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UT Arlington PHYS 3446 - Symmetries

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PHYS 3446 – Lecture #18Slide 2Slide 3Slide 4Slide 5Slide 6Slide 7Slide 8Slide 9Slide 10Slide 11Slide 12Slide 13Slide 14Slide 15Slide 16Slide 17Slide 18Slide 19Slide 20Wednesday, Nov. 8, 2006 PHYS 3446, Fall 2006Jae Yu1PHYS 3446 – Lecture #18Wednesday, Nov. 8, 2006Dr. Jae Yu1. Symmetries•Why do we care about the symmetry?•Symmetry in Lagrangian formalism•Symmetries in quantum mechanical system•Isospin symmetry•Local gauge symmetryWednesday, Nov. 8, 2006 PHYS 3446, Fall 2006Jae Yu2•No lecture next Monday, Nov. 13 but SH105 is reserved for your discussions concerning the projects•Quiz next Wednesday, Nov. 15 in class•2nd term exam–Wednesday, Nov. 22–Covers: Ch 4 – whatever we finish on Nov. 20•Reading assignments–10.3 and 10.4AnnouncementsWednesday, Nov. 8, 2006 PHYS 3446, Fall 2006Jae Yu3•We’ve learned about various newly introduced quantum numbers as a patch work to explain experimental observations–Lepton numbers–Baryon numbers–Isospin–Strangeness•Some of these numbers are conserved in certain situation but not in others–Very frustrating indeed….•These are due to lack of quantitative description by an elegant theoryQuantum NumbersWednesday, Nov. 8, 2006 PHYS 3446, Fall 2006Jae Yu4•Some quantum numbers are conserved in strong interactions but not in electromagnetic and weak interactions–Inherent reflection of underlying forces•Understanding conservation or violation of quantum numbers in certain situations is important for formulating quantitative theoretical frameworkWhy symmetry?Wednesday, Nov. 8, 2006 PHYS 3446, Fall 2006Jae Yu5•When is a quantum number conserved?–When there is an underlying symmetry in the system–When the quantum number is not affected (or is conserved) by (under) the changes in the physical system•Noether’s theoremNoether’s theorem: If there is a conserved quantity associated with a physical system, there exists an underlying invariance or symmetry principle responsible for this conservation.•Symmetries provide critical restrictions in formulating theoriesWhy symmetry?Wednesday, Nov. 8, 2006 PHYS 3446, Fall 2006Jae Yu6•Symmetry of a system is defined by any set of transformations that keep the equation of motion unchanged or invariant•Equations of motion can be obtained through –Lagrangian formalism: L=T-V where the Equation of motion is what minimizes the Lagrangian L under changes of coordinates–Hamiltonian formalism: H=T+V with the equation of motion that minimizes the Hamiltonian under changes of coordinates•Both these formalisms can be used to discuss symmetries in non-relativistic (or classical cases) or relativistic cases and quantum mechanical systemsSymmetries in Lagrangian FormalismWednesday, Nov. 8, 2006 PHYS 3446, Fall 2006Jae Yu7•Consider an isolated non-relativistic physical system of two particles interacting through a potential that only depends on the relative distance between them–EM and gravitational force•The total kinetic and potential energies of the system are: and •The equations of motion are thenSymmetries in Lagrangian Formalism?2 21 1 2 21 12 2T m r m r= +r r& &( )1 2V V r r= -r r2 2m r =r&&1 1m r =r&&( )1 2where iV r rr�- =�r rr( )1 1 2V r r- � - =rr r( )1 21V r rr�- -�r rr( )2 1 2V r r- � - =rr r( )1 22V r rr�- -�r rrˆ ˆˆi i ix V y V z Vx y x� � �+ +� � �Wednesday, Nov. 8, 2006 PHYS 3446, Fall 2006Jae Yu8•If we perform a linear translation of the origin of coordinate system by a constant vector–The position vectors of the two particles become –But the equations of motion do not change since is a constant vector–This is due to the invariance of the potential V under the translationSymmetries in Lagrangian Formalism1r �r'V =a-r2r �ra-r1r a-r r2r a-r r( )1 2' 'V r r- =r r( )1 2V r a r a- - + =r r r r( )1 2V r r-r rWednesday, Nov. 8, 2006 PHYS 3446, Fall 2006Jae Yu9•This means that the translation of the coordinate system for an isolated two particle system defines a symmetry of the system (remember Noether’s theorem?)•This particular physical system is invariant under spatial translation•What is the consequence of this invariance?–From the form of the potential, the total force is –Since Symmetries in Lagrangian Formalism?totF =r1 2V Vr r� �=-� �r r1 2F F+ =r r( ) ( )1 1 2 2 1 2V r r V r r- � - - � - =r rr r r r0Wednesday, Nov. 8, 2006 PHYS 3446, Fall 2006Jae Yu10Symmetries in Lagrangian Formalism?totF =rtotdPdt=r0•What does this mean?–Total momentum of the system is invariant under spatial translation•In other words, the translational symmetry results in linear momentum conservation•This holds for multi-particle system as wellWednesday, Nov. 8, 2006 PHYS 3446, Fall 2006Jae Yu11•For multi-particle system, using Lagrangian L=T-V, the equations of motion can be generalized•By construction,•As previously discussed, for the system with a potential that depends on the relative distance between particles, The Lagrangian is independent of particulars of the individual coordinate and thusSymmetries in Lagrangian Formalism0i iL Lddt rr� �- =��rr&iLr�=�r&0imLr�=�0i iidp Ldt r�= =�riTr�=�r&im r =r&ipr212i im rr�� �=� ��� �r&r&Wednesday, Nov. 8, 2006 PHYS 3446, Fall 2006Jae Yu12•Momentum pi can expanded to other kind of momenta for the given spatial translation–Rotational translation: Angular momentum–Time translation: Energy–Rotation in isospin space: Isospin•The equation says that if the Lagrangian of a physical system does not depend on specifics of a given coordinate, the conjugate momentum is conserved•One can turn this around and state that if a Lagrangian does not depend on some particular coordinate, it must be invariant under translations of this coordinate.0i iidp Ldt r�= =�rSymmetries in Lagrangian FormalismWednesday, Nov. 8, 2006 PHYS 3446, Fall 2006Jae Yu13•The translational symmetries of a physical system give invariance in the corresponding physical quantities–Symmetry under linear translation•Linear momentum conservation–Symmetry under spatial rotation •Angular momentum conservation–Symmetry under time translation•Energy conservation–Symmetry under isospin space rotation•Isospin conservationTranslational Symmetries & Conserved QuantitiesWednesday, Nov. 8, 2006 PHYS 3446, Fall 2006Jae Yu14•In


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UT Arlington PHYS 3446 - Symmetries

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