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2004 02 oct 2026 aug sep oct 23 2021 2022 2023 success fail about this capture timestamps the wayback machine http web archive org web 20220923094448 https en wikipedia org wiki rydberg_formula rydberg formula from wikipedia the free encyclopedia jump to navigation jump to search formula for spectral line wavelengths in alkali metals this article needs additional citations for verification please help improve this article by adding citations to reliable sources unsourced material may be challenged and removed find sources rydberg formula news newspapers books scholar jstor june 2017 learn how and when to remove this template message part of a series of articles about quantum mechanics i ℏ t ψ t h ψ t displaystyle i hbar frac partial partial t psi t rangle hat h psi t rangle schrödinger equation introduction glossary history background classical mechanics old quantum theory bra ket notation hamiltonian interference fundamentals complementarity decoherence entanglement energy level measurement nonlocality quantum number state superposition symmetry tunnelling uncertainty wave function collapse experiments bell s inequality davisson germer double slit elitzur vaidman franck hertz leggett garg inequality mach zehnder popper quantum eraser delayed choice schrödinger s cat stern gerlach wheeler s delayed choice formulations overview heisenberg interaction matrix phase space schrödinger sum over histories path integral equations dirac klein gordon pauli rydberg schrödinger interpretations overview bayesian consistent histories copenhagen de broglie bohm ensemble hidden variable local many worlds objective collapse quantum logic relational transactional advanced topics relativistic quantum mechanics quantum field theory quantum information science quantum computing quantum chaos density matrix scattering theory quantum statistical mechanics quantum machine learning scientists aharonov bell bethe blackett bloch bohm bohr born bose de broglie compton dirac davisson debye ehrenfest einstein everett fock fermi feynman glauber gutzwiller heisenberg hilbert jordan kramers pauli lamb landau laue moseley millikan onnes planck rabi raman rydberg schrödinger simmons sommerfeld von neumann weyl wien wigner zeeman zeilinger v t e rydberg s formula as it appears in a november 1888 record in atomic physics the rydberg formula calculates the wavelengths of a spectral line in many chemical elements the formula was primarily presented as a generalization of the balmer series for all atomic electron transitions of hydrogen it was first empirically stated in 1888 by the swedish physicist johannes rydberg 1 then theoretically by niels bohr in 1913 who used a primitive form of quantum mechanics the formula directly generalizes the equations used to calculate the wavelengths of the hydrogen spectral series contents 1 history 2 for hydrogen 3 for any hydrogen like element 4 see also 5 references history edit in 1880 rydberg worked on a formula describing the relation between the wavelengths in spectral lines of alkali metals he noticed that lines came in series and he found that he could simplify his calculations using the wavenumber the number of waves occupying the unit length equal to 1 λ the inverse of the wavelength as his unit of measurement he plotted the wavenumbers n of successive lines in each series against consecutive integers which represented the order of the lines in that particular series finding that the resulting curves were similarly shaped he sought a single function which could generate all of them when appropriate constants were inserted first he tried the formula n n 0 c 0 m m displaystyle textstyle n n_ 0 frac c_ 0 m m where n is the line s wavenumber n 0 is the series limit m is the line s ordinal number in the series m is a constant different for different series and c 0 is a universal constant this did not work very well rydberg was trying n n 0 c 0 m m 2 displaystyle textstyle n n_ 0 frac c_ 0 left m m right 2 when he became aware of balmer s formula for the hydrogen spectrum λ h m 2 m 2 4 displaystyle textstyle lambda hm 2 over m 2 4 in this equation m is an integer and h is a constant not to be confused with the later planck constant rydberg therefore rewrote balmer s formula in terms of wavenumbers as n n 0 4 n 0 m 2 displaystyle textstyle n n_ 0 4n_ 0 over m 2 this suggested that the balmer formula for hydrogen might be a special case with m 0 displaystyle textstyle m 0 and c 0 4 n 0 displaystyle text c _ 0 4n_ 0 where n 0 1 h displaystyle textstyle n_ 0 frac 1 h the reciprocal of balmer s constant this constant h is written b in the balmer equation article again to avoid confusion with planck s constant the term c 0 displaystyle text c _ 0 was found to be a universal constant common to all elements equal to 4 h this constant is now known as the rydberg constant and m is known as the quantum defect as stressed by niels bohr 2 expressing results in terms of wavenumber not wavelength was the key to rydberg s discovery the fundamental role of wavenumbers was also emphasized by the rydberg ritz combination principle of 1908 the fundamental reason for this lies in quantum mechanics light s wavenumber is proportional to frequency 1 λ f c displaystyle textstyle frac 1 lambda frac f c and therefore also proportional to light s quantum energy e thus 1 λ e h c displaystyle textstyle frac 1 lambda frac e hc in this formula the h represents planck s constant modern understanding is that rydberg s findings were a reflection of the underlying simplicity of the behavior of spectral lines in terms of fixed quantized energy difference between electron orbitals in atoms rydberg s 1888 classical expression for the form of the spectral series was not accompanied by a physical explanation walther ritz s pre quantum 1908 explanation for the mechanism underlying the spectral series was that atomic electrons behaved like magnets and that the magnets could vibrate with respect to the atomic nucleus at least temporarily to produce electromagnetic radiation 3 but this theory was superseded in 1913 by niels bohr s model of the atom in bohr s conception of the atom the integer rydberg and balmer n numbers represent electron orbitals at different integral distances from the atom a frequency or spectral energy emitted in a transition from n 1 to n 2 therefore represents the photon energy emitted or absorbed when an electron makes a jump from orbital 1 to orbital 2 later models found that the values for n 1 and n 2 corresponded to the principal quantum numbers of the two orbitals for hydrogen edit 1 λ v a c r h 1 n 1 2 1 n 2 2 displaystyle frac 1 lambda _ mathrm vac r_ text h left frac 1 n_ 1 2 frac 1 n_ 2 2 right where λ v a c displaystyle lambda _ mathrm vac is the wavelength of electromagnetic radiation emitted in vacuum r h displaystyle r_ text h is the rydberg constant for hydrogen approximately 1 096 775 83 10 7 m 1 n 1 displaystyle n_ 1 is the principal quantum number of an energy level and n 2 displaystyle n_ 2 is the principal quantum number of an energy level for the atomic electron transition note here n 2 n 1 displaystyle n_ 2 n_ 1 by setting n 1 displaystyle n_ 1 to 1 and letting n 2 displaystyle n_ 2 run from 2 to infinity the spectral lines known as the lyman series converging to 91 nm are obtained in the same manner n 1 n 2 name converge toward 1 2 lyman series 91 13 nm uv 2 3 balmer series 364 51 nm visible 3 4 paschen series 820 14 nm ir 4 5 brackett series 1458 03 nm far ir 5 6 pfund series 2278 17 nm far ir 6 7 humphreys series 3280 56 nm far ir a visual comparison of the hydrogen spectral series for n 1 1 to n 1 6 on a log scale for any hydrogen like element edit the formula above can be extended for use with any hydrogen like chemical elements with 1 λ r z 2 1 n 1 2 1 n 2 2 displaystyle frac 1 lambda rz 2 left frac 1 n_ 1 2 frac 1 n_ 2 2 right where λ displaystyle lambda is the wavelength in vacuum of the light emitted r displaystyle r is the rydberg constant for this element z displaystyle z is the atomic number i e the number of protons in the atomic nucleus of this element n 1 displaystyle n_ 1 is the principal quantum number of the lower energy level and n 2 displaystyle n_ 2 is the principal quantum number of the higher energy level for the atomic electron transition this formula can be directly applied only to hydrogen like also called hydrogenic atoms of chemical elements i e atoms with only one electron being affected by an effective nuclear charge which is easily estimated examples would include he li 2 be 3 etc where no other electrons exist in the atom but the rydberg formula also provides correct wavelengths for distant electrons where the effective nuclear charge can be estimated as the same as that for hydrogen since all but one of the nuclear charges have been screened by other electrons and the core of the atom has an effective positive charge of 1 finally with certain modifications replacement of z by z 1 and use of the integers 1 and 2 for the n s to give a numerical value of 3 4 for the difference of their inverse squares the rydberg formula provides correct values in the special case of k alpha lines since the transition in question is the k alpha transition of the electron from the 1s orbital to the 2p orbital this is analogous to the lyman alpha line transition for hydrogen and has the same frequency factor because the 2p electron is not screened by any other electrons in the atom from the nucleus the nuclear charge is diminished only by the single remaining 1s electron causing the system to be effectively a hydrogenic atom but with a diminished nuclear charge z 1 its frequency is thus the lyman alpha hydrogen frequency increased by a factor of z 1 2 this formula of f c λ lyman alpha frequency z 1 2 is historically known as moseley s law having added a factor c to convert wavelength to frequency and can be used to predict wavelengths of the k α k alpha x ray spectral emission lines of chemical elements from aluminum to gold see the biography of henry moseley for the historical importance of this law which was derived empirically at about the same time it was explained by the bohr model of the atom for other spectral transitions in multi electron atoms the rydberg formula generally provides incorrect results since the magnitude of the screening of inner electrons for outer electron transitions is variable and not possible to compensate for in the simple manner above the correction to the rydberg formula for these atoms is known as the quantum defect see also edit balmer series hydrogen line rydberg ritz combination principle references edit see rydberg j r 1889 researches sur la constitution des spectres d émission des éléments chimiques investigations of the composition of the emission spectra of chemical elements kongliga svenska vetenskaps akademiens handlingar proceedings of the royal swedish academy of science 2nd series in french 23 11 1 177 english summary rydberg j r 1890 on the structure of the line spectra of the chemical elements philosophical magazine 5th series 29 331 337 bohr n 1985 rydberg s discovery of the spectral laws in kalckar j ed collected works vol 10 amsterdam north holland publ cy pp 373 379 ritz w 1908 magnetische atomfelder und serienspektren the magnetic fields of atoms and spectral series annalen der physik in german 330 4 660 696 bibcode 1908anp 330 660r doi 10 1002 andp 19083300403 sutton mike july 2004 getting the numbers right the lonely struggle of the 19th century physicist chemist johannes rydberg chemistry world 1 7 38 41 issn 1473 7604 martinson i curtis l j 2005 janne rydberg his life and work nuclear instruments and methods in physics research section b 235 1 4 17 22 bibcode 2005nimpb 235 17m citeseerx 10 1 1 602 6210 doi 10 1016 j nimb 2005 03 137 v t e quantum mechanics background introduction history timeline classical mechanics old quantum theory glossary fundamentals born rule bra ket notation complementarity density matrix energy level ground state excited state degenerate levels zero point energy entanglement hamiltonian interference decoherence measurement nonlocality quantum state superposition tunnelling scattering theory symmetry in quantum mechanics uncertainty wave function collapse wave particle duality formulations formulations heisenberg interaction matrix mechanics schrödinger path integral formulation phase space equations dirac klein gordon pauli rydberg schrödinger interpretations interpretations bayesian consistent histories copenhagen de broglie bohm ensemble hidden variable local many worlds objective collapse quantum logic relational transactional von neumann wigner experiments bell s inequality davisson germer delayed choice quantum eraser double slit franck hertz mach zehnder interferometer elitzur vaidman popper quantum eraser stern gerlach wheeler s delayed choice science quantum biology quantum chemistry quantum chaos quantum cosmology quantum differential calculus quantum dynamics quantum geometry quantum measurement problem quantum stochastic calculus quantum spacetime technology quantum algorithms quantum amplifier quantum bus quantum cellular automata quantum finite automata quantum channel quantum circuit quantum complexity theory quantum computing timeline quantum cryptography quantum electronics quantum error correction quantum imaging quantum image processing quantum information quantum key distribution quantum logic quantum logic gates quantum machine quantum machine learning quantum metamaterial quantum metrology quantum network quantum neural network quantum optics quantum programming quantum sensing quantum simulator quantum teleportation extensions casimir effect quantum statistical mechanics quantum field theory history quantum gravity relativistic quantum mechanics related schrödinger s cat in popular culture quantum mysticism category physics portal commons retrieved from https en wikipedia org w 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