λ ψ ⟩ {\displaystyle \langle \psi ,P_{\lambda }\psi \rangle } , where P λ {\displaystyle P_{\lambda }} is the projector onto its associated eigenspace. In
3450-726: Is known as wave–particle duality . In addition to light, electrons , atoms , and molecules are all found to exhibit the same dual behavior when fired towards a double slit. Another non-classical phenomenon predicted by quantum mechanics is quantum tunnelling : a particle that goes up against a potential barrier can cross it, even if its kinetic energy is smaller than the maximum of the potential. In classical mechanics this particle would be trapped. Quantum tunnelling has several important consequences, enabling radioactive decay , nuclear fusion in stars, and applications such as scanning tunnelling microscopy , tunnel diode and tunnel field-effect transistor . When quantum systems interact,
3565-444: Is not possible for the solution to be a single momentum eigenstate, or a single position eigenstate, as these are not normalizable quantum states. Instead, we can consider a Gaussian wave packet : which has Fourier transform, and therefore momentum distribution We see that as we make a {\displaystyle a} smaller the spread in position gets smaller, but the spread in momentum gets larger. Conversely, by making
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3680-628: Is not sufficient for describing them at very small submicroscopic (atomic and subatomic ) scales. Most theories in classical physics can be derived from quantum mechanics as an approximation, valid at large (macroscopic/microscopic) scale. Quantum systems have bound states that are quantized to discrete values of energy , momentum , angular momentum , and other quantities, in contrast to classical systems where these quantities can be measured continuously. Measurements of quantum systems show characteristics of both particles and waves ( wave–particle duality ), and there are limits to how accurately
3795-815: Is part of quantum communication protocols, such as quantum key distribution and superdense coding . Contrary to popular misconception, entanglement does not allow sending signals faster than light , as demonstrated by the no-communication theorem . Another possibility opened by entanglement is testing for " hidden variables ", hypothetical properties more fundamental than the quantities addressed in quantum theory itself, knowledge of which would allow more exact predictions than quantum theory provides. A collection of results, most significantly Bell's theorem , have demonstrated that broad classes of such hidden-variable theories are in fact incompatible with quantum physics. According to Bell's theorem, if nature actually operates in accord with any theory of local hidden variables, then
3910-535: Is postulated to be normalized under the Hilbert space inner product, that is, it obeys ⟨ ψ , ψ ⟩ = 1 {\displaystyle \langle \psi ,\psi \rangle =1} , and it is well-defined up to a complex number of modulus 1 (the global phase), that is, ψ {\displaystyle \psi } and e i α ψ {\displaystyle e^{i\alpha }\psi } represent
4025-466: Is replaced by − i ℏ ∂ ∂ x {\displaystyle -i\hbar {\frac {\partial }{\partial x}}} , and in particular in the non-relativistic Schrödinger equation in position space the momentum-squared term is replaced with a Laplacian times − ℏ 2 {\displaystyle -\hbar ^{2}} . When two different quantum systems are considered together,
4140-454: Is represented by its numerical value, but ℓ {\displaystyle \ell } is represented by a letter as follows: 0 is represented by 's', 1 by 'p', 2 by 'd', 3 by 'f', and 4 by 'g'. For instance, one may speak of the subshell with n = 2 {\displaystyle n=2} and ℓ = 0 {\displaystyle \ell =0} as a '2s subshell'. Each electron also has angular momentum in
4255-418: Is the formation of such a wave packet which localizes the wave, and thus the particle, in space. In states where a quantum mechanical particle is bound, it must be localized as a wave packet, and the existence of the packet and its minimum size implies a spread and minimal value in particle wavelength, and thus also momentum and energy. In quantum mechanics, as a particle is localized to a smaller region in space,
4370-717: Is the real spherical harmonic related to either the real or imaginary part of the complex spherical harmonic Y ℓ m {\displaystyle Y_{\ell }^{m}} . Quantum mechanics Quantum mechanics is a fundamental theory that describes the behavior of nature at and below the scale of atoms . It is the foundation of all quantum physics , which includes quantum chemistry , quantum field theory , quantum technology , and quantum information science . Quantum mechanics can describe many systems that classical physics cannot. Classical physics can describe many aspects of nature at an ordinary ( macroscopic and (optical) microscopic ) scale, but
