Physics 334 Modern 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.

Relationship Between Energy and Momentum Compton Scattering Discovery of the Positron Special Theory of Relativity II CHAPTER 2 Special Theory of Relativity II Albert Einstein ( ) 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

v S’S’ v S’S’ Relativistic Momentum Because physicists believe that the conservation of momentum is fundamental, we begin by considering collisions without external forces: 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 x direction with velocity v with respect to system S. dP/dt = F ext = 0 S x z y u

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?

The conservation of linear momentum requires the total change in momentum of the collision, Δp F + Δp M, 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. Relativistic Momentum v S’S’ S

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! Relativistic Momentum v S’S’ S where: Important: note that we’re using  in this formula, but the v in  is really the velocity of the object, not necessarily that of its frame. We can reconcile this discrepancy by using Exercise 4-15: Using the above modification show that the initial y component of Frank ball cancels out with that of Mary’s

Relativistic momentum

Some physicists like to refer to the mass as the rest mass m 0 and call the term m =  m 0 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. At high velocity, does the mass increase or just the momentum?

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  in this formula, but it’s really the velocity of the object, not necessarily that of its frame.

Relativistic Energy Exercise 4-16: Show that the kinetic energy E k 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.

Relativistic Energy 1.Even an infinite amount of energy is not enough to achieve c. 2.For u<<c, the relativistic and non relativistic kinetic energies are almost identical. Electrons accelerated to high energies in an electric field

Total Energy and Rest Energy Manipulate the energy equation: The sum of the kinetic and rest energies is the total energy of the particle E and is given by: The term mc 2 is called the Rest Energy Becomes the famous

Square the momentum equation, p =  m u, and multiply by c 2 : Invariant Mass Substituting for u 2 using  2 = u 2 / c 2 : But And:

The first term on the right-hand side is just E 2, and the second is E 0 2 : Invariant Mass This equation relates the total energy of a particle with its momentum. The quantities ( E 2 – p 2 c 2 ) and m are invariant quantities. Note that when a particle’s velocity is zero and it has no momentum, this equation correctly gives E 0 as the particle’s total energy. or: Rearranging, we obtain a relation between energy and momentum.

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

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: This yields the change in wavelength of the scattered photon, known as the Compton effect: The electron energy is:

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 entirely into an electron and a positron in a process called pair production: Paul Dirac ( )

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

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:

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

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: