Physics:Quantum Runge–Lenz vector - Revision history

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	In classical mechanics, the '''Laplace–Runge–Lenz vector''' ('''LRL vector''') is a vector used chiefly to describe the shape and orientation of the orbit of one astronomical object around another, such as a binary star or a planet revolving around a star. For two bodies interacting by Newtonian gravity, the LRL vector is a constant of motion, meaning that it is the same no matter where it is calculated on the orbit;<ref name="goldstein_1980">{{cite book \| last=Goldstein \| first=H. \| author-link=Herbert Goldstein \| date=1980 \| title=Classical Mechanics \| edition=2nd \| publisher=Addison Wesley \| pages=102–105, 421–422}}</ref><ref name="taff_1985">{{cite book \| last = Taff \| first = L. G. \| author-link = Laurence G. Taff \| date = 1985 \| title = Celestial Mechanics: A Computational Guide for the Practitioner \| publisher = John Wiley and Sons \| location = New York \| pages = 42–43}}</ref> equivalently, the LRL vector is said to be ''conserved''. More generally, the LRL vector is conserved in all problems in which two bodies interact by a central force that varies as the inverse square of the distance between them; such problems are called Kepler problems.<ref name="goldstein_1980b">{{cite book \| last=Goldstein \| first=H. \| author-link=Herbert Goldstein \| date=1980 \| title=Classical Mechanics \| edition=2nd \| publisher=Addison Wesley \| pages=94–102}}</ref><ref name="arnold_1989">{{cite book \| last = Arnold \| first = V. I. \| author-link = Vladimir Arnold \| date = 1989 \| title = Mathematical Methods of Classical Mechanics \| edition = 2nd \| publisher = Springer-Verlag \| location = New York \| page = [https://archive.org/details/mathematicalmeth0000arno/page/38 38] \| isbn = 0-387-96890-3 \| url = https://archive.org/details/mathematicalmeth0000arno/page/38 }}</ref><ref name="sommerfeld_1989">{{cite book \| last = Sommerfeld \| first = A. \| author-link = Arnold Sommerfeld \| date = 1964 \| title = Mechanics \| series=Lectures on Theoretical Physics \| volume = 1 \| edition = 4th \| translator = Martin O. Stern \| publisher = Academic Press \| location = New York \| pages = 38–45}}</ref><ref name="lanczos_1970">{{cite book \| last = Lanczos \| first = C. \| author-link = Cornelius Lanczos \| date = 1970 \| title = The Variational Principles of Mechanics \| edition = 4th \| publisher = Dover Publications \| location = New York \| pages = 118, 129, 242, 248 }}</ref>		In classical mechanics, the '''Laplace–Runge–Lenz vector''' ('''LRL vector''') is a vector used chiefly to describe the shape and orientation of the orbit of one astronomical object around another, such as a binary star or a planet revolving around a star. For two bodies interacting by Newtonian gravity, the LRL vector is a constant of motion, meaning that it is the same no matter where it is calculated on the orbit;<ref name="goldstein_1980">{{cite book \| last=Goldstein \| first=H. \| author-link=Herbert Goldstein \| date=1980 \| title=Classical Mechanics \| edition=2nd \| publisher=Addison Wesley \| pages=102–105, 421–422}}</ref><ref name="taff_1985">{{cite book \| last = Taff \| first = L. G. \| author-link = Laurence G. Taff \| date = 1985 \| title = Celestial Mechanics: A Computational Guide for the Practitioner \| publisher = John Wiley and Sons \| location = New York \| pages = 42–43}}</ref> equivalently, the LRL vector is said to be ''conserved''. More generally, the LRL vector is conserved in all problems in which two bodies interact by a central force that varies as the inverse square of the distance between them; such problems are called Kepler problems.<ref name="goldstein_1980b">{{cite book \| last=Goldstein \| first=H. \| author-link=Herbert Goldstein \| date=1980 \| title=Classical Mechanics \| edition=2nd \| publisher=Addison Wesley \| pages=94–102}}</ref><ref name="arnold_1989">{{cite book \| last = Arnold \| first = V. I. \| author-link = Vladimir Arnold \| date = 1989 \| title = Mathematical Methods of Classical Mechanics \| edition = 2nd \| publisher = Springer-Verlag \| location = New York \| page = [https://archive.org/details/mathematicalmeth0000arno/page/38 38] \| isbn = 0-387-96890-3 \| url = https://archive.org/details/mathematicalmeth0000arno/page/38 }}</ref><ref name="sommerfeld_1989">{{cite book \| last = Sommerfeld \| first = A. \| author-link = Arnold Sommerfeld \| date = 1964 \| title = Mechanics \| series=Lectures on Theoretical Physics \| volume = 1 \| edition = 4th \| translator = Martin O. Stern \| publisher = Academic Press \| location = New York \| pages = 38–45}}</ref><ref name="lanczos_1970">{{cite book \| last = Lanczos \| first = C. \| author-link = Cornelius Lanczos \| date = 1970 \| title = The Variational Principles of Mechanics \| edition = 4th \| publisher = Dover Publications \| location = New York \| pages = 118, 129, 242, 248 }}</ref>

	Thus the [[Physics:~~Hydrogen atom~~\|hydrogen atom]] is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law of electrostatics, another inverse-square central force. The LRL vector was essential in the first [[Physics:Quantum mechanics\|quantum mechanical]] derivation of the spectrum of the hydrogen atom,<ref name="pauli_1926">{{cite journal \| last = Pauli \| first = W. \| author-link = Wolfgang Pauli \| date = 1926 \| title = Über das Wasserstoffspektrum vom Standpunkt der neuen Quantenmechanik \| journal = Zeitschrift für Physik \| volume = 36 \| issue = 5 \| pages = 336–363 \| doi = 10.1007/BF01450175 \| bibcode = 1926ZPhy...36..336P \| s2cid = 128132824 }}</ref><ref name="bohm_1993">{{cite book \| last = Bohm \| first = A. \| date = 1993 \| title = Quantum Mechanics: Foundations and Applications \| edition = 3rd \| publisher = Springer-Verlag \| location = New York \| pages = 205–222}}</ref> before the development of the Schrödinger equation. However, this approach is rarely used today.		Thus the [[Physics:Quantum atoms/hydrogen\|hydrogen atom]] is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law of electrostatics, another inverse-square central force. The LRL vector was essential in the first [[Physics:Quantum mechanics\|quantum mechanical]] derivation of the spectrum of the hydrogen atom,<ref name="pauli_1926">{{cite journal \| last = Pauli \| first = W. \| author-link = Wolfgang Pauli \| date = 1926 \| title = Über das Wasserstoffspektrum vom Standpunkt der neuen Quantenmechanik \| journal = Zeitschrift für Physik \| volume = 36 \| issue = 5 \| pages = 336–363 \| doi = 10.1007/BF01450175 \| bibcode = 1926ZPhy...36..336P \| s2cid = 128132824 }}</ref><ref name="bohm_1993">{{cite book \| last = Bohm \| first = A. \| date = 1993 \| title = Quantum Mechanics: Foundations and Applications \| edition = 3rd \| publisher = Springer-Verlag \| location = New York \| pages = 205–222}}</ref> before the development of the Schrödinger equation. However, this approach is rarely used today.

	In classical and quantum mechanics, conserved quantities generally correspond to a [[Physics:Quantum Symmetry in quantum mechanics\|symmetry]] of the system.<ref name="hanca_et_al_2004">{{cite journal \|author1=Hanca, J. \|author2=Tulejab, S. \|author3=Hancova, M. \|title=Symmetries and conservation laws: Consequences of Noether's theorem \|journal=American Journal of Physics \|volume=72 \|issue=4 \|pages=428–35 \|year=2004 \| doi = 10.1119/1.1591764 \| url=http://www.eftaylor.com/pub/symmetry.html \| bibcode = 2004AmJPh..72..428H }}</ref> The conservation of the LRL vector corresponds to an unusual symmetry; the Kepler problem is mathematically equivalent to a particle moving freely on the surface of a four-dimensional (hyper-)sphere<!--a 3-manifold, embedded in 4-space; the latter may be clearer to our readers-->,<ref name="fock_1935" >{{cite journal \| last = Fock \| first = V. \| author-link = Vladimir Fock \| date = 1935 \| title = Zur Theorie des Wasserstoffatoms \| journal = Zeitschrift für Physik \| volume = 98 \| issue = 3–4 \| pages = 145–154 \| doi = 10.1007/BF01336904\|bibcode = 1935ZPhy...98..145F \| s2cid = 123112334 }}</ref> so that the whole problem is symmetric under certain rotations of the four-dimensional space.<ref name="bargmann_1936" >{{cite journal \| last = Bargmann \| first = V. \| author-link = Valentine Bargmann \| date = 1936 \| title = Zur Theorie des Wasserstoffatoms: Bemerkungen zur gleichnamigen Arbeit von V. Fock \| journal = Zeitschrift für Physik \| volume = 99 \| issue = 7–8 \| pages = 576–582 \| doi = 10.1007/BF01338811 \| bibcode = 1936ZPhy...99..576B \| s2cid = 117461194 }}</ref> This higher symmetry results from two properties of the Kepler problem: the velocity vector always moves in a perfect circle and, for a given total energy, all such velocity circles intersect each other in the same two points.<ref name="hamilton_1847_hodograph">{{cite journal \| last = Hamilton \| first = W. R. \| author-link = William Rowan Hamilton \| date = 1847 \| title = The hodograph or a new method of expressing in symbolic language the Newtonian law of attraction \| journal = Proceedings of the Royal Irish Academy \| volume = 3 \| pages = 344–353 }}</ref>		In classical and quantum mechanics, conserved quantities generally correspond to a [[Physics:Quantum Symmetry in quantum mechanics\|symmetry]] of the system.<ref name="hanca_et_al_2004">{{cite journal \|author1=Hanca, J. \|author2=Tulejab, S. \|author3=Hancova, M. \|title=Symmetries and conservation laws: Consequences of Noether's theorem \|journal=American Journal of Physics \|volume=72 \|issue=4 \|pages=428–35 \|year=2004 \| doi = 10.1119/1.1591764 \| url=http://www.eftaylor.com/pub/symmetry.html \| bibcode = 2004AmJPh..72..428H }}</ref> The conservation of the LRL vector corresponds to an unusual symmetry; the Kepler problem is mathematically equivalent to a particle moving freely on the surface of a four-dimensional (hyper-)sphere<!--a 3-manifold, embedded in 4-space; the latter may be clearer to our readers-->,<ref name="fock_1935" >{{cite journal \| last = Fock \| first = V. \| author-link = Vladimir Fock \| date = 1935 \| title = Zur Theorie des Wasserstoffatoms \| journal = Zeitschrift für Physik \| volume = 98 \| issue = 3–4 \| pages = 145–154 \| doi = 10.1007/BF01336904\|bibcode = 1935ZPhy...98..145F \| s2cid = 123112334 }}</ref> so that the whole problem is symmetric under certain rotations of the four-dimensional space.<ref name="bargmann_1936" >{{cite journal \| last = Bargmann \| first = V. \| author-link = Valentine Bargmann \| date = 1936 \| title = Zur Theorie des Wasserstoffatoms: Bemerkungen zur gleichnamigen Arbeit von V. Fock \| journal = Zeitschrift für Physik \| volume = 99 \| issue = 7–8 \| pages = 576–582 \| doi = 10.1007/BF01338811 \| bibcode = 1936ZPhy...99..576B \| s2cid = 117461194 }}</ref> This higher symmetry results from two properties of the Kepler problem: the velocity vector always moves in a perfect circle and, for a given total energy, all such velocity circles intersect each other in the same two points.<ref name="hamilton_1847_hodograph">{{cite journal \| last = Hamilton \| first = W. R. \| author-link = William Rowan Hamilton \| date = 1847 \| title = The hodograph or a new method of expressing in symbolic language the Newtonian law of attraction \| journal = Proceedings of the Royal Irish Academy \| volume = 3 \| pages = 344–353 }}</ref>

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	In [[Physics:Classical mechanics\|classical mechanics]], the '''Laplace–Runge–Lenz vector''' ('''LRL vector''') is a [[Geometric algebra\|vector]] used chiefly to describe the shape and orientation of the [[Engineering:Orbit insertion\|orbit]] of one [[Astronomy:Astronomical object\|astronomical object]] around another, such as a [[Astronomy:Binary star\|binary star]] or a planet revolving around a star. For [[Two-body problem\|two bodies interacting]] by [[Physics:Newton's law of universal gravitation\|Newtonian gravity]], the LRL vector is a [[Physics:Constant of motion\|constant of motion]], meaning that it is the same no matter where it is calculated on the orbit;<ref name="goldstein_1980">{{cite book \| last=Goldstein \| first=H. \| author-link=Herbert Goldstein \| date=1980 \| title=Classical Mechanics \| edition=2nd \| publisher=Addison Wesley \| pages=102–105, 421–422}}</ref><ref name="taff_1985">{{cite book \| last = Taff \| first = L. G. \| author-link = Laurence G. Taff \| date = 1985 \| title = Celestial Mechanics: A Computational Guide for the Practitioner \| publisher = John Wiley and Sons \| location = New York \| pages = 42–43}}</ref> equivalently, the LRL vector is said to be ''[[Conservation law\|conserved]]''. More generally, the LRL vector is conserved in all problems in which two bodies interact by a [[Physics:Central force\|central force]] that varies as the [[Physics:Inverse-square law\|inverse square]] of the distance between them; such problems are called [[Physics:Kepler problem\|Kepler problem]]s.<ref name="goldstein_1980b">{{cite book \| last=Goldstein \| first=H. \| author-link=Herbert Goldstein \| date=1980 \| title=Classical Mechanics \| edition=2nd \| publisher=Addison Wesley \| pages=94–102}}</ref><ref name="arnold_1989">{{cite book \| last = Arnold \| first = V. I. \| author-link = Vladimir Arnold \| date = 1989 \| title = Mathematical Methods of Classical Mechanics \| edition = 2nd \| publisher = Springer-Verlag \| location = New York \| page = [https://archive.org/details/mathematicalmeth0000arno/page/38 38] \| isbn = 0-387-96890-3 \| url = https://archive.org/details/mathematicalmeth0000arno/page/38 }}</ref><ref name="sommerfeld_1989">{{cite book \| last = Sommerfeld \| first = A. \| author-link = Arnold Sommerfeld \| date = 1964 \| title = Mechanics \| series=Lectures on Theoretical Physics \| volume = 1 \| edition = 4th \| translator = Martin O. Stern \| publisher = Academic Press \| location = New York \| pages = 38–45}}</ref><ref name="lanczos_1970">{{cite book \| last = Lanczos \| first = C. \| author-link = Cornelius Lanczos \| date = 1970 \| title = The Variational Principles of Mechanics \| edition = 4th \| publisher = Dover Publications \| location = New York \| pages = 118, 129, 242, 248 }}</ref>		In [[Physics:Classical mechanics\|classical mechanics]], the '''Laplace–Runge–Lenz vector''' ('''LRL vector''') is a [[Geometric algebra\|vector]] used chiefly to describe the shape and orientation of the [[Engineering:Orbit insertion\|orbit]] of one [[Astronomy:Astronomical object\|astronomical object]] around another, such as a [[Astronomy:Binary star\|binary star]] or a planet revolving around a star. For [[Two-body problem\|two bodies interacting]] by [[Physics:Newton's law of universal gravitation\|Newtonian gravity]], the LRL vector is a [[Physics:Constant of motion\|constant of motion]], meaning that it is the same no matter where it is calculated on the orbit;<ref name="goldstein_1980">{{cite book \| last=Goldstein \| first=H. \| author-link=Herbert Goldstein \| date=1980 \| title=Classical Mechanics \| edition=2nd \| publisher=Addison Wesley \| pages=102–105, 421–422}}</ref><ref name="taff_1985">{{cite book \| last = Taff \| first = L. G. \| author-link = Laurence G. Taff \| date = 1985 \| title = Celestial Mechanics: A Computational Guide for the Practitioner \| publisher = John Wiley and Sons \| location = New York \| pages = 42–43}}</ref> equivalently, the LRL vector is said to be ''[[Conservation law\|conserved]]''. More generally, the LRL vector is conserved in all problems in which two bodies interact by a [[Physics:Central force\|central force]] that varies as the [[Physics:Inverse-square law\|inverse square]] of the distance between them; such problems are called [[Physics:Kepler problem\|Kepler problem]]s.<ref name="goldstein_1980b">{{cite book \| last=Goldstein \| first=H. \| author-link=Herbert Goldstein \| date=1980 \| title=Classical Mechanics \| edition=2nd \| publisher=Addison Wesley \| pages=94–102}}</ref><ref name="arnold_1989">{{cite book \| last = Arnold \| first = V. I. \| author-link = Vladimir Arnold \| date = 1989 \| title = Mathematical Methods of Classical Mechanics \| edition = 2nd \| publisher = Springer-Verlag \| location = New York \| page = [https://archive.org/details/mathematicalmeth0000arno/page/38 38] \| isbn = 0-387-96890-3 \| url = https://archive.org/details/mathematicalmeth0000arno/page/38 }}</ref><ref name="sommerfeld_1989">{{cite book \| last = Sommerfeld \| first = A. \| author-link = Arnold Sommerfeld \| date = 1964 \| title = Mechanics \| series=Lectures on Theoretical Physics \| volume = 1 \| edition = 4th \| translator = Martin O. Stern \| publisher = Academic Press \| location = New York \| pages = 38–45}}</ref><ref name="lanczos_1970">{{cite book \| last = Lanczos \| first = C. \| author-link = Cornelius Lanczos \| date = 1970 \| title = The Variational Principles of Mechanics \| edition = 4th \| publisher = Dover Publications \| location = New York \| pages = 118, 129, 242, 248 }}</ref>

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	The Laplace–Runge–Lenz vector is named after [[Biography:Pierre-Simon Laplace\|Pierre-Simon de Laplace]], [[Biography:Carl David Tolmé Runge\|Carl Runge]] and [[Biography:Wilhelm Lenz\|Wilhelm Lenz]]. It is also known as the '''Laplace vector''',<ref name="goldstein_1980c">{{cite book \| last=Goldstein \| first=H. \| author-link=Herbert Goldstein \| date=1980 \| title=Classical Mechanics \| edition=2nd \| publisher=Addison Wesley \| page=421}}</ref><ref name="arnold_1989b">{{cite book \| last = Arnold \| first = V. I. \| author-link = Vladimir Arnold \| date = 1989 \| title = Mathematical Methods of Classical Mechanics \| edition = 2nd \| publisher = Springer-Verlag \| location = New York \| pages = 413–415 \| isbn = 0-387-96890-3 }}</ref> the '''Runge–Lenz vector'''<ref name="goldstein_1975_1976">{{cite journal \| last=Goldstein \| first=H. \| author-link=Herbert Goldstein \| date=1975 \| title=Prehistory of the Runge–Lenz vector \| journal=[[American Journal of Physics]] \| volume=43 \| issue=8 \| pages=737–738 \| doi=10.1119/1.9745\|bibcode = 1975AmJPh..43..737G }}<br />{{cite journal \| last=Goldstein \| first=H. \| author-link=Herbert Goldstein \| date=1976 \| title=More on the prehistory of the Runge–Lenz vector \| journal=[[American Journal of Physics]] \| volume=44 \| issue=11 \| pages=1123–1124 \| doi=10.1119/1.10202\|bibcode = 1976AmJPh..44.1123G }}</ref> and the '''Lenz vector'''.<ref name="bohm_1993" /> Ironically, [[Stigler's law of eponymy\|none of those scientists]] discovered it.<ref name="goldstein_1975_1976" /> The LRL vector has been re-discovered and re-formulated several times;<ref name="goldstein_1975_1976" /> for example, it is equivalent to the dimensionless [[eccentricity vector]] of [[Physics:Celestial mechanics\|celestial mechanics]].<ref name="taff_1985" /><ref name="arnold_1989b" /><ref name="hamilton_1847_quaternions">{{cite journal \| last = Hamilton \| first = W. R. \| author-link = William Rowan Hamilton \| date = 1847 \| title = Applications of Quaternions to Some Dynamical Questions \| journal = Proceedings of the Royal Irish Academy \| volume = 3 \| pages = Appendix III}}</ref> Various generalizations of the LRL vector have been defined, which incorporate the effects of [[Physics:Special relativity\|special relativity]], [[Physics:Electromagnetic field\|electromagnetic field]]s and even different types of central forces.<ref name="landau_lifshitz_1976">{{cite book \| last=Landau \| first=L. D. \| author-link=Lev Landau \| author2=Lifshitz E. M. \| author-link2=Evgeny Lifshitz \| date=1976 \| title=Mechanics \| edition=3rd \| publisher=Pergamon Press \| page=[https://archive.org/details/mechanics00land/page/154 154] \| isbn=0-08-021022-8 \| url=https://archive.org/details/mechanics00land/page/154 }}</ref><ref name="fradkin_1967">{{cite journal \| last = Fradkin \| first = D. M. \| date = 1967 \| title = Existence of the Dynamic Symmetries O<sub>4</sub> and SU<sub>3</sub> for All Classical Central Potential Problems \| journal = Progress of Theoretical Physics \| volume = 37 \| issue = 5 \| pages = 798–812 \| doi = 10.1143/PTP.37.798\|bibcode = 1967PThPh..37..798F \| doi-access = free }}</ref><ref name="yoshida_1987">{{cite journal \| last = Yoshida \| first = T. \| date = 1987 \| title = Two methods of generalisation of the Laplace–Runge–Lenz vector \| journal = European Journal of Physics \| volume = 8 \| issue = 4 \| pages = 258–259 \| doi = 10.1088/0143-0807/8/4/005\|bibcode = 1987EJPh....8..258Y \| s2cid = 250843588 }}</ref>		The Laplace–Runge–Lenz vector is named after [[Biography:Pierre-Simon Laplace\|Pierre-Simon de Laplace]], [[Biography:Carl David Tolmé Runge\|Carl Runge]] and [[Biography:Wilhelm Lenz\|Wilhelm Lenz]]. It is also known as the '''Laplace vector''',<ref name="goldstein_1980c">{{cite book \| last=Goldstein \| first=H. \| author-link=Herbert Goldstein \| date=1980 \| title=Classical Mechanics \| edition=2nd \| publisher=Addison Wesley \| page=421}}</ref><ref name="arnold_1989b">{{cite book \| last = Arnold \| first = V. I. \| author-link = Vladimir Arnold \| date = 1989 \| title = Mathematical Methods of Classical Mechanics \| edition = 2nd \| publisher = Springer-Verlag \| location = New York \| pages = 413–415 \| isbn = 0-387-96890-3 }}</ref> the '''Runge–Lenz vector'''<ref name="goldstein_1975_1976">{{cite journal \| last=Goldstein \| first=H. \| author-link=Herbert Goldstein \| date=1975 \| title=Prehistory of the Runge–Lenz vector \| journal=[[American Journal of Physics]] \| volume=43 \| issue=8 \| pages=737–738 \| doi=10.1119/1.9745\|bibcode = 1975AmJPh..43..737G }}<br />{{cite journal \| last=Goldstein \| first=H. \| author-link=Herbert Goldstein \| date=1976 \| title=More on the prehistory of the Runge–Lenz vector \| journal=[[American Journal of Physics]] \| volume=44 \| issue=11 \| pages=1123–1124 \| doi=10.1119/1.10202\|bibcode = 1976AmJPh..44.1123G }}</ref> and the '''Lenz vector'''.<ref name="bohm_1993" /> Ironically, [[Stigler's law of eponymy\|none of those scientists]] discovered it.<ref name="goldstein_1975_1976" /> The LRL vector has been re-discovered and re-formulated several times;<ref name="goldstein_1975_1976" /> for example, it is equivalent to the dimensionless [[eccentricity vector]] of [[Physics:Celestial mechanics\|celestial mechanics]].<ref name="taff_1985" /><ref name="arnold_1989b" /><ref name="hamilton_1847_quaternions">{{cite journal \| last = Hamilton \| first = W. R. \| author-link = William Rowan Hamilton \| date = 1847 \| title = Applications of Quaternions to Some Dynamical Questions \| journal = Proceedings of the Royal Irish Academy \| volume = 3 \| pages = Appendix III}}</ref> Various generalizations of the LRL vector have been defined, which incorporate the effects of [[Physics:Special relativity\|special relativity]], [[Physics:Electromagnetic field\|electromagnetic field]]s and even different types of central forces.<ref name="landau_lifshitz_1976">{{cite book \| last=Landau \| first=L. D. \| author-link=Lev Landau \| author2=Lifshitz E. M. \| author-link2=Evgeny Lifshitz \| date=1976 \| title=Mechanics \| edition=3rd \| publisher=Pergamon Press \| page=[https://archive.org/details/mechanics00land/page/154 154] \| isbn=0-08-021022-8 \| url=https://archive.org/details/mechanics00land/page/154 }}</ref><ref name="fradkin_1967">{{cite journal \| last = Fradkin \| first = D. M. \| date = 1967 \| title = Existence of the Dynamic Symmetries O<sub>4</sub> and SU<sub>3</sub> for All Classical Central Potential Problems \| journal = Progress of Theoretical Physics \| volume = 37 \| issue = 5 \| pages = 798–812 \| doi = 10.1143/PTP.37.798\|bibcode = 1967PThPh..37..798F \| doi-access = free }}</ref><ref name="yoshida_1987">{{cite journal \| last = Yoshida \| first = T. \| date = 1987 \| title = Two methods of generalisation of the Laplace–Runge–Lenz vector \| journal = European Journal of Physics \| volume = 8 \| issue = 4 \| pages = 258–259 \| doi = 10.1088/0143-0807/8/4/005\|bibcode = 1987EJPh....8..258Y \| s2cid = 250843588 }}</ref>
	[[File:Runge-Lenz vector diagram.png\|thumb]]		</div>

			<div style="width:300px;">
			[[File:Runge-Lenz vector diagram.png\|thumb\|280px\|Quantum Runge–Lenz vector.]]
			</div>

			</div>

	==Context==		==Context==

imported>WikiHarold: Repair Quantum Collection B backlink template

2026-05-08T19:49:51Z

Repair Quantum Collection B backlink template

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	In classical mechanics, the '''Laplace–Runge–Lenz vector''' ('''LRL vector''') is a vector used chiefly to describe the shape and orientation of the orbit of one astronomical object around another, such as a binary star or a planet revolving around a star. For two bodies interacting by Newtonian gravity, the LRL vector is a constant of motion, meaning that it is the same no matter where it is calculated on the orbit;<ref name="goldstein_1980">{{cite book \| last=Goldstein \| first=H. \| author-link=Herbert Goldstein \| date=1980 \| title=Classical Mechanics \| edition=2nd \| publisher=Addison Wesley \| pages=102–105, 421–422}}</ref><ref name="taff_1985">{{cite book \| last = Taff \| first = L. G. \| author-link = Laurence G. Taff \| date = 1985 \| title = Celestial Mechanics: A Computational Guide for the Practitioner \| publisher = John Wiley and Sons \| location = New York \| pages = 42–43}}</ref> equivalently, the LRL vector is said to be ''conserved''. More generally, the LRL vector is conserved in all problems in which two bodies interact by a central force that varies as the inverse square of the distance between them; such problems are called Kepler problems.<ref name="goldstein_1980b">{{cite book \| last=Goldstein \| first=H. \| author-link=Herbert Goldstein \| date=1980 \| title=Classical Mechanics \| edition=2nd \| publisher=Addison Wesley \| pages=94–102}}</ref><ref name="arnold_1989">{{cite book \| last = Arnold \| first = V. I. \| author-link = Vladimir Arnold \| date = 1989 \| title = Mathematical Methods of Classical Mechanics \| edition = 2nd \| publisher = Springer-Verlag \| location = New York \| page = [https://archive.org/details/mathematicalmeth0000arno/page/38 38] \| isbn = 0-387-96890-3 \| url = https://archive.org/details/mathematicalmeth0000arno/page/38 }}</ref><ref name="sommerfeld_1989">{{cite book \| last = Sommerfeld \| first = A. \| author-link = Arnold Sommerfeld \| date = 1964 \| title = Mechanics \| series=Lectures on Theoretical Physics \| volume = 1 \| edition = 4th \| translator = Martin O. Stern \| publisher = Academic Press \| location = New York \| pages = 38–45}}</ref><ref name="lanczos_1970">{{cite book \| last = Lanczos \| first = C. \| author-link = Cornelius Lanczos \| date = 1970 \| title = The Variational Principles of Mechanics \| edition = 4th \| publisher = Dover Publications \| location = New York \| pages = 118, 129, 242, 248 }}</ref>		In classical mechanics, the '''Laplace–Runge–Lenz vector''' ('''LRL vector''') is a vector used chiefly to describe the shape and orientation of the orbit of one astronomical object around another, such as a binary star or a planet revolving around a star. For two bodies interacting by Newtonian gravity, the LRL vector is a constant of motion, meaning that it is the same no matter where it is calculated on the orbit;<ref name="goldstein_1980">{{cite book \| last=Goldstein \| first=H. \| author-link=Herbert Goldstein \| date=1980 \| title=Classical Mechanics \| edition=2nd \| publisher=Addison Wesley \| pages=102–105, 421–422}}</ref><ref name="taff_1985">{{cite book \| last = Taff \| first = L. G. \| author-link = Laurence G. Taff \| date = 1985 \| title = Celestial Mechanics: A Computational Guide for the Practitioner \| publisher = John Wiley and Sons \| location = New York \| pages = 42–43}}</ref> equivalently, the LRL vector is said to be ''conserved''. More generally, the LRL vector is conserved in all problems in which two bodies interact by a central force that varies as the inverse square of the distance between them; such problems are called Kepler problems.<ref name="goldstein_1980b">{{cite book \| last=Goldstein \| first=H. \| author-link=Herbert Goldstein \| date=1980 \| title=Classical Mechanics \| edition=2nd \| publisher=Addison Wesley \| pages=94–102}}</ref><ref name="arnold_1989">{{cite book \| last = Arnold \| first = V. I. \| author-link = Vladimir Arnold \| date = 1989 \| title = Mathematical Methods of Classical Mechanics \| edition = 2nd \| publisher = Springer-Verlag \| location = New York \| page = [https://archive.org/details/mathematicalmeth0000arno/page/38 38] \| isbn = 0-387-96890-3 \| url = https://archive.org/details/mathematicalmeth0000arno/page/38 }}</ref><ref name="sommerfeld_1989">{{cite book \| last = Sommerfeld \| first = A. \| author-link = Arnold Sommerfeld \| date = 1964 \| title = Mechanics \| series=Lectures on Theoretical Physics \| volume = 1 \| edition = 4th \| translator = Martin O. Stern \| publisher = Academic Press \| location = New York \| pages = 38–45}}</ref><ref name="lanczos_1970">{{cite book \| last = Lanczos \| first = C. \| author-link = Cornelius Lanczos \| date = 1970 \| title = The Variational Principles of Mechanics \| edition = 4th \| publisher = Dover Publications \| location = New York \| pages = 118, 129, 242, 248 }}</ref>

	Thus the [[Physics:~~Hydrogen atom~~\|hydrogen atom]] is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law of electrostatics, another inverse-square central force. The LRL vector was essential in the first [[Physics:Quantum mechanics\|quantum mechanical]] derivation of the spectrum of the hydrogen atom,<ref name="pauli_1926">{{cite journal \| last = Pauli \| first = W. \| author-link = Wolfgang Pauli \| date = 1926 \| title = Über das Wasserstoffspektrum vom Standpunkt der neuen Quantenmechanik \| journal = Zeitschrift für Physik \| volume = 36 \| issue = 5 \| pages = 336–363 \| doi = 10.1007/BF01450175 \| bibcode = 1926ZPhy...36..336P \| s2cid = 128132824 }}</ref><ref name="bohm_1993">{{cite book \| last = Bohm \| first = A. \| date = 1993 \| title = Quantum Mechanics: Foundations and Applications \| edition = 3rd \| publisher = Springer-Verlag \| location = New York \| pages = 205–222}}</ref> before the development of the Schrödinger equation. However, this approach is rarely used today.		Thus the [[Physics:Quantum atoms/hydrogen\|hydrogen atom]] is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law of electrostatics, another inverse-square central force. The LRL vector was essential in the first [[Physics:Quantum mechanics\|quantum mechanical]] derivation of the spectrum of the hydrogen atom,<ref name="pauli_1926">{{cite journal \| last = Pauli \| first = W. \| author-link = Wolfgang Pauli \| date = 1926 \| title = Über das Wasserstoffspektrum vom Standpunkt der neuen Quantenmechanik \| journal = Zeitschrift für Physik \| volume = 36 \| issue = 5 \| pages = 336–363 \| doi = 10.1007/BF01450175 \| bibcode = 1926ZPhy...36..336P \| s2cid = 128132824 }}</ref><ref name="bohm_1993">{{cite book \| last = Bohm \| first = A. \| date = 1993 \| title = Quantum Mechanics: Foundations and Applications \| edition = 3rd \| publisher = Springer-Verlag \| location = New York \| pages = 205–222}}</ref> before the development of the Schrödinger equation. However, this approach is rarely used today.

	In classical and quantum mechanics, conserved quantities generally correspond to a [[Physics:Quantum Symmetry in quantum mechanics\|symmetry]] of the system.<ref name="hanca_et_al_2004">{{cite journal \|author1=Hanca, J. \|author2=Tulejab, S. \|author3=Hancova, M. \|title=Symmetries and conservation laws: Consequences of Noether's theorem \|journal=American Journal of Physics \|volume=72 \|issue=4 \|pages=428–35 \|year=2004 \| doi = 10.1119/1.1591764 \| url=http://www.eftaylor.com/pub/symmetry.html \| bibcode = 2004AmJPh..72..428H }}</ref> The conservation of the LRL vector corresponds to an unusual symmetry; the Kepler problem is mathematically equivalent to a particle moving freely on the surface of a four-dimensional (hyper-)sphere<!--a 3-manifold, embedded in 4-space; the latter may be clearer to our readers-->,<ref name="fock_1935" >{{cite journal \| last = Fock \| first = V. \| author-link = Vladimir Fock \| date = 1935 \| title = Zur Theorie des Wasserstoffatoms \| journal = Zeitschrift für Physik \| volume = 98 \| issue = 3–4 \| pages = 145–154 \| doi = 10.1007/BF01336904\|bibcode = 1935ZPhy...98..145F \| s2cid = 123112334 }}</ref> so that the whole problem is symmetric under certain rotations of the four-dimensional space.<ref name="bargmann_1936" >{{cite journal \| last = Bargmann \| first = V. \| author-link = Valentine Bargmann \| date = 1936 \| title = Zur Theorie des Wasserstoffatoms: Bemerkungen zur gleichnamigen Arbeit von V. Fock \| journal = Zeitschrift für Physik \| volume = 99 \| issue = 7–8 \| pages = 576–582 \| doi = 10.1007/BF01338811 \| bibcode = 1936ZPhy...99..576B \| s2cid = 117461194 }}</ref> This higher symmetry results from two properties of the Kepler problem: the velocity vector always moves in a perfect circle and, for a given total energy, all such velocity circles intersect each other in the same two points.<ref name="hamilton_1847_hodograph">{{cite journal \| last = Hamilton \| first = W. R. \| author-link = William Rowan Hamilton \| date = 1847 \| title = The hodograph or a new method of expressing in symbolic language the Newtonian law of attraction \| journal = Proceedings of the Royal Irish Academy \| volume = 3 \| pages = 344–353 }}</ref>		In classical and quantum mechanics, conserved quantities generally correspond to a [[Physics:Quantum Symmetry in quantum mechanics\|symmetry]] of the system.<ref name="hanca_et_al_2004">{{cite journal \|author1=Hanca, J. \|author2=Tulejab, S. \|author3=Hancova, M. \|title=Symmetries and conservation laws: Consequences of Noether's theorem \|journal=American Journal of Physics \|volume=72 \|issue=4 \|pages=428–35 \|year=2004 \| doi = 10.1119/1.1591764 \| url=http://www.eftaylor.com/pub/symmetry.html \| bibcode = 2004AmJPh..72..428H }}</ref> The conservation of the LRL vector corresponds to an unusual symmetry; the Kepler problem is mathematically equivalent to a particle moving freely on the surface of a four-dimensional (hyper-)sphere<!--a 3-manifold, embedded in 4-space; the latter may be clearer to our readers-->,<ref name="fock_1935" >{{cite journal \| last = Fock \| first = V. \| author-link = Vladimir Fock \| date = 1935 \| title = Zur Theorie des Wasserstoffatoms \| journal = Zeitschrift für Physik \| volume = 98 \| issue = 3–4 \| pages = 145–154 \| doi = 10.1007/BF01336904\|bibcode = 1935ZPhy...98..145F \| s2cid = 123112334 }}</ref> so that the whole problem is symmetric under certain rotations of the four-dimensional space.<ref name="bargmann_1936" >{{cite journal \| last = Bargmann \| first = V. \| author-link = Valentine Bargmann \| date = 1936 \| title = Zur Theorie des Wasserstoffatoms: Bemerkungen zur gleichnamigen Arbeit von V. Fock \| journal = Zeitschrift für Physik \| volume = 99 \| issue = 7–8 \| pages = 576–582 \| doi = 10.1007/BF01338811 \| bibcode = 1936ZPhy...99..576B \| s2cid = 117461194 }}</ref> This higher symmetry results from two properties of the Kepler problem: the velocity vector always moves in a perfect circle and, for a given total energy, all such velocity circles intersect each other in the same two points.<ref name="hamilton_1847_hodograph">{{cite journal \| last = Hamilton \| first = W. R. \| author-link = William Rowan Hamilton \| date = 1847 \| title = The hodograph or a new method of expressing in symbolic language the Newtonian law of attraction \| journal = Proceedings of the Royal Irish Academy \| volume = 3 \| pages = 344–353 }}</ref>

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	{{~~short~~ description\|Vector used in astronomy}}		{{Short description\|Vector used in astronomy}}
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	{{hatnote\|Throughout this article, vectors and their magnitudes are indicated by boldface and italic type, respectively. For example, <math>\left\| \mathbf{A} \right\| = A</math>.}}		{{hatnote\|Throughout this article, vectors and their magnitudes are indicated by boldface and italic type, respectively. For example, <math>\left\| \mathbf{A} \right\| = A</math>.}}

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	{{short description\|Vector used in astronomy}}		{{short description\|Vector used in astronomy}}

	{{Quantum book backlink\|Mathematical structure and systems}}		{{Quantum book backlink\|Mathematical structure and systems}}
	{{Quantum article nav\|previous=Physics:Quantum Angular momentum operator\|previous label=Angular momentum operator\|next=Physics:Quantum Approximation Methods\|next label=Approximation Methods}}		{{Quantum article nav\|previous=Physics:Quantum Angular momentum operator\|previous label=Angular momentum operator\|next=Physics:Quantum Approximation Methods\|next label=Approximation Methods}}

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	{{short description\|Vector used in astronomy}}		{{short description\|Vector used in astronomy}}

	{{Quantum book backlink\|Mathematical structure and systems}}		{{Quantum book backlink\|Mathematical structure and systems}}
			{{Quantum article nav\|previous=Physics:Quantum Angular momentum operator\|previous label=Angular momentum operator\|next=Physics:Quantum Approximation Methods\|next label=Approximation Methods}}
	{{hatnote\|Throughout this article, vectors and their magnitudes are indicated by boldface and italic type, respectively. For example, <math>\left\| \mathbf{A} \right\| = A</math>.}}		{{hatnote\|Throughout this article, vectors and their magnitudes are indicated by boldface and italic type, respectively. For example, <math>\left\| \mathbf{A} \right\| = A</math>.}}