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{{Short description|Different states of quantum systems}} {{Use American English|date=January 2019}}[[Image:Energy levels.svg|thumb|right|Energy levels for an [[electron]] in an [[atom]]: ground state and [[excited state]]s. After absorbing [[energy]], an electron may "jump" from the ground state to a higher energy excited state.]] {{Quantum mechanics|cTopic=Fundamental concepts}} A [[quantum mechanics|quantum mechanical]] system or [[particle]] that is [[bound state|bound]]—that is, confined spatially—can only take on certain discrete values of energy, called '''energy levels'''. This contrasts with [[classical mechanics|classical]] particles, which can have any amount of energy. The term is commonly used for the energy levels of the electrons in atoms, [[ion]]s, or [[molecule]]s, which are bound by the electric field of the [[atomic nucleus|nucleus]], but can also refer to energy levels of nuclei or [[molecular vibration|vibrational]] or [[rotational energy]] levels in molecules. The energy spectrum of a system with such discrete energy levels is said to be [[Quantization (physics)|quantized]]. In [[chemistry]] and [[atomic physics]], an electron shell, or principal energy level, may be thought of as the [[orbit]] of one or more [[electrons]] around an [[atom]]'s [[atomic nucleus|nucleus]]. The closest shell to the nucleus is called the "1 shell" (also called "K shell"), followed by the "2 shell" (or "L shell"), then the "3 shell" (or "M shell"), and so on further and further from the nucleus. The shells correspond with the [[principal quantum number]]s ({{math|''n''}} = 1, 2, 3, 4, ...) or are labeled alphabetically with letters used in the [[X-ray notation]] (K, L, M, N, ...). Each shell can contain only a fixed number of electrons: The first shell can hold up to two electrons, the second shell can hold up to eight (2 + 6) electrons, the third shell can hold up to 18 (2 + 6 + 10) and so on. The general formula is that the ''n''th shell can in principle hold up to 2''n''<sup>2</sup> electrons.<ref name="madsci">[http://www.madsci.org/posts/archives/1999-03/921736624.Ch.r.html Re: Why do electron shells have set limits ?] madsci.org, 17 March 1999, Dan Berger, Faculty Chemistry/Science, Bluffton College</ref> Since electrons are [[electrostatic field|electrically attracted]] to the nucleus, an atom's electrons will generally occupy outer shells only if the more inner shells have already been completely filled by other electrons. However, this is not a strict requirement: atoms may have two or even three incomplete outer shells. (See [[Madelung rule]] for more details.) For an explanation of why electrons exist in these shells see [[electron configuration]].<ref name="corrosionsource.com">[http://www.corrosionsource.com/handbook/periodic/e_subshells.htm Electron Subshells]. Corrosion Source. Retrieved on 1 December 2011.</ref> If the [[potential energy]] is set to zero at [[infinity|infinite]] distance from the atomic nucleus or molecule, the usual convention, then [[bound states|bound electron states]] have negative potential energy. If an atom, ion, or molecule is at the lowest possible energy level, it and its electrons are said to be in the ''[[ground state]]''. If it is at a higher energy level, it is said to be ''[[excited state|excited]]'', or any electrons that have higher energy than the ground state are ''excited''. An energy level is regarded as [[degenerate energy level|degenerate]] if there is more than one measurable quantum mechanical [[quantum state|state]] associated with it. == Explanation == [[File:Hydrogen Density Plots.png|thumb|[[Wavefunction]]s of a [[hydrogen]] atom, showing the probability of finding the electron in the space around the nucleus. Each stationary state defines a specific energy level of the atom.]] Quantized energy levels result from the wave behavior of particles, which gives a relationship between a particle's energy and its [[wavelength]]. For a confined particle such as an [[electron]] in an atom, the [[wave function]]s that have well defined energies have the form of a [[standing wave]].