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95-431: The Zeeman effect ( / ˈ z eɪ m ə n / ZAY -mən , Dutch: [ˈzeːmɑn] ) is the splitting of a spectral line into several components in the presence of a static magnetic field . It is caused by interaction of the magnetic field with the magnetic moment of the atomic electron associated to its orbital motion and spin ; this interaction shifts some orbital energies more than others, resulting in

190-527: A combination of the thermal Doppler broadening and the impact pressure broadening yields a Voigt profile . However, the different line broadening mechanisms are not always independent. For example, the collisional effects and the motional Doppler shifts can act in a coherent manner, resulting under some conditions even in a collisional narrowing , known as the Dicke effect . The phrase "spectral lines", when not qualified, usually refers to lines having wavelengths in

285-416: A finite line-of-sight velocity projection. If different parts of the emitting body have different velocities (along the line of sight), the resulting line will be broadened, with the line width proportional to the width of the velocity distribution. For example, radiation emitted from a distant rotating body, such as a star , will be broadened due to the line-of-sight variations in velocity on opposite sides of

380-564: A general angular momentum operator L {\displaystyle L} as These ladder operators have the property as long as m L {\displaystyle m_{L}} lies in the range − L , … . . . , L {\displaystyle {-L,\dots ...,L}} (otherwise, they return zero). Using ladder operators J ± {\displaystyle J_{\pm }} and I ± {\displaystyle I_{\pm }} We can rewrite

475-473: A given level. To get the complete picture, including intermediate field strengths, we must consider eigenstates which are superpositions of the | F , m F ⟩ {\displaystyle |F,m_{F}\rangle } and | m I , m J ⟩ {\displaystyle |m_{I},m_{J}\rangle } basis states. For J = 1 / 2 {\displaystyle J=1/2} ,

570-581: A good approximation to the true solution if it is truncated at a point at which its elements are minimum. This is called an asymptotic series . If the perturbation series is divergent or not a power series (for example, if the asymptotic expansion must include non-integer powers   ε ( 1 / 2 )   {\displaystyle \ \varepsilon ^{\left(1/2\right)}\ } or negative powers   ε − 2   {\displaystyle \ \varepsilon ^{-2}\ } ) then

665-401: A good approximation, precisely because the parts that were ignored were of size   ε 2   . {\displaystyle \ \varepsilon ^{2}~.} The process can then be repeated, to obtain corrections   A 2   , {\displaystyle \ A_{2}\ ,} and so on. In practice, this process rapidly explodes into

760-445: A hot material are detected, perhaps in the presence of a broad spectrum from a cooler source. The intensity of light, over a narrow frequency range, is increased due to emission by the hot material. Spectral lines are highly atom-specific, and can be used to identify the chemical composition of any medium. Several elements, including helium , thallium , and caesium , were discovered by spectroscopic means. Spectral lines also depend on

855-429: A magnetic field is where H 0 {\displaystyle H_{0}} is the unperturbed Hamiltonian of the atom, and V M {\displaystyle V_{\rm {M}}} is the perturbation due to the magnetic field: where μ → {\displaystyle {\vec {\mu }}} is the magnetic moment of the atom. The magnetic moment consists of

950-434: A particle would be emitted in radioactive elements. This was later named Fermi's golden rule . Perturbation theory in quantum mechanics is fairly accessible, mainly because quantum mechanics is limited to linear wave equations, but also since the quantum mechanical notation allows expressions to be written in fairly compact form, thus making them easier to comprehend. This resulted in an explosion of applications, ranging from

1045-412: A problem, by starting from the exact solution of a related, simpler problem. A critical feature of the technique is a middle step that breaks the problem into "solvable" and "perturbative" parts. In regular perturbation theory , the solution is expressed as a power series in a small parameter ε {\displaystyle \varepsilon } . The first term is the known solution to

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1140-447: A profusion of terms, which become extremely hard to manage by hand. Isaac Newton is reported to have said, regarding the problem of the Moon 's orbit, that "It causeth my head to ache." This unmanageability has forced perturbation theory to develop into a high art of managing and writing out these higher order terms. One of the fundamental breakthroughs in quantum mechanics for controlling

1235-464: A result of which the computations could be performed with a very high accuracy. The discovery of the planet Neptune in 1848 by Urbain Le Verrier , based on the deviations in motion of the planet Uranus . He sent the coordinates to J.G. Galle who successfully observed Neptune through his telescope – a triumph of perturbation theory. The standard exposition of perturbation theory is given in terms of