4485-415: Is the reduced Planck constant . The constant i ℏ {\displaystyle i\hbar } is introduced so that the Hamiltonian is reduced to the classical Hamiltonian in cases where the quantum system can be approximated by a classical system; the ability to make such an approximation in certain limits is called the correspondence principle . The solution of this differential equation
4600-469: Is then If the state for the first system is the vector ψ A {\displaystyle \psi _{A}} and the state for the second system is ψ B {\displaystyle \psi _{B}} , then the state of the composite system is Not all states in the joint Hilbert space H A B {\displaystyle {\mathcal {H}}_{AB}} can be written in this form, however, because
4715-498: Is therefore a key concept for visualizing the excitation process associated with a given transition . For example, one can say for a given transition that it corresponds to the excitation of an electron from an occupied orbital to a given unoccupied orbital. Nevertheless, one has to keep in mind that electrons are fermions ruled by the Pauli exclusion principle and cannot be distinguished from each other. Moreover, it sometimes happens that
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4830-427: The n = 2 shell has only orbitals with ℓ = 0 {\displaystyle \ell =0} , and ℓ = 1 {\displaystyle \ell =1} . The set of orbitals associated with a particular value of ℓ are sometimes collectively called a subshell . The magnetic quantum number , m ℓ {\displaystyle m_{\ell }} , describes
4945-520: The Bohr model where it determines the radius of each circular electron orbit. In modern quantum mechanics however, n determines the mean distance of the electron from the nucleus; all electrons with the same value of n lie at the same average distance. For this reason, orbitals with the same value of n are said to comprise a " shell ". Orbitals with the same value of n and also the same value of ℓ are even more closely related, and are said to comprise
5060-501: The Born rule : in the simplest case the eigenvalue λ {\displaystyle \lambda } is non-degenerate and the probability is given by | ⟨ λ → , ψ ⟩ | 2 {\displaystyle |\langle {\vec {\lambda }},\psi \rangle |^{2}} , where λ → {\displaystyle {\vec {\lambda }}}
5175-3110: The Condon–Shortley phase convention , real orbitals are related to complex orbitals in the same way that the real spherical harmonics are related to complex spherical harmonics. Letting ψ n , ℓ , m {\displaystyle \psi _{n,\ell ,m}} denote a complex orbital with quantum numbers n , ℓ , and m , the real orbitals ψ n , ℓ , m real {\displaystyle \psi _{n,\ell ,m}^{\text{real}}} may be defined by ψ n , ℓ , m real = { 2 ( − 1 ) m Im { ψ n , ℓ , | m | } for m < 0 ψ n , ℓ , | m | for m = 0 2 ( − 1 ) m Re { ψ n , ℓ , | m | } for m > 0 = { i 2 ( ψ n , ℓ , − | m | − ( − 1 ) m ψ n , ℓ , | m | ) for m < 0 ψ n , ℓ , | m | for m = 0 1 2 ( ψ n , ℓ , − | m | + ( − 1 ) m ψ n , ℓ , | m | ) for m > 0 {\displaystyle {\begin{aligned}\psi _{n,\ell ,m}^{\text{real}}&={\begin{cases}{\sqrt {2}}(-1)^{m}{\text{Im}}\left\{\psi _{n,\ell ,|m|}\right\}&{\text{ for }}m<0\\[2pt]\psi _{n,\ell ,|m|}&{\text{ for }}m=0\\[2pt]{\sqrt {2}}(-1)^{m}{\text{Re}}\left\{\psi _{n,\ell ,|m|}\right\}&{\text{ for }}m>0\end{cases}}\\[4pt]&={\begin{cases}{\frac {i}{\sqrt {2}}}\left(\psi _{n,\ell ,-|m|}-(-1)^{m}\psi _{n,\ell ,|m|}\right)&{\text{ for }}m<0\\[2pt]\psi _{n,\ell ,|m|}&{\text{ for }}m=0\\[4pt]{\frac {1}{\sqrt {2}}}\left(\psi _{n,\ell ,-|m|}+(-1)^{m}\psi _{n,\ell ,|m|}\right)&{\text{ for }}m>0\end{cases}}\end{aligned}}} If ψ n , ℓ , m ( r , θ , ϕ ) = R n l ( r ) Y ℓ m ( θ , ϕ ) {\displaystyle \psi _{n,\ell ,m}(r,\theta ,\phi )=R_{nl}(r)Y_{\ell }^{m}(\theta ,\phi )} , with R n l ( r ) {\displaystyle R_{nl}(r)}
5290-580: The Hartree–Fock approximation, which is one way to reduce the complexities of molecular orbital theory . Atomic orbitals can be the hydrogen-like "orbitals" which are exact solutions to the Schrödinger equation for a hydrogen-like "atom" (i.e., atom with one electron). Alternatively, atomic orbitals refer to functions that depend on the coordinates of one electron (i.e., orbitals) but are used as starting points for approximating wave functions that depend on
5405-418: The Pauli exclusion principle . Thus the n = 1 state can hold one or two electrons, while the n = 2 state can hold up to eight electrons in 2s and 2p subshells. In helium, all n = 1 states are fully occupied; the same is true for n = 1 and n = 2 in neon. In argon, the 3s and 3p subshells are similarly fully occupied by eight electrons; quantum mechanics also allows
5520-507: The atomic orbital model (or electron cloud or wave mechanics model), a modern framework for visualizing submicroscopic behavior of electrons in matter. In this model, the electron cloud of an atom may be seen as being built up (in approximation) in an electron configuration that is a product of simpler hydrogen-like atomic orbitals. The repeating periodicity of blocks of 2, 6, 10, and 14 elements within sections of periodic table arises naturally from total number of electrons that occupy
5635-713: The canonical commutation relation : Given a quantum state, the Born rule lets us compute expectation values for both X {\displaystyle X} and P {\displaystyle P} , and moreover for powers of them. Defining the uncertainty for an observable by a standard deviation , we have and likewise for the momentum: The uncertainty principle states that Either standard deviation can in principle be made arbitrarily small, but not both simultaneously. This inequality generalizes to arbitrary pairs of self-adjoint operators A {\displaystyle A} and B {\displaystyle B} . The commutator of these two operators
5750-456: The emission and absorption spectra of atoms became an increasingly useful tool in the understanding of electrons in atoms. The most prominent feature of emission and absorption spectra (known experimentally since the middle of the 19th century), was that these atomic spectra contained discrete lines. The significance of the Bohr model was that it related the lines in emission and absorption spectra to
5865-530: The orbital bone Orbitofrontal cortex , a part of the brain used for decision making Business [ edit ] Orbital Corporation , an Australian engine technology company Orbital Sciences Corporation , a U.S. satellite launch and defense systems corporation Orbital ATK , American aerospace manufacturer formed from the merger of Orbital Sciences Corporation and parts of Alliant Techsystems Transportation [ edit ] Ring road (or orbital road in some regions) Orbital (metro) ,
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#17328510960715980-426: The periodic table . The stationary states ( quantum states ) of a hydrogen-like atom are its atomic orbitals. However, in general, an electron's behavior is not fully described by a single orbital. Electron states are best represented by time-depending "mixtures" ( linear combinations ) of multiple orbitals. See Linear combination of atomic orbitals molecular orbital method . The quantum number n first appeared in
6095-423: The photoelectric effect . These early attempts to understand microscopic phenomena, now known as the " old quantum theory ", led to the full development of quantum mechanics in the mid-1920s by Niels Bohr , Erwin Schrödinger , Werner Heisenberg , Max Born , Paul Dirac and others. The modern theory is formulated in various specially developed mathematical formalisms . In one of them, a mathematical entity called
6210-716: The probability of finding an electron in a specific region around the nucleus. Each orbital in an atom is characterized by a set of values of three quantum numbers n , ℓ , and m ℓ , which respectively correspond to electron's energy, its orbital angular momentum , and its orbital angular momentum projected along a chosen axis ( magnetic quantum number ). The orbitals with a well-defined magnetic quantum number are generally complex-valued. Real-valued orbitals can be formed as linear combinations of m ℓ and −m ℓ orbitals, and are often labeled using associated harmonic polynomials (e.g., xy , x − y ) which describe their angular structure. An orbital can be occupied by
6325-502: The uncertainty principle . One should remember that these orbital 'states', as described here, are merely eigenstates of an electron in its orbit. An actual electron exists in a superposition of states, which is like a weighted average , but with complex number weights. So, for instance, an electron could be in a pure eigenstate (2, 1, 0), or a mixed state 1 / 2 (2, 1, 0) + 1 / 2 i {\displaystyle i} (2, 1, 1), or even
6440-562: The wave function provides information, in the form of probability amplitudes , about what measurements of a particle's energy, momentum, and other physical properties may yield. Quantum mechanics allows the calculation of properties and behaviour of physical systems. It is typically applied to microscopic systems: molecules, atoms and sub-atomic particles. It has been demonstrated to hold for complex molecules with thousands of atoms, but its application to human beings raises philosophical problems, such as Wigner's friend , and its application to
6555-458: The 2p subshell of an atom contains 4 electrons. This subshell has 3 orbitals, each with n = 2 and ℓ = 1. There is also another, less common system still used in X-ray science known as X-ray notation , which is a continuation of the notations used before orbital theory was well understood. In this system, the principal quantum number is given a letter associated with it. For n = 1, 2, 3, 4, 5, ... ,
6670-402: The Bohr model is called a semi-classical model because of its quantization of angular momentum, not primarily because of its relationship with electron wavelength, which appeared in hindsight a dozen years after the Bohr model was proposed. The Bohr model was able to explain the emission and absorption spectra of hydrogen . The energies of electrons in the n = 1, 2, 3, etc. states in
6785-423: The Bohr model match those of current physics. However, this did not explain similarities between different atoms, as expressed by the periodic table, such as the fact that helium (two electrons), neon (10 electrons), and argon (18 electrons) exhibit similar chemical inertness. Modern quantum mechanics explains this in terms of electron shells and subshells which can each hold a number of electrons determined by
6900-431: The Hilbert space for the spin of a single proton is simply the space of two-dimensional complex vectors C 2 {\displaystyle \mathbb {C} ^{2}} with the usual inner product. Physical quantities of interest – position, momentum, energy, spin – are represented by observables, which are Hermitian (more precisely, self-adjoint ) linear operators acting on
7015-411: The Hilbert space of the combined system is the tensor product of the Hilbert spaces of the two components. For example, let A and B be two quantum systems, with Hilbert spaces H A {\displaystyle {\mathcal {H}}_{A}} and H B {\displaystyle {\mathcal {H}}_{B}} , respectively. The Hilbert space of the composite system
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#17328510960717130-432: The Hilbert space. A quantum state can be an eigenvector of an observable, in which case it is called an eigenstate , and the associated eigenvalue corresponds to the value of the observable in that eigenstate. More generally, a quantum state will be a linear combination of the eigenstates, known as a quantum superposition . When an observable is measured, the result will be one of its eigenvalues with probability given by
7245-489: The Schrödinger equation are known for very few relatively simple model Hamiltonians including the quantum harmonic oscillator , the particle in a box , the dihydrogen cation , and the hydrogen atom . Even the helium atom – which contains just two electrons – has defied all attempts at a fully analytic treatment, admitting no solution in closed form . However, there are techniques for finding approximate solutions. One method, called perturbation theory , uses
7360-662: The accuracy of hydrogen-like orbitals. The term orbital was introduced by Robert S. Mulliken in 1932 as short for one-electron orbital wave function . Niels Bohr explained around 1913 that electrons might revolve around a compact nucleus with definite angular momentum. Bohr's model was an improvement on the 1911 explanations of Ernest Rutherford , that of the electron moving around a nucleus. Japanese physicist Hantaro Nagaoka published an orbit-based hypothesis for electron behavior as early as 1904. These theories were each built upon new observations starting with simple understanding and becoming more correct and complex. Explaining
7475-403: The analytic result for a simple quantum mechanical model to create a result for a related but more complicated model by (for example) the addition of a weak potential energy . Another approximation method applies to systems for which quantum mechanics produces only small deviations from classical behavior. These deviations can then be computed based on the classical motion. One consequence of
7590-461: The associated compressed wave packet requires a larger and larger range of momenta, and thus larger kinetic energy. Thus the binding energy to contain or trap a particle in a smaller region of space increases without bound as the region of space grows smaller. Particles cannot be restricted to a geometric point in space, since this would require infinite particle momentum. In chemistry, Erwin Schrödinger , Linus Pauling , Mulliken and others noted that
7705-399: The atom fixed the problem of energy loss from radiation from a ground state (by declaring that there was no state below this), and more importantly explained the origin of spectral lines. After Bohr's use of Einstein 's explanation of the photoelectric effect to relate energy levels in atoms with the wavelength of emitted light, the connection between the structure of electrons in atoms and
7820-547: The atomic orbitals are the eigenstates of the Hamiltonian operator for the energy. They can be obtained analytically, meaning that the resulting orbitals are products of a polynomial series, and exponential and trigonometric functions . (see hydrogen atom ). For atoms with two or more electrons, the governing equations can be solved only with the use of methods of iterative approximation. Orbitals of multi-electron atoms are qualitatively similar to those of hydrogen, and in
7935-606: The basic quantum formalism is the uncertainty principle. In its most familiar form, this states that no preparation of a quantum particle can imply simultaneously precise predictions both for a measurement of its position and for a measurement of its momentum. Both position and momentum are observables, meaning that they are represented by Hermitian operators . The position operator X ^ {\displaystyle {\hat {X}}} and momentum operator P ^ {\displaystyle {\hat {P}}} do not commute, but rather satisfy
8050-503: The behavior of these electron "orbits" was one of the driving forces behind the development of quantum mechanics . With J. J. Thomson 's discovery of the electron in 1897, it became clear that atoms were not the smallest building blocks of nature , but were rather composite particles. The newly discovered structure within atoms tempted many to imagine how the atom's constituent parts might interact with each other. Thomson theorized that multiple electrons revolve in orbit-like rings within
8165-557: The bulk of the atomic mass was tightly condensed into a nucleus, which was also found to be positively charged. It became clear from his analysis in 1911 that the plum pudding model could not explain atomic structure. In 1913, Rutherford's post-doctoral student, Niels Bohr , proposed a new model of the atom, wherein electrons orbited the nucleus with classical periods, but were permitted to have only discrete values of angular momentum, quantized in units ħ . This constraint automatically allowed only certain electron energies. The Bohr model of
8280-404: The collection of probability amplitudes that pertain to another. One consequence of the mathematical rules of quantum mechanics is a tradeoff in predictability between measurable quantities. The most famous form of this uncertainty principle says that no matter how a quantum particle is prepared or how carefully experiments upon it are arranged, it is impossible to have a precise prediction for
8395-465: The configuration interaction expansion converges very slowly and that one cannot speak about simple one-determinant wave function at all. This is the case when electron correlation is large. Fundamentally, an atomic orbital is a one-electron wave function, even though many electrons are not in one-electron atoms, and so the one-electron view is an approximation. When thinking about orbitals, we are often given an orbital visualization heavily influenced by
8510-440: The consequence of Heisenberg's relation was that the electron, as a wave packet, could not be considered to have an exact location in its orbital. Max Born suggested that the electron's position needed to be described by a probability distribution which was connected with finding the electron at some point in the wave-function which described its associated wave packet. The new quantum mechanics did not give exact results, but only
8625-626: The continuous case, these formulas give instead the probability density . After the measurement, if result λ {\displaystyle \lambda } was obtained, the quantum state is postulated to collapse to λ → {\displaystyle {\vec {\lambda }}} , in the non-degenerate case, or to P λ ψ / ⟨ ψ , P λ ψ ⟩ {\textstyle P_{\lambda }\psi {\big /}\!{\sqrt {\langle \psi ,P_{\lambda }\psi \rangle }}} , in
8740-431: The dependence in position means that the momentum operator is equivalent (up to an i / ℏ {\displaystyle i/\hbar } factor) to taking the derivative according to the position, since in Fourier analysis differentiation corresponds to multiplication in the dual space . This is why in quantum equations in position space, the momentum p i {\displaystyle p_{i}}
8855-587: The electrons into circular orbits reminiscent of Saturn's rings. Few people took notice of Nagaoka's work at the time, and Nagaoka himself recognized a fundamental defect in the theory even at its conception, namely that a classical charged object cannot sustain orbital motion because it is accelerating and therefore loses energy due to electromagnetic radiation. Nevertheless, the Saturnian model turned out to have more in common with modern theory than any of its contemporaries. In 1909, Ernest Rutherford discovered that
8970-419: The energy differences between the orbits that electrons could take around an atom. This was, however, not achieved by Bohr through giving the electrons some kind of wave-like properties, since the idea that electrons could behave as matter waves was not suggested until eleven years later. Still, the Bohr model's use of quantized angular momenta and therefore quantized energy levels was a significant step toward
9085-411: The form of quantum mechanical spin given by spin s = 1 / 2 . Its projection along a specified axis is given by the spin magnetic quantum number , m s , which can be + 1 / 2 or − 1 / 2 . These values are also called "spin up" or "spin down" respectively. The Pauli exclusion principle states that no two electrons in an atom can have
9200-415: The general case. The probabilistic nature of quantum mechanics thus stems from the act of measurement. This is one of the most difficult aspects of quantum systems to understand. It was the central topic in the famous Bohr–Einstein debates , in which the two scientists attempted to clarify these fundamental principles by way of thought experiments . In the decades after the formulation of quantum mechanics,
9315-413: The individual numbers and letters: "'one' 'ess'") is the lowest energy level ( n = 1 ) and has an angular quantum number of ℓ = 0 , denoted as s. Orbitals with ℓ = 1, 2 and 3 are denoted as p, d and f respectively. The set of orbitals for a given n and ℓ is called a subshell , denoted The superscript y shows the number of electrons in the subshell. For example, the notation 2p indicates that
9430-665: The integer values in the range − ℓ ≤ m ℓ ≤ ℓ {\displaystyle -\ell \leq m_{\ell }\leq \ell } . The above results may be summarized in the following table. Each cell represents a subshell, and lists the values of m ℓ {\displaystyle m_{\ell }} available in that subshell. Empty cells represent subshells that do not exist. Subshells are usually identified by their n {\displaystyle n} - and ℓ {\displaystyle \ell } -values. n {\displaystyle n}
9545-462: The interference pattern appears via the varying density of these particle hits on the screen. Furthermore, versions of the experiment that include detectors at the slits find that each detected photon passes through one slit (as would a classical particle), and not through both slits (as would a wave). However, such experiments demonstrate that particles do not form the interference pattern if one detects which slit they pass through. This behavior
9660-439: The letters associated with those numbers are K, L, M, N, O, ... respectively. The simplest atomic orbitals are those that are calculated for systems with a single electron, such as the hydrogen atom . An atom of any other element ionized down to a single electron (He , Li , etc.) is very similar to hydrogen, and the orbitals take the same form. In the Schrödinger equation for this system of one negative and one positive particle,
9775-430: The light passing through the slits is observed on a screen behind the plate. The wave nature of light causes the light waves passing through the two slits to interfere , producing bright and dark bands on the screen – a result that would not be expected if light consisted of classical particles. However, the light is always found to be absorbed at the screen at discrete points, as individual particles rather than waves;
9890-410: The math. You can choose a different basis of eigenstates by superimposing eigenstates from any other basis (see Real orbitals below). Atomic orbitals may be defined more precisely in formal quantum mechanical language. They are approximate solutions to the Schrödinger equation for the electrons bound to the atom by the electric field of the atom's nucleus . Specifically, in quantum mechanics,
10005-404: The mixed state 2 / 5 (2, 1, 0) + 3 / 5 i {\displaystyle i} (2, 1, 1). For each eigenstate, a property has an eigenvalue . So, for the three states just mentioned, the value of n {\displaystyle n} is 2, and the value of l {\displaystyle l} is 1. For the second and third states,
10120-411: The model is most useful when applied to physical systems that share these symmetries. The Stern–Gerlach experiment —where an atom is exposed to a magnetic field—provides one such example. Instead of the complex orbitals described above, it is common, especially in the chemistry literature, to use real atomic orbitals. These real orbitals arise from simple linear combinations of complex orbitals. Using
10235-432: The momentum operator with momentum p = ℏ k {\displaystyle p=\hbar k} . The coefficients of the superposition are ψ ^ ( k , 0 ) {\displaystyle {\hat {\psi }}(k,0)} , which is the Fourier transform of the initial quantum state ψ ( x , 0 ) {\displaystyle \psi (x,0)} . It
10350-413: The oldest and most common is the " transformation theory " proposed by Paul Dirac , which unifies and generalizes the two earliest formulations of quantum mechanics – matrix mechanics (invented by Werner Heisenberg ) and wave mechanics (invented by Erwin Schrödinger ). An alternative formulation of quantum mechanics is Feynman 's path integral formulation , in which a quantum-mechanical amplitude
10465-412: The one-dimensional case in the x {\displaystyle x} direction, the time-independent Schrödinger equation may be written With the differential operator defined by with state ψ {\displaystyle \psi } in this case having energy E {\displaystyle E} coincident with the kinetic energy of the particle. The general solutions of
10580-473: The orbital angular momentum of each electron and is a non-negative integer. Within a shell where n is some integer n 0 , ℓ ranges across all (integer) values satisfying the relation 0 ≤ ℓ ≤ n 0 − 1 {\displaystyle 0\leq \ell \leq n_{0}-1} . For instance, the n = 1 shell has only orbitals with ℓ = 0 {\displaystyle \ell =0} , and
10695-396: The original quantum system ceases to exist as an independent entity (see Measurement in quantum mechanics ). The time evolution of a quantum state is described by the Schrödinger equation: Here H {\displaystyle H} denotes the Hamiltonian , the observable corresponding to the total energy of the system, and ℏ {\displaystyle \hbar }
