UTA McNair Scholars Program Extends ANNUAL Application deadline

1. UTA McNair Scholars Program Extends ANNUAL Application deadline The UTA McNair Scholars Program has extended its 2008 recruitment period through November 25.  The McNair program is a federal TRIO program that offers opportunities to selected undergraduates hoping to pursue graduate studies leading to the Ph.D. and a career in the academy.  We appreciate faculty recommendations of talented undergraduates to this program. The McNair program works with at least 30 students each academic year from low-income/first-generation orunderrepresented (African American, Hispanic, Native American, Native Hawaiian) backgrounds.  Two-thirds of our participants must be first-generation/low-income students.  Applicants must be (or have): U.S. citizens or permanent residents. sophomores (by spring 2009), juniors or seniors (not graduating before at least December 2009) a minimum 2.9 GPA , able to engage in a summer of faculty-mentored research between their junior and senior years.  All majors are eligible, although we especially encourage those in STEM fields to apply.  The program focuses on all aspects of graduate school preparation:  faculty-mentored summer research (including a $3000 stipend and three credit hours of independent study), publication of research abstract in the UTA McNair Research Journal, GRE prep classes, diverse seminars related to program goals, conference participation, graduate school visits, and other benefits.  Application packets are available in 122 Hammond Hall, may be downloaded from http://www.uta.edu/soar/  (Click on TRIO Programs, then McNair), or requested by telephone (817-272-3715) or via e-mail (mcnair@uta.edu). PHYS 3446, Fall 2008 Andrew Brandt

2. Strangeness • Strangeness quantum number • Murray Gell-Mann and Abraham Pais proposed a new additive quantum number that are carried by these particles • Conserved in strong interactions • Violated in weak decays • S=0 for all ordinary mesons and baryons as well as photons and leptons • For any strong associated-production reaction w/ the initial state S=0, the total strangeness of particles in the final state should add up to 0. PHYS 3446, Fall 2008 Andrew Brandt

3. More on Strangeness • Let’s look at the reactions again • This is a strong interaction • Strangeness must be conserved • S: 0 + 0  +1 -1 • How about the decays of the final state particles? • and • These decays are weak interactions so S is not conserved • S: -1  0 + 0 and +1  0 + 0 • A not-really-elegant solution • S only conserved in Strong and EM interactions  Unique strangeness quantum numbers cannot be assigned to leptons • Leads to the hypothesis of strange quarks PHYS 3446, Fall 2008 Andrew Brandt

4. Isospin Quantum Number • Strong force does not depend on the charge of the particle • Nuclear properties of protons and neutrons are very similar • From the studies of mirror nuclei, the strengths of p-p, p-n and n-n strong interactions are essentially the same • If corrected by EM interactions, the x-sec between n-n and p-p are the same • Since strong force is much stronger than any other forces, we could imagine a new quantum number that applies to all particles • Protons and neutrons are two orthogonal mass eigenstates of the same particle like spin up and down states PHYS 3446, Fall 2008 Andrew Brandt

5. Isospin Quantum Number • Protons and neutrons are degenerate in mass because of some symmetry of the strong force • Isospin symmetry  Under the strong force these two particles appear identical • Presence of Electromagnetic or Weak forces breaks this symmetry, distinguishing p from n • Isospin works just like spin • Protons and neutrons have isospin ½  Isospin doublet • Three pions, p+, p- and p0, have almost the same masses • X-sec by these particles are almost the same after correcting for EM effects • Strong force does not distinguish these particles  Isospin triplet PHYS 3446, Fall 2008 Andrew Brandt

6. Isospin Quantum Number • This QN is found to be conserved in strong interactions • But not conserved in EM or Weak interactions • Isospin no longer used, replaced by quark model PHYS 3446, Fall 2008 Andrew Brandt

7. Violation of Quantum Numbers • The QN we learned are conserved in strong interactions are but many of them are violated in EM or weak interactions • 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 present PHYS 3446, Fall 2008 Andrew Brandt

8. Quantum Numbers • Baryon Number • An additive and conserved quantum number, Baryon number (B) • This number is conserved in general but not absolute • Lepton Number • Quantum number assigned to leptons • Lepton numbers by species and the total lepton numbers must be conserved • Strangeness Numbers • Conserved in strong interactions • But violated in weak interactions • Isospin Quantum Numbers • Conserved in strong interactions • But violated in weak and EM interactions PHYS 3446, Fall 2008 Andrew Brandt