1330-412: A result, only three spectral lines will be visible, corresponding to the Δ m l = 0 , ± 1 {\displaystyle \Delta m_{l}=0,\pm 1} selection rule. The splitting Δ E = B μ B Δ m l {\displaystyle \Delta E=B\mu _{\rm {B}}\Delta m_{l}} is independent of

1425-496: A simple Keplerian ellipse because of the competing gravitation of the Earth and the Sun . Perturbation methods start with a simplified form of the original problem, which is simple enough to be solved exactly. In celestial mechanics , this is usually a Keplerian ellipse . Under Newtonian gravity , an ellipse is exactly correct when there are only two gravitating bodies (say, the Earth and

1520-453: A term, a denominator, an integral, and so on; thus complex integrals can be written as simple diagrams, with absolutely no ambiguity as to what they mean. The one-to-one correspondence between the diagrams, and specific integrals is what gives them their power. Although originally developed for quantum field theory, it turns out the diagrammatic technique is broadly applicable to many other perturbative series (although not always worthwhile). In

1615-423: A tiny spectral band with a nonzero range of frequencies, not a single frequency (i.e., a nonzero spectral width ). In addition, its center may be shifted from its nominal central wavelength. There are several reasons for this broadening and shift. These reasons may be divided into two general categories – broadening due to local conditions and broadening due to extended conditions. Broadening due to local conditions

1710-429: Is broadened because the photons at the line center have a greater reabsorption probability than the photons at the line wings. Indeed, the reabsorption near the line center may be so great as to cause a self reversal in which the intensity at the center of the line is less than in the wings. This process is also sometimes called self-absorption . Radiation emitted by a moving source is subject to Doppler shift due to

1805-566: Is called the Paschen–Back effect . In the modern scientific literature, these terms are rarely used, with a tendency to use just the "Zeeman effect". Another rarely used obscure term is inverse Zeeman effect , referring to the Zeeman effect in an absorption spectral line. A similar effect, splitting of the nuclear energy levels in the presence of a magnetic field, is referred to as the nuclear Zeeman effect . The total Hamiltonian of an atom in

1900-533: Is due to effects which hold in a small region around the emitting element, usually small enough to assure local thermodynamic equilibrium . Broadening due to extended conditions may result from changes to the spectral distribution of the radiation as it traverses its path to the observer. It also may result from the combining of radiation from a number of regions which are far from each other. The lifetime of excited states results in natural broadening, also known as lifetime broadening. The uncertainty principle relates

1995-444: Is inserted into   ε D 1 {\displaystyle \ \varepsilon D_{1}} . This results in an equation for   A 1   , {\displaystyle \ A_{1}\ ,} which, in the general case, can be written in closed form as a sum over integrals over   A 0   . {\displaystyle \ A_{0}~.} Thus, one has obtained

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2090-460: Is known, and one seeks the general solution   A   {\displaystyle \ A\ } to   D = D 0 + ε D 1   . {\displaystyle \ D=D_{0}+\varepsilon D_{1}~.} Next the approximation   A ≈ A 0 + ε A 1   {\displaystyle \ A\approx A_{0}+\varepsilon A_{1}\ }

2185-414: Is observed depends on the type of material and its temperature relative to another emission source. An absorption line is produced when photons from a hot, broad spectrum source pass through a cooler material. The intensity of light, over a narrow frequency range, is reduced due to absorption by the material and re-emission in random directions. By contrast, a bright emission line is produced when photons from

2280-610: Is small (less than the fine structure ), it can be treated as a perturbation; this is the Zeeman effect proper. In the Paschen–Back effect, described below, V M {\displaystyle V_{M}} exceeds the LS coupling significantly (but is still small compared to H 0 {\displaystyle H_{0}} ). In ultra-strong magnetic fields, the magnetic-field interaction may exceed H 0 {\displaystyle H_{0}} , in which case

2375-477: Is small, causing the perturbative correction to " blow up ", becoming as large or maybe larger than the zeroth order term. This situation signals a breakdown of perturbation theory: It stops working at this point, and cannot be expanded or summed any further. In formal terms, the perturbative series is an asymptotic series : A useful approximation for a few terms, but at some point becomes less accurate if even more terms are added. The breakthrough from chaos theory

2470-456: Is the Landé g-factor . A more accurate approach is to take into account that the operator of the magnetic moment of an electron is a sum of the contributions of the orbital angular momentum L → {\displaystyle {\vec {L}}} and the spin angular momentum S → {\displaystyle {\vec {S}}} , with each multiplied by

2565-478: Is the solution of the exactly solvable problem, while further terms describe the deviation in the solution, due to the deviation from the initial problem. Formally, we have for the approximation to the full solution   A   , {\displaystyle \ A\ ,} a series in the small parameter (here called ε ), like the following: In this example,   A 0   {\displaystyle \ A_{0}\ } would be

2660-422: Is the splitting of atomic energy levels in the presence of a strong magnetic field. This occurs when an external magnetic field is sufficiently strong to disrupt the coupling between orbital ( L → {\displaystyle {\vec {L}}} ) and spin ( S → {\displaystyle {\vec {S}}} ) angular momenta. This effect is the strong-field limit of

2755-463: Is the z-component of the total angular momentum. For a single electron above filled shells s = 1 / 2 {\displaystyle s=1/2} and j = l ± s {\displaystyle j=l\pm s} , the Landé g-factor can be simplified into: Taking V m {\displaystyle V_{m}} to be the perturbation, the Zeeman correction to

2850-794: Is then the projection of the spin onto the direction of J → {\displaystyle {\vec {J}}} : and for the (time-)"averaged" orbital vector: Thus, Using L → = J → − S → {\displaystyle {\vec {L}}={\vec {J}}-{\vec {S}}} and squaring both sides, we get and: using S → = J → − L → {\displaystyle {\vec {S}}={\vec {J}}-{\vec {L}}} and squaring both sides, we get Combining everything and taking J z = ℏ m j {\displaystyle J_{z}=\hbar m_{j}} , we obtain

2945-941: The Bohr magneton and nuclear magneton respectively, J → {\displaystyle {\vec {J}}} and I → {\displaystyle {\vec {I}}} are the electron and nuclear angular momentum operators and g J {\displaystyle g_{J}} is the Landé g-factor : g J = g L J ( J + 1 ) + L ( L + 1 ) − S ( S + 1 ) 2 J ( J + 1 ) + g S J ( J + 1 ) − L ( L + 1 ) + S ( S + 1 ) 2 J ( J + 1 ) . {\displaystyle g_{J}=g_{L}{\frac {J(J+1)+L(L+1)-S(S+1)}{2J(J+1)}}+g_{S}{\frac {J(J+1)-L(L+1)+S(S+1)}{2J(J+1)}}.} In

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3040-547: The Lyman series or Balmer series . Originally all spectral lines were classified into series: the principal series , sharp series , and diffuse series . These series exist across atoms of all elements, and the patterns for all atoms are well-predicted by the Rydberg-Ritz formula . These series were later associated with suborbitals. There are a number of effects which control spectral line shape . A spectral line extends over

3135-484: The Moon ) but not quite correct when there are three or more objects (say, the Earth, Moon , Sun , and the rest of the Solar System ) and not quite correct when the gravitational interaction is stated using formulations from general relativity . Keeping the above example in mind, one follows a general recipe to obtain the perturbation series. The perturbative expansion is created by adding successive corrections to

3230-448: The Zeeman effect to the hyperfine splitting in the hydrogen atom . Despite the simpler notation, perturbation theory applied to quantum field theory still easily gets out of hand. Richard Feynman developed the celebrated Feynman diagrams by observing that many terms repeat in a regular fashion. These terms can be replaced by dots, lines, squiggles and similar marks, each standing for

3325-686: The dipole approximation), as governed by the selection rules . Since the distance between the Zeeman sub-levels is a function of magnetic field strength, this effect can be used to measure magnetic field strength, e.g. that of the Sun and other stars or in laboratory plasmas . In 1896 Zeeman learned that his laboratory had one of Henry Augustus Rowland 's highest resolving Rowland grating , an imaging spectrographic mirror. Zeeman had read James Clerk Maxwell 's article in Encyclopædia Britannica describing Michael Faraday 's failed attempts to influence light with magnetism. Zeeman wondered if

3420-400: The first-order correction   A 1   {\displaystyle \ A_{1}\ } and thus   A ≈ A 0 + ε A 1   {\displaystyle \ A\approx A_{0}+\varepsilon A_{1}\ } is a good approximation to   A   . {\displaystyle \ A~.} It is

3515-538: The infrared spectral lines include the Paschen series of hydrogen. At even longer wavelengths, the radio spectrum includes the 21-cm line used to detect neutral hydrogen throughout the cosmos . For each element, the following table shows the spectral lines which appear in the visible spectrum at about 400-700 nm. Perturbation theory In mathematics and applied mathematics , perturbation theory comprises methods for finding an approximate solution to