<ref name="Tipler"> {{cite book | last1 = Tipler | first1 = Paul A. | last2 = Mosca | first2 = Gene | title = Physics for Scientists and Engineers, 5th Ed. | publisher = W. H. Freeman and Co. | volume = 2 | date = 2004 | pages = 1129 | url = https://books.google.com/books?id=R2Nuh3Ux1AwC&dq=%22energy+level%22+%22standing+waves%22&pg=PA1129 | isbn = 0716708108 }}</ref> States having well-defined energies are called [[stationary state]]s because they are the states that do not change in time. Informally, these states correspond to a whole number of wavelengths of the [[wavefunction]] along a closed path (a path that ends where it started), such as a circular orbit around an atom, where the number of wavelengths gives the type of [[atomic orbital]] (0 for s-orbitals, 1 for p-orbitals and so on). Elementary examples that show mathematically how energy levels come about are the [[particle in a box]] and the [[quantum harmonic oscillator]]. Any [[quantum superposition|superposition]] ([[linear combination]]) of energy states is also a quantum state, but such states change with time and do not have well-defined energies. A measurement of the energy results in the [[wavefunction collapse|collapse]] of the wavefunction, which results in a new state that consists of just a single energy state. Measurement of the possible energy levels of an object is called [[spectroscopy]]. == History == The first evidence of quantization in atoms was the observation of [[spectral lines]] in light from the sun in the early 1800s by [[Joseph von Fraunhofer]] and [[William Hyde Wollaston]]. The notion of energy levels was proposed in 1913 by Danish physicist [[Niels Bohr]] in the [[Bohr theory]] of the atom. The modern quantum mechanical theory giving an explanation of these energy levels in terms of the [[Schrödinger equation]] was advanced by [[Erwin Schrödinger]] and [[Werner Heisenberg]] in 1926. == Atoms == === Intrinsic energy levels === In the formulas for energy of electrons at various levels given below in an atom, the zero point for energy is set when the electron in question has completely left the atom; i.e. when the electron's principal quantum number {{math|1=''n'' = ∞}}. When the electron is bound to the atom in any closer value of {{mvar|n}}, the electron's energy is lower and is considered negative. ==== Orbital state energy level: atom/ion with nucleus + one electron ==== Assume there is one electron in a given atomic orbital in a [[Hydrogen-like atom|hydrogen-like atom (ion)]]. The energy of its state is mainly determined by the electrostatic interaction of the (negative) electron with the (positive) nucleus. The energy levels of an electron around a nucleus are given by: : <math>E_n = - h c R_{\infty} \frac{Z^2}{n^2}</math> (typically between 1 [[electronvolt|eV]] and 10<sup>3</sup> eV), where {{math|''R''<sub>∞</sub>}} is the [[Rydberg constant]], {{mvar|Z}} is the [[atomic number]], {{mvar|n}} is the principal quantum number, {{math|''h''}} is the [[Planck constant]], and {{math|''c''}} is the [[speed of light]]. For hydrogen-like atoms (ions) only, the Rydberg levels depend only on the principal quantum number {{mvar|n}}. This equation is obtained from combining the [[Rydberg formula#Rydberg formula for any hydrogen-like element|Rydberg formula for any hydrogen-like element]] (shown below) with {{math|1=''E'' = ''hν'' = ''hc'' / ''λ''}} assuming that the principal quantum number {{mvar|n}} above = {{math|''n''<sub>1</sub>}} in the Rydberg formula and {{math|1=''n''<sub>2</sub> = ∞}} (principal quantum number of the energy level the electron descends from, when emitting a [[photon]]). The [[Rydberg formula]] was derived from empirical [[Emission spectrum|spectroscopic emission]] data. : <math>\frac{1}{\lambda} = RZ^2 \left(\frac{1}{n_1^2}-\frac{1}{n_2^2}\right)</math> An equivalent formula can be derived quantum mechanically from the time-independent [[Schrödinger equation]] with a kinetic energy [[Hamiltonian operator]] using a [[wave function]] as an [[eigenfunction]] to obtain the energy levels as [[Eigenvalue#Schrödinger equation|eigenvalues]], but the Rydberg constant would be replaced by other fundamental physics constants. ==== Electron–electron interactions in atoms ==== If there is more than one electron around the atom, electron–electron interactions raise the energy level. These interactions are often neglected if the spatial overlap of the electron wavefunctions is low. For multi-electron atoms, interactions between electrons cause the preceding equation to be no longer accurate as stated simply with {{mvar|Z}} as the [[atomic number]]. A simple (though not complete<!