10810-428: The position becomes more and more uncertain. The uncertainty in momentum, however, stays constant. The particle in a one-dimensional potential energy box is the most mathematically simple example where restraints lead to the quantization of energy levels. The box is defined as having zero potential energy everywhere inside a certain region, and therefore infinite potential energy everywhere outside that region. For
10925-424: The probabilities for the occurrence of a variety of possible such results. Heisenberg held that the path of a moving particle has no meaning if we cannot observe it, as we cannot with electrons in an atom. In the quantum picture of Heisenberg, Schrödinger and others, the Bohr atom number n for each orbital became known as an n-sphere in a three-dimensional atom and was pictured as the most probable energy of
11040-423: The probability cloud of the electron's wave packet which surrounded the atom. Orbitals have been given names, which are usually given in the form: where X is the energy level corresponding to the principal quantum number n ; type is a lower-case letter denoting the shape or subshell of the orbital, corresponding to the angular momentum quantum number ℓ . For example, the orbital 1s (pronounced as
11155-450: The projection of the orbital angular momentum along a chosen axis. It determines the magnitude of the current circulating around that axis and the orbital contribution to the magnetic moment of an electron via the Ampèrian loop model. Within a subshell ℓ {\displaystyle \ell } , m ℓ {\displaystyle m_{\ell }} obtains
11270-400: The question of what constitutes a "measurement" has been extensively studied. Newer interpretations of quantum mechanics have been formulated that do away with the concept of " wave function collapse " (see, for example, the many-worlds interpretation ). The basic idea is that when a quantum system interacts with a measuring apparatus, their respective wave functions become entangled so that
11385-545: The radial functions R ( r ) which can be chosen as a starting point for the calculation of the properties of atoms and molecules with many electrons: Although hydrogen-like orbitals are still used as pedagogical tools, the advent of computers has made STOs preferable for atoms and diatomic molecules since combinations of STOs can replace the nodes in hydrogen-like orbitals. Gaussians are typically used in molecules with three or more atoms. Although not as accurate by themselves as STOs, combinations of many Gaussians can attain
11500-498: The radial part of the orbital, this definition is equivalent to ψ n , ℓ , m real ( r , θ , ϕ ) = R n l ( r ) Y ℓ m ( θ , ϕ ) {\displaystyle \psi _{n,\ell ,m}^{\text{real}}(r,\theta ,\phi )=R_{nl}(r)Y_{\ell m}(\theta ,\phi )} where Y ℓ m {\displaystyle Y_{\ell m}}
11615-413: The result can be the creation of quantum entanglement : their properties become so intertwined that a description of the whole solely in terms of the individual parts is no longer possible. Erwin Schrödinger called entanglement "... the characteristic trait of quantum mechanics, the one that enforces its entire departure from classical lines of thought". Quantum entanglement enables quantum computing and
11730-566: The results of a Bell test will be constrained in a particular, quantifiable way. Many Bell tests have been performed and they have shown results incompatible with the constraints imposed by local hidden variables. It is not possible to present these concepts in more than a superficial way without introducing the mathematics involved; understanding quantum mechanics requires not only manipulating complex numbers, but also linear algebra , differential equations , group theory , and other more advanced subjects. Accordingly, this article will present
11845-463: The same physical system. In other words, the possible states are points in the projective space of a Hilbert space, usually called the complex projective space . The exact nature of this Hilbert space is dependent on the system – for example, for describing position and momentum the Hilbert space is the space of complex square-integrable functions L 2 ( C ) {\displaystyle L^{2}(\mathbb {C} )} , while
11960-530: The same values of all four quantum numbers. If there are two electrons in an orbital with given values for three quantum numbers, ( n , ℓ , m ), these two electrons must differ in their spin projection m s . The above conventions imply a preferred axis (for example, the z direction in Cartesian coordinates), and they also imply a preferred direction along this preferred axis. Otherwise there would be no sense in distinguishing m = +1 from m = −1 . As such,
12075-406: The simplest models, they are taken to have the same form. For more rigorous and precise analysis, numerical approximations must be used. A given (hydrogen-like) atomic orbital is identified by unique values of three quantum numbers: n , ℓ , and m ℓ . The rules restricting the values of the quantum numbers, and their energies (see below), explain the electron configuration of the atoms and