3610-418: The selection rules for an electric dipole transition , i.e., Δ s = 0 , Δ m s = 0 , Δ l = ± 1 , Δ m l = 0 , ± 1 {\displaystyle \Delta s=0,\Delta m_{s}=0,\Delta l=\pm 1,\Delta m_{l}=0,\pm 1} this allows to ignore the spin degree of freedom altogether. As

3705-431: The temperature and density of the material, so they are widely used to determine the physical conditions of stars and other celestial bodies that cannot be analyzed by other means. Depending on the material and its physical conditions, the energy of the involved photons can vary widely, with the spectral lines observed across the electromagnetic spectrum , from radio waves to gamma rays . Strong spectral lines in

3800-509: The visible band of the full electromagnetic spectrum . Many spectral lines occur at wavelengths outside this range. At shorter wavelengths, which correspond to higher energies, ultraviolet spectral lines include the Lyman series of hydrogen . At the much shorter wavelengths of X-rays , the lines are known as characteristic X-rays because they remain largely unchanged for a given chemical element, independent of their chemical environment. Longer wavelengths correspond to lower energies, where

3895-460: The visible part of the electromagnetic spectrum often have a unique Fraunhofer line designation, such as K for a line at 393.366 nm emerging from singly-ionized calcium atom, Ca , though some of the Fraunhofer "lines" are blends of multiple lines from several different species . In other cases, the lines are designated according to the level of ionization by adding a Roman numeral to

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3990-430: The 2P 3/2 level into 4 states ( m j = 3 / 2 , 1 / 2 , − 1 / 2 , − 3 / 2 {\displaystyle m_{j}=3/2,1/2,-1/2,-3/2} ). The Landé g-factors for the three levels are: Note in particular that the size of the energy splitting is different for the different orbitals, because the g J values are different. On

4085-977: The Hamiltonian as We can now see that at all times, the total angular momentum projection m F = m J + m I {\displaystyle m_{F}=m_{J}+m_{I}} will be conserved. This is because both J z {\displaystyle J_{z}} and I z {\displaystyle I_{z}} leave states with definite m J {\displaystyle m_{J}} and m I {\displaystyle m_{I}} unchanged, while J + I − {\displaystyle J_{+}I_{-}} and J − I + {\displaystyle J_{-}I_{+}} either increase m J {\displaystyle m_{J}} and decrease m I {\displaystyle m_{I}} or vice versa, so

4180-494: The Hamiltonian can be solved analytically, resulting in the Breit–Rabi formula (named after Gregory Breit and Isidor Isaac Rabi ). Notably, the electric quadrupole interaction is zero for L = 0 {\displaystyle L=0} ( J = 1 / 2 {\displaystyle J=1/2} ), so this formula is fairly accurate. We now utilize quantum mechanical ladder operators , which are defined for

4275-482: The Zeeman effect. When s = 0 {\displaystyle s=0} , the two effects are equivalent. The effect was named after the German physicists Friedrich Paschen and Ernst E. A. Back . When the magnetic-field perturbation significantly exceeds the spin–orbit interaction, one can safely assume [ H 0 , S ] = 0 {\displaystyle [H_{0},S]=0} . This allows

4370-521: The accuracy of solutions to Newton's gravitational equations, which led many eminent 18th and 19th century mathematicians, notably Joseph-Louis Lagrange and Pierre-Simon Laplace , to extend and generalize the methods of perturbation theory. These well-developed perturbation methods were adopted and adapted to solve new problems arising during the development of quantum mechanics in 20th century atomic and subatomic physics. Paul Dirac developed quantum perturbation theory in 1927 to evaluate when

4465-443: The appropriate gyromagnetic ratio : where g l = 1 {\displaystyle g_{l}=1} and g s ≈ 2.0023193 {\displaystyle g_{s}\approx 2.0023193} (the latter is called the anomalous gyromagnetic ratio ; the deviation of the value from 2 is due to the effects of quantum electrodynamics ). In the case of the LS coupling , one can sum over all electrons in

4560-463: The atom can no longer exist in its normal meaning, and one talks about Landau levels instead. There are intermediate cases which are more complex than these limit cases. If the spin–orbit interaction dominates over the effect of the external magnetic field, L → {\displaystyle {\vec {L}}} and S → {\displaystyle {\vec {S}}} are not separately conserved, only

4655-397: The atom: where L → {\displaystyle {\vec {L}}} and S → {\displaystyle {\vec {S}}} are the total spin momentum and spin of the atom, and averaging is done over a state with a given value of the total angular momentum. If the interaction term V M {\displaystyle V_{M}}

4750-415: The basis: Spectral line Spectral lines are the result of interaction between a quantum system (usually atoms , but sometimes molecules or atomic nuclei ) and a single photon . When a photon has about the right amount of energy (which is connected to its frequency) to allow a change in the energy state of the system (in the case of an atom this is usually an electron changing orbitals ),

4845-756: The case of weak magnetic fields, the Zeeman interaction can be treated as a perturbation to the | F , m f ⟩ {\displaystyle |F,m_{f}\rangle } basis. In the high field regime, the magnetic field becomes so strong that the Zeeman effect will dominate, and one must use a more complete basis of | I , J , m I , m J ⟩ {\displaystyle |I,J,m_{I},m_{J}\rangle } or just | m I , m J ⟩ {\displaystyle |m_{I},m_{J}\rangle } since I {\displaystyle I} and J {\displaystyle J} will be constant within

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4940-519: The complete Hamiltonian from the very beginning and never specifies a perturbation operator as such. Møller–Plesset perturbation theory uses the difference between the Hartree–;Fock Hamiltonian and the exact non-relativistic Hamiltonian as the perturbation. The zero-order energy is the sum of orbital energies. The first-order energy is the Hartree–;Fock energy and electron correlation

5035-573: The designation of the chemical element . Neutral atoms are denoted with the Roman numeral I, singly ionized atoms with II, and so on, so that, for example: Cu II — copper ion with +1 charge, Cu Fe III — iron ion with +2 charge, Fe More detailed designations usually include the line wavelength and may include a multiplet number (for atomic lines) or band designation (for molecular lines). Many spectral lines of atomic hydrogen also have designations within their respective series , such as

5130-420: The effect. Wolfgang Pauli recalled that when asked by a colleague as to why he looked unhappy, he replied, "How can one look happy when he is thinking about the anomalous Zeeman effect?" At higher magnetic field strength the effect ceases to be linear. At even higher field strengths, comparable to the strength of the atom's internal field, the electron coupling is disturbed and the spectral lines rearrange. This

5225-500: The effects of inhomogeneous broadening is sometimes reduced by a process called motional narrowing . Certain types of broadening are the result of conditions over a large region of space rather than simply upon conditions that are local to the emitting particle. Opacity broadening is an example of a non-local broadening mechanism. Electromagnetic radiation emitted at a particular point in space can be reabsorbed as it travels through space. This absorption depends on wavelength. The line

5320-410: The electronic and nuclear parts; however, the latter is many orders of magnitude smaller and will be neglected here. Therefore, where μ B {\displaystyle \mu _{\rm {B}}} is the Bohr magneton , J → {\displaystyle {\vec {J}}} is the total electronic angular momentum , and g {\displaystyle g}

5415-409: The energy is The Lyman-alpha transition in hydrogen in the presence of the spin–orbit interaction involves the transitions In the presence of an external magnetic field, the weak-field Zeeman effect splits the 1S 1/2 and 2P 1/2 levels into 2 states each ( m j = 1 / 2 , − 1 / 2 {\displaystyle m_{j}=1/2,-1/2} ) and

5510-466: The equations of motion, interactions between particles, terms of higher powers in the Hamiltonian/free energy. For physical problems involving interactions between particles, the terms of the perturbation series may be displayed (and manipulated) using Feynman diagrams . Perturbation theory was first devised to solve otherwise intractable problems in the calculation of the motions of planets in

5605-492: The equations of the two-body problem , the two bodies being the planet and the Sun. Since astronomic data came to be known with much greater accuracy, it became necessary to consider how the motion of a planet around the Sun is affected by other planets. This was the origin of the three-body problem ; thus, in studying the system Moon-Earth-Sun, the mass ratio between the Moon and the Earth

5700-642: The expansion are the Feynman diagrams , which allow quantum mechanical perturbation series to be represented by a sketch. Perturbation theory has been used in a large number of different settings in physics and applied mathematics. Examples of the "collection of equations" D {\displaystyle D} include algebraic equations , differential equations (e.g., the equations of motion and commonly wave equations ), thermodynamic free energy in statistical mechanics , radiative transfer, and Hamiltonian operators in quantum mechanics . Examples of

5795-645: The expectation values of L z {\displaystyle L_{z}} and S z {\displaystyle S_{z}} to be easily evaluated for a state | ψ ⟩ {\displaystyle |\psi \rangle } . The energies are simply The above may be read as implying that the LS-coupling is completely broken by the external field. However m l {\displaystyle m_{l}} and m s {\displaystyle m_{s}} are still "good" quantum numbers. Together with

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5890-450: The extent that decay rates can be artificially suppressed or enhanced. The atoms in a gas which are emitting radiation will have a distribution of velocities. Each photon emitted will be "red"- or "blue"-shifted by the Doppler effect depending on the velocity of the atom relative to the observer. The higher the temperature of the gas, the wider the distribution of velocities in the gas. Since

5985-491: The flame he observed a slight broadening of the sodium images. When Zeeman switched to cadmium at the source he observed the images split when the magnet was energized. These splitting could be analyzed with Hendrik Lorentz 's then-new electron theory . In retrospect we now know that the magnetic effects on sodium require quantum mechanical treatment. Zeeman and Lorentz were awarded the 1902 Nobel prize; in his acceptance speech Zeeman explained his apparatus and showed slides of

6080-638: The following formula for the hydrogen atom in the Paschen–;Back limit: In this example, the fine-structure corrections are ignored. ( n = 2 , l = 1 {\displaystyle n=2,l=1} ) ∣ m l , m s ⟩ {\displaystyle \mid m_{l},m_{s}\rangle } ( n = 1 , l = 0 {\displaystyle n=1,l=0} ) ∣ m l , m s ⟩ {\displaystyle \mid m_{l},m_{s}\rangle } In

6175-764: The kinds of solutions that are found perturbatively include the solution of the equation of motion ( e.g. , the trajectory of a particle), the statistical average of some physical quantity ( e.g. , average magnetization), and the ground state energy of a quantum mechanical problem. Examples of exactly solvable problems that can be used as starting points include linear equations , including linear equations of motion ( harmonic oscillator , linear wave equation ), statistical or quantum-mechanical systems of non-interacting particles (or in general, Hamiltonians or free energies containing only terms quadratic in all degrees of freedom). Examples of systems that can be solved with perturbations include systems with nonlinear contributions to

6270-543: The known solution to the exactly solvable initial problem, and the terms   A 1 , A 2 , A 3 , …   {\displaystyle \ A_{1},A_{2},A_{3},\ldots \ } represent the first-order , second-order , third-order , and higher-order terms , which may be found iteratively by a mechanistic but increasingly difficult procedure. For small   ε   {\displaystyle \ \varepsilon \ } these higher-order terms in

6365-703: The left, fine structure splitting is depicted. This splitting occurs even in the absence of a magnetic field, as it is due to spin–orbit coupling. Depicted on the right is the additional Zeeman splitting, which occurs in the presence of magnetic fields. [REDACTED] ( n = 2 , l = 1 {\displaystyle n=2,l=1} ) ∣ j , m j ⟩ {\displaystyle \mid j,m_{j}\rangle } ( n = 1 , l = 0 {\displaystyle n=1,l=0} ) ∣ j , m j ⟩ {\displaystyle \mid j,m_{j}\rangle } The Paschen–Back effect

6460-825: The letter "D". The process is generally mechanical, if laborious. One begins by writing the equations   D   {\displaystyle \ D\ } so that they split into two parts: some collection of equations   D 0   {\displaystyle \ D_{0}\ } which can be solved exactly, and some additional remaining part   ε D 1   {\displaystyle \ \varepsilon D_{1}\ } for some small   ε ≪ 1   . {\displaystyle \ \varepsilon \ll 1~.} The solution   A 0   {\displaystyle \ A_{0}\ } (to   D 0   {\displaystyle \ D_{0}\ } )

6555-533: The lifetime of an excited state (due to spontaneous radiative decay or the Auger process ) with the uncertainty of its energy. Some authors use the term "radiative broadening" to refer specifically to the part of natural broadening caused by the spontaneous radiative decay. A short lifetime will have a large energy uncertainty and a broad emission. This broadening effect results in an unshifted Lorentzian profile . The natural broadening can be experimentally altered only to

6650-456: The magnetic dipole approximation, the Hamiltonian which includes both the hyperfine and Zeeman interactions is where A {\displaystyle A} is the hyperfine splitting (in Hz) at zero applied magnetic field, μ B {\displaystyle \mu _{\rm {B}}} and μ N {\displaystyle \mu _{\rm {N}}} are

6745-394: The magnetic potential energy of the atom in the applied external magnetic field, where the quantity in square brackets is the Landé g-factor g J of the atom ( g L = 1 {\displaystyle g_{L}=1} and g S ≈ 2 {\displaystyle g_{S}\approx 2} ) and m j {\displaystyle m_{j}}

6840-424: The nature of the perturbing force as follows: Inhomogeneous broadening is a general term for broadening because some emitting particles are in a different local environment from others, and therefore emit at a different frequency. This term is used especially for solids, where surfaces, grain boundaries, and stoichiometry variations can create a variety of local environments for a given atom to occupy. In liquids,

6935-420: The new spectrographic techniques could succeed where early efforts had not. When illuminated by a slit shaped source, the grating produces a long array of slit images corresponding to different wavelengths. Zeeman placed a piece of asbestos soaked in salt water into a Bunsen burner flame at the source of the grating: he could easily see two lines for sodium light emission. Energizing a 10 kilogauss magnet around

7030-457: The order to which the perturbation is carried out: first-order perturbation theory or second-order perturbation theory, and whether the perturbed states are degenerate, which requires singular perturbation . In the singular case extra care must be taken, and the theory is slightly more elaborate. Many of the ab initio quantum chemistry methods use perturbation theory directly or are closely related methods. Implicit perturbation theory works with

7125-420: The perturbation problem is called a singular perturbation problem . Many special techniques in perturbation theory have been developed to analyze singular perturbation problems. The earliest use of what would now be called perturbation theory was to deal with the otherwise unsolvable mathematical problems of celestial mechanics : for example the orbit of the Moon , which moves noticeably differently from

7220-738: The perturbative series have "small denominators": That is, they have the general form     ψ n V ϕ m     ( ω n − ω m )     {\displaystyle \ {\frac {\ \psi _{n}V\phi _{m}\ }{\ (\omega _{n}-\omega _{m})\ }}\ } where   ψ n   , {\displaystyle \ \psi _{n}\ ,}   V   , {\displaystyle \ V\ ,} and   ϕ m   {\displaystyle \ \phi _{m}\ } are some complicated expressions pertinent to

7315-414: The photon is absorbed. Then the energy will be spontaneously re-emitted, either as one photon at the same frequency as the original one or in a cascade, where the sum of the energies of the photons emitted will be equal to the energy of the one absorbed (assuming the system returns to its original state). A spectral line may be observed either as an emission line or an absorption line . Which type of line

7410-414: The power series in   ε   {\displaystyle \ \varepsilon \ } converges with a nonzero radius of convergence, the perturbation problem is called a regular perturbation problem. In regular perturbation problems, the asymptotic solution smoothly approaches the exact solution. However, the perturbation series can also diverge, and the truncated series can still be

7505-502: The problem to be solved, and   ω n   {\displaystyle \ \omega _{n}\ } and   ω m   {\displaystyle \ \omega _{m}\ } are real numbers; very often they are the energy of normal modes . The small divisor problem arises when the difference   ω n − ω m   {\displaystyle \ \omega _{n}-\omega _{m}\ }

7600-567: The second half of the 20th century, as chaos theory developed, it became clear that unperturbed systems were in general completely integrable systems , while the perturbed systems were not. This promptly lead to the study of "nearly integrable systems", of which the KAM torus is the canonical example. At the same time, it was also discovered that many (rather special) non-linear systems , which were previously approachable only through perturbation theory, are in fact completely integrable. This discovery

7695-704: The series generally (but not always) become successively smaller. An approximate "perturbative solution" is obtained by truncating the series, often by keeping only the first two terms, expressing the final solution as a sum of the initial (exact) solution and the "first-order" perturbative correction Some authors use big O notation to indicate the order of the error in the approximate solution: A = A 0 + ε A 1 + O (   ε 2   )   . {\displaystyle \;A=A_{0}+\varepsilon A_{1}+{\mathcal {O}}{\bigl (}\ \varepsilon ^{2}\ {\bigr )}~.} If

7790-454: The simplified problem. The corrections are obtained by forcing consistency between the unperturbed solution, and the equations describing the system in full. Write   D   {\displaystyle \ D\ } for this collection of equations; that is, let the symbol   D   {\displaystyle \ D\ } stand in for the problem to be solved. Quite often, these are differential equations, thus,

