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crop.unl/claes/PHYS926Lectures.html

The tentative schedule of lectures for the semester with links to posted electronic versions of my notes can be found at: . http://crop.unl.edu/claes/PHYS926Lectures.html. 1936 Millikan ’s group shows at earth’s surface cosmic ray showers

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crop.unl/claes/PHYS926Lectures.html

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  1. The tentative schedule of lectures for the semester with links to posted electronic versions of my notes can be found at: http://crop.unl.edu/claes/PHYS926Lectures.html

  2. 1936 Millikan’s group shows at earth’s surface cosmic ray showers are dominated by electrons, gammas, and X-particles capable of penetrating deep underground (to lake bottom and deep tunnel experiments) and yielding isolated single cloud chamber tracks

  3. 1937 Street and Stevenson • 1938 Andersonand Neddermeyer • determine X-particles • are charged • have 206× the electron’s mass • decay to electrons with • a mean lifetime of 2msec 0.000002 sec

  4. Schrödinger’s Equation Based on the constant (conserved) value of the Hamiltonian expression total energy  sum of KE + PE with the replacement of variables by “operators” As enormously powerful and successful as this equation is, what are its flaws? Its limitations?

  5. We could attempt a RELATIVISTIC FORM of Schrödinger: What is the relativistic expression for energy? relativistic energy-momentum relation As you’ll appreciate LATER this simple form (devoid of spin factors) describes spin-less (scalar) bosons For m=0 this yields the homogeneous differential equation: Which you solved in E&M to find that wave equations for these fields were possible (electromagnetic radiation).

  6. (1935)Hideki Yukawa saw the inhomogeneous equation as possibly descriptive of a scalar particle mediating SHORT-RANGE forces like the “strong” nuclear force between nucleons (ineffective much beyond the typical 10-15 meter extent of a nucleus For a static potential drop and assuming a spherically symmetric potential, can cast this equation in the form: with a solution (you will verify for homework): h mc whereR=

  7. h mc Let’s compare: whereR= to the potential of electromagnetic fields: with e-r/R=1 its like R orm = 0! For a range something like 10-15 m Yukawa hypothesized the existence of a new (spinless) boson with mc2 ~ 100+ MeV. In 1947 the spin 0 pion was identified with a mass ~140 MeV/c2

  8. 1947 Lattes, Muirhead, Occhialini and Powell observe pion decay Cecil Powell (1947) Bristol University 

  9. C.F.Powell, P.H. Fowler, D.H.Perkins Nature 159, 694 (1947) Nature 163, 82 (1949)

  10. Quantum Field Theory Not only is energy & momentum QUANTIZED (energy levels/orbitals) but like photons are quanta of electromagnetic energy, all particle states are the physical manifestation of quantum mechanical wave functions (fields). Not only does each atomic electron exist trapped within quantized energy levels or spin states, but its mass, its physical existence, is a quantum state of a matter field. the quanta of the em potential  virtual photons as opposed to observable photons These are not physical photons in orbitals about the electron. They are continuously and spontaneously being emitted/reabsorbed. e-

  11. The Boson Propagator What is the momentum spectrum of Yukawa’s massive (spin 0) relativistic boson? Remember it was proposed in analogy to the E&M wave functions of a photon. What distribution of momentum (available to transfer) does a quantum wave packet of this potential field carry? qr = qrcos dV = r2 d sin d dr Integrating the angular part: 2 The more massive the mediating boson, the smaller this distribution…

  12. Consistently ~600 microns (0.6 mm) 

  13. pdg.lbl.gov/pdgmail

  14. BraKet notation We generalize the definitions of vectors and inner products ("dot" products) to extend the formalism to functions (like QM wavefunctions) and differential operators. v = vxx + vyy + vzzSnvnn then the inner product is denoted by vu = ^ ^ ^ ^ Snvnun ^ ^ Remember: nm = nm sometimes represented by row and column matrices: [vxvyvz] ux uy= [ ] uz vxux + vyuy + vzuz

  15. We most often think of "vectors" in ordinary 3-dim space, but can immediately and easily generalize to COMPLEX numbers: vu =Sn [vxvyvz] ux uy = [ ] uz Snvn*un * * * vx*ux + vy*uy + vz*uz transpose column into row and take complex conjugate and by the requirement < v| u> = < v| u>* we guarantee that the “dot product” is real

  16. Every “vector” is a ket : |v1> |v2> including the unit “basis” vectors. We write: | v > = Sn| > and the scalar product by the symbol < | > and the orthonormal condition on basis vectors can be stated as < | > = d Now if we write |v1> = SC1n|n>and |v2> = SC2n|n> then “we know”: < v2 | v1 > = SnC2n*C1n = S because of orthonormality < v2 | | v1 > = Sm “bra” Cnn v u m n mn Sn,mC2m* C1n<m|n> SmC2m* <m|SmC1n|n>

  17. C1n So what should this give? < n | v1 > = ?? So if we write |v > = SCn|n> = Sn|n> = Sn = {Sn } = <n|v> |n><n|v> 1 |v> |v> |n><n| |v> Remember: < m | n > gives a single element 1 x 1 matrix but: | m > < n | gives a???

  18. Sn|n><n| In the case of ordinary 3-dim vectors, this is a sum over the products: [ 1 0 0 ] [ 0 1 0 ] [ 0 0 1 ] 1 0 0 0 1 0 0 0 1 + + 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 = + + 1 0 0 0 1 0 0 0 1 =

  19. Two important BASIC CONCEPTS • The “coupling” of a fermion • (fundamental constituent of matter) • to a vector boson • (the carrier or intermediary of interactions) e- • Recognized symmetries • are intimately related to CONSERVED quantities in nature • which fix the QUANTUM numbers describing quantum states • and help us characterize the basic, fundamental interactions • between particles

  20. Should the selected orientation of the x-axis matter? As far as the form of the equations of motion? (all derivable from a Lagrangian) As far as the predictions those equations make? Any calculable quantities/outcpome/results? Should the selected position of the coordinate origin matter? If it “doesn’t matter” then we have a symmetry: the x-axis can be rotated through any direction of 3-dimensional space or slid around to any arbitrary location and the basic form of the equations…and, more importantly, all the predictions of those equations are unaffected.

  21. If a coordinate axis’ orientation or origin’s exact location “doesn’t matter” then it shouldn’t appear explicitly in the Lagrangian! EXAMPLE:TRANSLATION Moving every position (vector) in space by a fixed a (equivalent to “dropping the origin back” –a) –a original description of position r r' or new description of position

  22. For a system of particles: function of separation acted on only by CENTAL FORCES: no forces external to the system

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