Physics 334 Modern Physics

1. Physics 334Modern Physics Credits: Material for this PowerPoint was adopted from Rick Trebino’s lectures from Georgia Tech which were based on the textbook “Modern Physics” by Thornton and Rex. Many of the images have been used also from “Modern Physics” by Tipler and Llewellyn, others from a variety of sources (PowerPoint clip art, Wikipedia encyclopedia etc), and contributions are noted wherever possible in the PowerPoint file. The PDF handouts are intended for my Modern Physics class, as a study aid only.

2. CHAPTER 2Special Theory of Relativity II • Relationship Between Energy and Momentum • Compton Scattering • Discovery of the Positron Albert Einstein (1879-1955) If you are out to describe the truth, leave elegance to the tailor. The most incomprehen-sible thing about the world is that it is at all comprehensible. - Albert Einstein

3. y z x v v S’ S’ Relativistic Momentum Frank is at rest in S and throws a ball of mass m in the -y-direction. Mary (in the moving system) similarly throws a ball in system S’ that’s moving in the xdirection with velocity vwith respect to system S. S • Because physicists believe that the conservation of momentum is fundamental, we begin by considering collisions without external forces: u dP/dt = Fext = 0

4. Relativistic Momentum • What does Frank measure for the change in momentum of his own ball? What does Frank measure for the change in momentum of Mary’s ball?

5. S v S’ Relativistic Momentum • The conservation of linear momentum requires the total change in momentum of the collision, ΔpF+ ΔpM, to be zero. The addition of these y-momenta is clearly not zero. • Linear momentum is not conserved if we use the conventions for momentum from classical physics—even if we use the velocity transformation equations from special relativity. • There is no problem with the x direction, but there is a problem with the y direction the ball is thrown in each system.

6. S v S’ Relativistic Momentum • The failure of the conservation of momentum in the collision cannot be due to the velocities, because we used the Lorentz transformation to find the y components. It must have something to do with mass! We can reconcile this discrepancy by using where: Important: note that we’re using g in this formula, but the v in g is really the velocity of the object, not necessarily that of its frame. Exercise 4-15: Using the above modification show that the initial y component of Frank ball cancels out with that of Mary’s

7. Relativistic momentum

8. At high velocity, does the mass increase or just the momentum? • Some physicists like to refer to the mass as the rest massm0 and call the term m = gm0 the relativistic mass. In this manner the classical form of momentum, m, is retained. The mass is then imagined to increase at high speeds. • Most physicists prefer to keep the concept of mass as an invariant, intrinsic property of an object. We adopt this latter approach and will use the term mass exclusively to mean rest mass. Although we may use the terms mass and rest mass synonymously, we will not use the term relativistic mass.

9. Relativistic Energy • We must now redefine the concepts of work and energy. • So we modify Newton’s second law to include our new definition of linear momentum, and force becomes: where, again, we’re using g in this formula, but it’s really the velocity of the object, not necessarily that of its frame.

10. Relativistic Energy Exercise 4-16: Show that the kinetic energy Ek and hence the work done by a net force in accelerating a particle from rest to some velocity u is; Exercise 3: Show that for u=v<<c the relativistic kinetic energy and non relativistic kinetic energy are indistinguishable.

11. Relativistic Energy • Even an infinite amount of energy is not enough to achieve c. • For u<<c, the relativistic and non relativistic kinetic energies are almost identical. Electrons accelerated to high energies in an electric field

12. Total Energy and Rest Energy • Manipulate the energy equation: The term mc2 is called theRest Energy The sum of the kinetic and rest energies is thetotal energyof the particle E and is given by: Becomes the famous

13. Invariant Mass Square the momentum equation, p = g m u, and multiply by c2: Substituting for u2 using b 2 = u2 / c2 : But And:

14. Invariant Mass The first term on the right-hand side is just E2, and the second is E02: Rearranging, we obtain a relation between energy and momentum. or: This equation relates the total energy of a particle with its momentum. The quantities (E2 – p2c2) and m are invariant quantities. Note that when a particle’s velocity is zero and it has no momentum, this equation correctly gives E0 as the particle’s total energy.

15. Massless particles Exercise 4-17 : Starting from; Show that any massless particle must travel at the speed of light.

16. Compton Effect • When a photon enters matter, it can interact with one of the electrons. The laws of conservation of energy and momentum apply, as in any elastic collision between two particles. The momentum of a particle moving at the speed of light is: The electron energy is: This yields the change in wavelength of the scattered photon, known as the Compton effect:

17. Pair Production and Annihilation • If a photon can create an electron, it must also create a positive charge to balance charge conservation. • In 1932, C. D. Anderson observed a positively charged electron (e+) in cosmic radiation. This particle, called a positron, had been predicted to exist several years earlier by P. A. M. Dirac. • A photon’s energy can be converted entirelyinto an electron and a positron in a process called pair production: Paul Dirac (1902 - 1984)

18. Pair Production in Empty Space E- hn • Conservation of energy for pair production in empty space is: E+ The total energy for a particle is: So: This yields a lower limit on the photon energy: Momentum conservation yields: This yields an upper limit on the photon energy: A contradiction! And hence the conversion of energy and momentum for pair production in empty space is impossible!

19. Pair Production in Matter • In the presence of matter, the nucleus absorbs some energy and momentum. • The photon energy required for pair production in the presence of matter is:

20. Pair Annihilation • A positron passing through matter will likely annihilate with an electron. The electron and positron can form an atom-like configuration first, calledpositronium. • Pair annihilation in empty space produces two photons to conserve momentum. Annihilation near a nucleus can result in a single photon.

21. Pair Annihilation • Conservation of energy: • Conservation of momentum: So the two photons will have the same frequency: The two photons from positronium annihilation will move in opposite directions with an energy: