PHY 711 Classical Mechanics and Mathematical Methods 11

PHY 711 Classical Mechanics and Mathematical Methods 11 -11: 50 AM MWF Olin 107 Plan for Lecture 2: 1. Brief comment on quiz 2. Particle interactions 3. Notion of center of mass reference fame 4. Introduction to scattering theory 9/02/2016 PHY 711 Fall 2016 -- Lecture 2 1

9/02/2016 PHY 711 Fall 2016 -- Lecture 2 2

9/02/2016 PHY 711 Fall 2016 -- Lecture 2 3

Schedule additional office hours by email: natalie@wfu. edu 9/02/2016 PHY 711 Fall 2016 -- Lecture 2 4

Comment on quiz questions 1. 3. 9/02/2016 PHY 711 Fall 2016 -- Lecture 2 5

Scattering theory: 9/02/2016 detector PHY 711 Fall 2016 -- Lecture 2 6

b Figure from Marion & Thorton, Classical Dynamics 9/02/2016 PHY 711 Fall 2016 -- Lecture 2 7

Note: Notion of cross section is common to many areas of physics including classical mechanics, quantum mechanics, optics, etc. Only in the classical mechanics can we calculate it using geometric considerations b Figure from Marion & Thorton, Classical Dynamics Note: We are assuming that the process is isotropic in f 9/02/2016 PHY 711 Fall 2016 -- Lecture 2 8

Simple example – collision of hard spheres Microscopic view: 9/02/2016 PHY 711 Fall 2016 -- Lecture 2 9

Simple example – collision of hard spheres -- continued Hard sphere: 9/02/2016 PHY 711 Fall 2016 -- Lecture 2 10

Relationship of scattering cross-section to particle interactions -Classical mechanics of a conservative 2 -particle system. 1 2 9/02/2016 PHY 711 Fall 2016 -- Lecture 2 11

Typical two-particle interactions – 9/02/2016 PHY 711 Fall 2016 -- Lecture 2 12

Relationship between center of mass and laboratory frames of reference 9/02/2016 PHY 711 Fall 2016 -- Lecture 2 13

Classical mechanics of a conservative 2 -particle system -continued 9/02/2016 PHY 711 Fall 2016 -- Lecture 2 14

Note: The following analysis will be carried out in the center of mass frame of reference. In laboratory frame: V 1 In center-of-mass frame: m v 1 m 1 r origin v. CM mtarget Also note: We are assuming that the interaction between particle and target V(r) conserves energy and angular momentum. 9/02/2016 PHY 711 Fall 2016 -- Lecture 2 15

For a continuous potential interaction in center of mass reference frame: Need to relate these parameters to differential cross section to be discussed next Wednesday V(r) Erel rmin 9/02/2016 PHY 711 Fall 2016 -- Lecture 2 16

The results above were derived in the center of mass reference frame; relationship between normal laboratory reference and center of mass: Laboratory reference frame: Before m 1 u 1 After m 2 u 2=0 v 1 m 2 V 1 y v 2 z Center of mass reference frame: Before After m 1 U 1 9/02/2016 U 2 q q V 2 PHY 711 Fall 2016 -- Lecture 2 17

Relationship between center of mass and laboratory frames of reference -- continued 9/02/2016 PHY 711 Fall 2016 -- Lecture 2 18

Relationship between center of mass and laboratory frames of reference VCM V 1 q y v 1 For elastic scattering 9/02/2016 PHY 711 Fall 2016 -- Lecture 2 19

Digression – elastic scattering Also note: 9/02/2016 PHY 711 Fall 2016 -- Lecture 2 20

Relationship between center of mass and laboratory frames of reference – continued (elastic scattering) VCM V 1 9/02/2016 qy PHY 711 Fall 2016 -- Lecture 2 v 1 21

Differential cross sections in different reference frames 9/02/2016 PHY 711 Fall 2016 -- Lecture 2 22

Differential cross sections in different reference frames – continued: 9/02/2016 PHY 711 Fall 2016 -- Lecture 2 23

Example: 9/02/2016 suppose m 1 = m 2 PHY 711 Fall 2016 -- Lecture 2 24