9. Quantum Number Conservation • 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 framework PHYS 3446, Fall 2008 Andrew Brandt

10. Symmetry • When is a quantum number conserved? • When there is an underlying symmetry in the system • When the quantum number is not affected by the changes in the physical system • Noether’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 theories PHYS 3446, Fall 2008 Andrew Brandt

11. Symmetries in Lagrangian Formalism • 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 classical cases or relativistic cases and quantum mechanical systems PHYS 3446, Fall 2008 Andrew Brandt

12. Symmetries in Lagrangian Formalism? • 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 then PHYS 3446, Fall 2008 Andrew Brandt

13. Symmetries in Lagrangian Formalism • 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 translation PHYS 3446, Fall 2008 Andrew Brandt

14. Symmetries in Lagrangian Formalism? • This means that 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 PHYS 3446, Fall 2008 Andrew Brandt

15. Symmetries in Lagrangian Formalism? • 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 well PHYS 3446, Fall 2008 Andrew Brandt

16. Symmetries in Lagrangian Formalism • 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 thus PHYS 3446, Fall 2008 Andrew Brandt

17. Translational Symmetries & Conserved Quantities • The translational symmetries of a physical system gives 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 conservation PHYS 3446, Fall 2008 Andrew Brandt

18. Symmetries in Quantum Mechanics • In quantum mechanics, an observable physical quantity corresponds to the expectation value of the corresponding operator on a given quantum state • The expectation value is given as a product of wave function vectors about the physical quantity (operator) • The wave function ( )is the probability distribution function of a quantum state at any given space-time coordinates • The observable is invariant or conserved if the operator Q commutes with Hamiltonian PHYS 3446, Fall 2008 Andrew Brandt

19. Types of Symmetry • All symmetry transformations of the theory can be divided in two categories • Continuous symmetry: Symmetry under continuous transformation • Spatial translation • Time translation • Rotation • Discrete symmetry: Symmetry under discrete transformation • Transformation in discrete quantum mechanical system PHYS 3446, Fall 2008 Andrew Brandt

20. Local Symmetries • All continuous symmetries can be classified as • Global symmetry: Parameters of transformation are constant • Transformation is the same throughout the entire space-time points • All continuous transformations we discussed so far are global symmetries • Local symmetry: Parameters of transformation depend on space-time coordinates • The magnitude of transformation is different from point to point • How do we preserve a symmetry in this situation? • Real forces must be introduced!! PHYS 3446, Fall 2008 Andrew Brandt

21. Local Symmetries • Let’s consider the time-independent Schrödinger Eq. • If is a solution, should also be a solution for a constant a • Any quantum mechanical wave functions can be defined up to a constant phase • A transformation involving a constant phase is a symmetry of any quantum mechanical system • Conserves probability density  Conservation of electrical charge is associated w/ this kind of global transformation. PHYS 3446, Fall 2008 Andrew Brandt

22. Local Symmetries • Let’s consider a local phase transformation • How can we make this transformation local? • Multiplying a phase parameter with an explicit dependence on the position vector • This does not mean that we are transforming positions but just that the phase is dependent on the position • Thus under local transformation, we obtain PHYS 3446, Fall 2008 Andrew Brandt

23. Local Symmetries • Thus, Schrödinger equation • is not invariant (or a symmetry) under local phase transformation • What does this mean? • Energy conservation is no longer valid. • What can we do to conserve the energy? • Consider an arbitrary modification of a gradient operator PHYS 3446, Fall 2008 Andrew Brandt

24. Additional Field Local Symmetries • Now requiring the vector potential to change under transformation as • Makes • And the local symmetry of the modified Schrödinger equation is preserved under the transformation PHYS 3446, Fall 2008 Andrew Brandt

25. Local Symmetries • The invariance under a local phase transformation requires the introduction of additional fields • These fields are called gauge fields • Leads to the introduction of a definite physical force • The potential can be interpreted as the EM vector potential • The symmetry group associated with the single parameter phase transformation in the previous slides is called Abelian or commuting symmetry and is called U(1) gauge group Electromagnetic force group PHYS 3446, Fall 2008 Andrew Brandt