12190-683: The simultaneous coordinates of all the electrons in an atom or molecule. The coordinate systems chosen for orbitals are usually spherical coordinates ( r , θ , φ ) in atoms and Cartesian ( x , y , z ) in polyatomic molecules. The advantage of spherical coordinates here is that an orbital wave function is a product of three factors each dependent on a single coordinate: ψ ( r , θ , φ ) = R ( r ) Θ( θ ) Φ( φ ) . The angular factors of atomic orbitals Θ( θ ) Φ( φ ) generate s, p, d, etc. functions as real combinations of spherical harmonics Y ℓm ( θ , φ ) (where ℓ and m are quantum numbers). There are typically three mathematical forms for
12305-639: The solutions to the hydrogen atom, where orbitals are given by the product between a radial function and a pure spherical harmonic . The quantum numbers, together with the rules governing their possible values, are as follows: The principal quantum number n describes the energy of the electron and is always a positive integer . In fact, it can be any positive integer, but for reasons discussed below, large numbers are seldom encountered. Each atom has, in general, many orbitals associated with each value of n ; these orbitals together are sometimes called electron shells . The azimuthal quantum number ℓ describes
12420-450: The state of an atom, i.e., an eigenstate of the atomic Hamiltonian , is approximated by an expansion (see configuration interaction expansion and basis set ) into linear combinations of anti-symmetrized products ( Slater determinants ) of one-electron functions. The spatial components of these one-electron functions are called atomic orbitals. (When one considers also their spin component, one speaks of atomic spin orbitals .) A state
12535-625: The superposition principle implies that linear combinations of these "separable" or "product states" are also valid. For example, if ψ A {\displaystyle \psi _{A}} and ϕ A {\displaystyle \phi _{A}} are both possible states for system A {\displaystyle A} , and likewise ψ B {\displaystyle \psi _{B}} and ϕ B {\displaystyle \phi _{B}} are both possible states for system B {\displaystyle B} , then
12650-441: The theory is that it usually cannot predict with certainty what will happen, but only give probabilities. Mathematically, a probability is found by taking the square of the absolute value of a complex number , known as a probability amplitude. This is known as the Born rule , named after physicist Max Born . For example, a quantum particle like an electron can be described by a wave function, which associates to each point in space
12765-472: The two slit diffraction of electrons), it was found that the electrons orbiting a nucleus could not be fully described as particles, but needed to be explained by wave–particle duality . In this sense, electrons have the following properties: Wave-like properties: Particle-like properties: Thus, electrons cannot be described simply as solid particles. An analogy might be that of a large and often oddly shaped "atmosphere" (the electron), distributed around
12880-411: The understanding of electrons in atoms, and also a significant step towards the development of quantum mechanics in suggesting that quantized restraints must account for all discontinuous energy levels and spectra in atoms. With de Broglie 's suggestion of the existence of electron matter waves in 1924, and for a short time before the full 1926 Schrödinger equation treatment of hydrogen-like atoms ,
12995-437: The universe as a whole remains speculative. Predictions of quantum mechanics have been verified experimentally to an extremely high degree of accuracy . For example, the refinement of quantum mechanics for the interaction of light and matter, known as quantum electrodynamics (QED), has been shown to agree with experiment to within 1 part in 10 when predicting the magnetic properties of an electron. A fundamental feature of
13110-579: The value for m l {\displaystyle m_{l}} is a superposition of 0 and 1. As a superposition of states, it is ambiguous—either exactly 0 or exactly 1—not an intermediate or average value like the fraction 1 / 2 . A superposition of eigenstates (2, 1, 1) and (3, 2, 1) would have an ambiguous n {\displaystyle n} and l {\displaystyle l} , but m l {\displaystyle m_{l}} would definitely be 1. Eigenstates make it easier to deal with
13225-519: The value of a physical quantity can be predicted prior to its measurement, given a complete set of initial conditions (the uncertainty principle ). Quantum mechanics arose gradually from theories to explain observations that could not be reconciled with classical physics, such as Max Planck 's solution in 1900 to the black-body radiation problem, and the correspondence between energy and frequency in Albert Einstein 's 1905 paper , which explained
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