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UT Arlington PHYS 3446 - Lecture Notes

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Monday, Nov. 22, 2010 PHYS 3446, Fall 2010 Andrew Brandt 1PHYS 3446 – Lecture #21Monday, Nov. 22, 2010Dr. Andrew Brandt•Quantum Numbers•Symmetries•Note HW8 is optional, will replace lowest grade•Good rough draft of talk due Dec.1Project Assignments and Updated AgendaMonday Dec. 6• Long baseline neutrino experiments (neutrino mass) Butler-Mayfield• Solar Neutrino Deficit Baral-Lord• G-2 experiments Ouyang-Wright• HERA experiments: diffraction/large rapidity gaps Gray-WooWeds Dec. 8• Quark-Gluon Plasma (RHIC) Byrd-Pryor• Higgs Boson Theory Bridges-Corbin• Standard Model+Beyond the Standard Model Higgs Boson Searches at DzeroAbsher-Dean-Shumate• Supersymmetry Theory Ibarra• Extra Dimension Searches at ATLAS Contreras-LaRoqueHW8 (optional) due 12/1/101) What was the typical beam energy of early accelerators?2) Which accelerator can produce higher accelerating potenials, Cockroft-Walton or Van de Graff? What is the ratio of the maximum voltages of the two accelerators?3) Is the LHC a cyclotron, a synchrotron, or a synchrocyclotron?4) For a 7 TeV beam and a 10Tesla magnet, what radius does this imply (Eq. 8.9’)? Look up the actual dipole magnet strength and radius of the LHC. How do these compare to the rule of thumb? 5) What is strong focussing? What kind of magnets does it involve?Monday, Nov. 15, 2010 PHYS 3446, Fall 2010 Andrew Brandt 3Monday, Nov. 22, 2010 PHYS 3446, Fall 2010 Andrew Brandt 4• Baryon Number– An additive and conserved quantum number, Baryon number (B)– This number is conserved in strong interactions and EM but not necessarily in weak interactions• Lepton Number– Quantum number assigned to leptons– Lepton numbers by species and the total lepton numbers must be conserved (EM+EW)• Strangeness Numbers– Conserved in strong interactions– But violated in weak interactions• Isospin Quantum Numbers– Conserved in strong interactions– But violated in weak and EM interactionsQuantum NumbersMonday, Nov. 22, 2010 PHYS 3446, Fall 2010 Andrew Brandt 5• 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 frameworkQuantum Number ConservationMonday, Nov. 22, 2010 PHYS 3446, Fall 2010 Andrew Brandt 6• Three types of weak interactions– Hadronic decays: Only hadrons in the final state– Semi-leptonic decays: both hadrons and leptons are present– Leptonic decays: only leptons are presentWeak Interactions0npepeeeMonday, Nov. 22, 2010 PHYS 3446, Fall 2010 Andrew Brandt 7• When is a quantum number conserved?– When there is an underlying symmetry in the system– When the quantum number is not affected by changes in the physical system• Noether’s theorem: If there is a conserved quantityassociated with a physical system, there exists an underlying invariance or symmetry principle responsible for this conservation.• Symmetries provide critical restrictions in formulating theoriesSymmetryMonday, Nov. 22, 2010 PHYS 3446, Fall 2010 Andrew Brandt 8• 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?221 1 2 21122T m r m r12V V r r22mr11mr12where iV r rr1 1 2V r r121V r rr2 1 2V r r122V r rrˆˆˆi i ix V y V z Vx y xMonday, Nov. 22, 2010 PHYS 3446, Fall 2010 Andrew Brandt 9• 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'Va2ra1ra2ra12''V r r12V r a r a12V r rMonday, Nov. 22, 2010 PHYS 3446, Fall 2010 Andrew Brandt 10• A 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 an equation of motion that minimizes the Hamiltonian under changes of coordinates• Either of these notations can be used to discuss symmetries in classical cases or relativistic cases and quantum mechanical systemsSymmetries in Lagrangian FormalismMonday, Nov. 22, 2010 PHYS 3446, Fall 2010 Andrew Brandt 11• The translation of the coordinate system for an isolated two particle system defines a symmetry of the system (recall 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?totF12VVrr12FF1 1 2 2 1 2V r r V r r0Monday, Nov. 22, 2010 PHYS 3446, Fall 2010 Andrew Brandt 12Symmetries in Lagrangian Formalism?totFtotdPdt0• What does this mean?– Total momentum of the system is invariant under spatial translation• In other words, translational symmetry results in linear momentum conservation• This holds for multi-particle system as wellMonday, Nov. 22, 2010 PHYS 3446, Fall 2010 Andrew Brandt 13• 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 Formalism0iiLLddt rriLr0imLr0iiidp Ldt riTrimrip212iimrrMonday, Nov. 22, 2010 PHYS 3446, Fall 2010 Andrew Brandt 14– 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 conservationSymmetries & Conserved QuantitiesMonday, Nov. 22, 2010 PHYS 3446, Fall 2010 Andrew Brandt 15• In quantum mechanics, an observable physical quantity corresponds to the


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UT Arlington PHYS 3446 - Lecture Notes

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