-- **Note** what is hinted at here is that screening is only a mean-field effect. Electron-electron interactions also lead to dynamic correlation-exchange energy shifts. If strong enough, correlation-exchange can prevent us from being able to look at the atom in terms of orbitals at all, leaving only the consideration of many-body states. However, in the case of atoms the correlation-exchange seems to be a small perturbation (usually). -->) way to understand this is as a [[shielding effect]], where the outer electrons see an effective nucleus of reduced charge, since the inner electrons are bound tightly to the nucleus and partially cancel its charge. This leads to an approximate correction where {{mvar|Z}} is substituted with an [[effective nuclear charge]] symbolized as {{math|''Z''<sub>eff</sub>}} that depends strongly on the principal quantum number. <math display="block">E_{n,\ell} = - h c R_{\infty} \frac{{Z_{\rm eff}}^2}{n^2}</math> In such cases, the orbital types (determined by the [[azimuthal quantum number]] {{mvar|ℓ}}) as well as their levels within the molecule affect {{math|''Z''<sub>eff</sub>}} and therefore also affect the various atomic electron energy levels. The [[Aufbau principle]] of filling an atom with electrons for an [[electron configuration]] takes these differing energy levels into account. For filling an atom with electrons in the ground state, the lowest energy levels are filled first and consistent with the [[Pauli exclusion principle]], the [[Aufbau principle]], and [[Hund's rule]]. ==== Fine structure splitting ==== [[Fine structure]] arises from relativistic kinetic energy corrections, [[spin–orbit coupling]] (an electrodynamic interaction between the electron's [[spin (physics)|spin]] and motion and the nucleus's electric field) and the Darwin term (contact term interaction of {{serif|s}} shell{{which|reason=of which principal q.n.?|date=January 2014}} electrons inside the nucleus). These affect the levels by a typical order of magnitude of 10<sup>−3</sup> eV. ==== Hyperfine structure ==== {{Main|Hyperfine structure}} This even finer structure is due to electron–nucleus [[Angular momentum coupling#Spin–spin coupling|spin–spin interaction]],<!--might be better to link to [[J-coupling]]. Will check later--> resulting in a typical change in the energy levels by a typical order of magnitude of 10<sup>−4</sup> eV. === Energy levels due to external fields === ==== Zeeman effect ==== {{Main|Zeeman effect}} There is an interaction energy associated with the magnetic dipole moment, {{math|'''''μ'''''<sub>''L''</sub>}}, arising from the electronic orbital angular momentum, {{math|''L''}}, given by : <math>U = -\boldsymbol{\mu}_L\cdot\mathbf{B}</math> with : <math>-\boldsymbol{\mu}_L = \dfrac{e\hbar}{2m}\mathbf{L} = \mu_B\mathbf{L}</math>. Additionally taking into account the magnetic momentum arising from the electron spin. Due to relativistic effects ([[Dirac equation]]), there is a magnetic momentum, {{math|'''''μ'''''<sub>''S''</sub>}}, arising from the electron spin : <math>-\boldsymbol{\mu}_S = -\mu_\text{B} g_S \mathbf{S}</math>, with {{math|''g''<sub>''S''</sub>}} the electron-spin [[g-factor (physics)|g-factor]] (about 2), resulting in a total magnetic moment, {{math|'''''μ'''''}}, : <math>\boldsymbol{\mu} = \boldsymbol{\mu}_L + \boldsymbol{\mu}_S</math>. The interaction energy therefore becomes : <math>U_B = -\boldsymbol{\mu}\cdot\mathbf{B} = \mu_\text{B} B (M_L + g_S M_S)</math>. ==== Stark effect ==== {{Main|Stark effect}} == Molecules == [[Chemical bond]]s between atoms in a molecule form because they make the situation more stable for the involved atoms, which generally means the sum energy level for the involved atoms in the molecule is lower than if the atoms were not so bonded. As separate atoms approach each other to [[covalent bond|covalently bond]], their orbitals affect each other's energy levels to form bonding and antibonding [[molecular orbital]]s. The energy level of the [[bonding molecular orbital|bonding orbitals]] is lower, and the energy level of the [[antibonding molecular orbital|antibonding orbitals]] is higher. For the bond in the molecule to be stable, the covalent bonding electrons occupy the lower energy bonding orbital, which may be signified by such symbols as σ or π depending on the situation. Corresponding anti-bonding orbitals can be signified by adding an asterisk to get σ* or π* orbitals. A [[non-bonding orbital]] in a molecule is an orbital with electrons in outer [[electron shell|shell]]s which do not participate in bonding and its energy level is the same as that of the constituent atom. Such orbitals can be designated as '''n''' orbitals. The electrons in an n orbital are typically [[lone pair]]s. <ref name="chemguide">[http://www.chemguide.co.uk/analysis/uvvisible/theory.html#top UV-Visible Absorption Spectra]</ref> In polyatomic molecules, different vibrational and rotational energy levels are also involved. Roughly speaking, a molecular energy state (i.e., an [[eigenstate]] of the [[molecular Hamiltonian]]) is the sum of the electronic, vibrational, rotational, nuclear, and translational components, such that: <math display="block">E = E_\text{electronic} + E_\text{vibrational} + E_\text{rotational} + E_\text{nuclear} + E_\text{translational}</math> where {{math|''E''<sub>electronic</sub>}} is an [[eigenvalue]] of the [[electronic molecular Hamiltonian]] (the value of the [[potential energy surface]]) at the [[molecular geometry|equilibrium geometry of the molecule]]. The molecular energy levels are labelled by the [[molecular term symbol]]s. The specific energies of these components vary with the specific energy state and the substance. === Energy level diagrams === There are various types of energy level diagrams for bonds between atoms in a molecule. ; Examples : ''[[Molecular orbital diagram]]s'', ''[[Jablonski diagram]]s'', and ''[[Franck–Condon principle|Franck–Condon]]'' diagrams. == Energy level transitions == {{Further|atomic electron transition|molecular electron transition}} [[File:Atomic Absorption (hv corrected).png|thumb|right|200px|An increase in energy level from {{math|''E''<sub>1</sub>}} to {{math|''E''<sub>2</sub>}} resulting from absorption of a photon represented by the red squiggly arrow, and whose energy is {{math|''[[Planck constant|h]] [[Frequency|ν]]''}}.]] [[File:Schematic diagram of atomic line spontaneous emission (hv corrected).png|thumb|left|200px|A decrease in energy level from {{math|''E''<sub>2</sub>}} to {{math|''E''<sub>1</sub>}} resulting in emission of a photon represented by the red squiggly arrow, and whose energy is {{math|''h ν''}}.]] Electrons in atoms and molecules can change (make ''[[Atomic electron transition|transitions]]'' in) energy levels by emitting or absorbing a [[photon]] (of [[electromagnetic radiation]]), whose energy must be exactly equal to the energy difference between the two levels. Electrons can also be completely removed from a chemical species such as an atom, molecule, or [[ion]]. Complete removal of an electron from an atom can be a form of [[ionization]], which is effectively moving the electron out to an [[atomic orbital|orbital]] with an infinite [[principal quantum number]], in effect so far away so as to have practically no more effect on the remaining atom (ion). For various types of atoms, there are 1st, 2nd, 3rd, etc. [[ionization energy|ionization energies]] for removing the 1st, then the 2nd, then the 3rd, etc. of the highest energy electrons, respectively, from the atom originally in the [[ground state]]. Energy in corresponding opposite quantities can also be released, sometimes in the form of [[photon energy]], when electrons are added to positively charged ions or sometimes atoms. Molecules can also undergo transitions in their [[molecular vibration|vibrational]] or rotational energy levels. Energy level transitions can also be nonradiative, meaning emission or absorption of a photon is not involved. If an atom, ion, or molecule is at the lowest possible energy level, it and its electrons are said to be in the ''ground state''. If it is at a higher energy level, it is said to be ''excited'', or any electrons that have higher energy than the ground state are ''excited''. Such a species can be excited to a higher energy level by [[light absorption|absorbing]] a photon whose energy is equal to the energy difference between the levels. Conversely, an excited species can go to a lower energy level by spontaneously emitting a photon equal to the energy difference. A photon's energy is equal to the Planck constant ({{math|''h''}}) times its [[frequency]] ({{mvar|f}}) and thus is proportional to its frequency, or inversely to its [[wavelength]] ({{mvar|λ}}).<ref name="chemguide" /> : {{math|1=Δ''E'' = ''hf'' = ''hc'' / ''λ''}}, since {{math|''c''}}, the speed of light, equals to {{math|''fλ''}}<ref name="chemguide" /> Correspondingly, many kinds of [[spectroscopy]] are based on detecting the frequency or wavelength of the emitted or [[Absorption spectroscopy|absorbed]] photons to provide information on the material analyzed, including information on the energy levels and electronic structure of materials obtained by analyzing the [[spectrum]]. An asterisk is commonly used to designate an excited state. An electron transition in a molecule's bond from a ground state to an excited state may have a designation such as σ → σ*, π → π*, or n → π* meaning excitation of an electron from a σ bonding to a σ [[antibonding]] orbital, from a π bonding to a π antibonding orbital, or from an n non-bonding to a π antibonding orbital.