7885-442: The solar system. For instance, Newton's law of universal gravitation explained the gravitation between two astronomical bodies, but when a third body is added, the problem was, "How does each body pull on each?" Kepler's orbital equations only solve Newton's gravitational equations when the latter are limited to just two bodies interacting. The gradually increasing accuracy of astronomical observations led to incremental demands in

7980-548: The solvable problem. Successive terms in the series at higher powers of ε {\displaystyle \varepsilon } usually become smaller. An approximate 'perturbation solution' is obtained by truncating the series, often keeping only the first two terms, the solution to the known problem and the 'first order' perturbation correction. Perturbation theory is used in a wide range of fields and reaches its most sophisticated and advanced forms in quantum field theory . Perturbation theory (quantum mechanics) describes

8075-448: The spectral line is a combination of all of the emitted radiation, the higher the temperature of the gas, the broader the spectral line emitted from that gas. This broadening effect is described by a Gaussian profile and there is no associated shift. The presence of nearby particles will affect the radiation emitted by an individual particle. There are two limiting cases by which this occurs: Pressure broadening may also be classified by

8170-472: The spectrographic images. Historically, one distinguishes between the normal and an anomalous Zeeman effect (discovered by Thomas Preston in Dublin, Ireland). The anomalous effect appears on transitions where the net spin of the electrons is non-zero. It was called "anomalous" because the electron spin had not yet been discovered, and so there was no good explanation for it at the time that Zeeman observed

8265-653: The split spectrum. The effect is named after the Dutch physicist Pieter Zeeman , who discovered it in 1896 and received a Nobel Prize in Physics for this discovery. It is analogous to the Stark effect , the splitting of a spectral line into several components in the presence of an electric field . Also similar to the Stark effect, transitions between different components have, in general, different intensities, with some being entirely forbidden (in

8360-399: The star (this effect usually referred to as rotational broadening). The greater the rate of rotation, the broader the line. Another example is an imploding plasma shell in a Z-pinch . Each of these mechanisms can act in isolation or in combination with others. Assuming each effect is independent, the observed line profile is a convolution of the line profiles of each mechanism. For example,

8455-469: The sum is always unaffected. Furthermore, since J = 1 / 2 {\displaystyle J=1/2} there are only two possible values of m J {\displaystyle m_{J}} which are ± 1 / 2 {\displaystyle \pm 1/2} . Therefore, for every value of m F {\displaystyle m_{F}} there are only two possible states, and we can define them as

8550-443: The total angular momentum J → = L → + S → {\displaystyle {\vec {J}}={\vec {L}}+{\vec {S}}} is. The spin and orbital angular momentum vectors can be thought of as precessing about the (fixed) total angular momentum vector J → {\displaystyle {\vec {J}}} . The (time-)"averaged" spin vector

8645-445: The unperturbed energies and electronic configurations of the levels being considered. More precisely, if s ≠ 0 {\displaystyle s\neq 0} , each of these three components is actually a group of several transitions due to the residual spin–orbit coupling and relativistic corrections (which are of the same order, known as 'fine structure'). The first-order perturbation theory with these corrections yields

8740-405: The use of this method in quantum mechanics . The field in general remains actively and heavily researched across multiple disciplines. Perturbation theory develops an expression for the desired solution in terms of a formal power series known as a perturbation series in some "small" parameter, that quantifies the deviation from the exactly solvable problem. The leading term in this power series

8835-477: Was an explanation of why this happened: The small divisors occur whenever perturbation theory is applied to a chaotic system. The one signals the presence of the other. Since the planets are very remote from each other, and since their mass is small as compared to the mass of the Sun, the gravitational forces between the planets can be neglected, and the planetary motion is considered, to a first approximation, as taking place along Kepler's orbits, which are defined by

8930-459: Was chosen as the "small parameter". Lagrange and Laplace were the first to advance the view that the so-called "constants" which describe the motion of a planet around the Sun gradually change: They are "perturbed", as it were, by the motion of other planets and vary as a function of time; hence the name "perturbation theory". Perturbation theory was investigated by the classical scholars – Laplace, Siméon Denis Poisson , Carl Friedrich Gauss – as

9025-511: Was quite dramatic, as it allowed exact solutions to be given. This, in turn, helped clarify the meaning of the perturbative series, as one could now compare the results of the series to the exact solutions. The improved understanding of dynamical systems coming from chaos theory helped shed light on what was termed the small denominator problem or small divisor problem . In the 19th century Poincaré observed (as perhaps had earlier mathematicians) that sometimes 2nd and higher order terms in

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