<ref name="chemguide" /><ref>[http://www.chem.ucla.edu/~bacher/UV-vis/uv_vis_tetracyclone.html.html Theory of Ultraviolet-Visible (UV-Vis) Spectroscopy]</ref> Reverse electron transitions for all these types of excited molecules are also possible to return to their ground states, which can be designated as σ* → σ, π* → π, or π* → n. A transition in an energy level of an electron in a molecule may be combined with a [[vibrational transition]] and called a [[vibronic transition]]. A vibrational and [[rotational transition]] may be combined by [[rotational–vibrational coupling|rovibrational coupling]]. In [[rovibronic coupling]], electron transitions are simultaneously combined with both vibrational and rotational transitions. Photons involved in transitions may have energy of various ranges in the electromagnetic spectrum, such as [[X-ray]], [[ultraviolet]], [[visible light]], [[infrared]], or [[microwave]] radiation, depending on the type of transition. In a very general way, energy level differences between electronic states are larger, differences between vibrational levels are intermediate, and differences between rotational levels are smaller, although there can be overlap. [[Translation (physics)|Translational]] energy levels are practically continuous and can be calculated as kinetic energy using [[classical mechanics]]. Higher [[temperature]] causes fluid atoms and molecules to move faster increasing their translational energy, and thermally excites molecules to higher average amplitudes of vibrational and rotational modes (excites the molecules to higher internal energy levels). This means that as temperature rises, translational, vibrational, and rotational contributions to molecular [[heat capacity]] let molecules absorb heat and hold more [[internal energy]]. [[Conduction (heat)|Conduction of heat]] typically occurs as molecules or atoms collide [[Heat transfer|transferring the heat]] between each other. At even higher temperatures, electrons can be thermally excited to higher energy orbitals in atoms or molecules. A subsequent drop of an electron to a lower energy level can release a photon, causing a possibly coloured glow. An electron further from the nucleus has higher potential energy than an electron closer to the nucleus, thus it becomes less bound to the nucleus, since its potential energy is negative and inversely dependent on its distance from the nucleus.<ref>{{cite web |url=http://chemed.chem.wisc.edu/chempaths/GenChem-Textbook/Electron-Density-and-Potential-Energy-899.html |title=Electron Density and Potential Energy |access-date=2010-10-07 |url-status=dead |archive-url=https://web.archive.org/web/20100718111313/http://chemed.chem.wisc.edu/chempaths/GenChem-Textbook/Electron-Density-and-Potential-Energy-899.html |archive-date=2010-07-18 }}</ref> == Crystalline materials == [[Crystal|Crystalline solid]]s are found to have [[energy band]]s, instead of or in addition to energy levels. Electrons can take on any energy within an unfilled band. At first this appears to be an exception to the requirement for energy levels. However, as shown in [[band theory]], energy bands are actually made up of many discrete energy levels which are too close together to resolve. Within a band the number of levels is of the order of the number of atoms in the crystal, so although electrons are actually restricted to these energies, they appear to be able to take on a continuum of values. The important energy levels in a crystal are the top of the [[valence band]], the bottom of the [[conduction band]], the [[Fermi level]], the [[vacuum level]], and the energy levels of any [[defect states]] in the crystal. == See also == * [[Perturbation theory (quantum mechanics)]] * [[Atomic clock]] * [[Computational chemistry]] == References == {{reflist}} {{Quantum mechanics topics}} {{Authority control}} {{DEFAULTSORT:Energy Level}} [[Category:Chemical properties]] [[Category:Atomic physics]] [[Category:Molecular physics]] [[Category:Quantum chemistry]] [[Category:Theoretical chemistry]] [[Category:Computational chemistry]] [[Category:Spectroscopy]] [[pl:Powłoka elektronowa]]
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