Section I: (Chapter 1) Review of Classical Mechanics Newtonian mechanics Coordinate transformations Lagrangian approach Hamiltonian with generalized momenta

Session 1. (Chapter 1) Review of Classical Mechanics Newtonian Mechanics Given force F, determine position of an object at anytime: F ~ d 2 r/dt 2 Proportionality constant = m, property of the object. Integration of eq. (1) gives r=r(t) ---the solution: prediction of the motion. In Cartesian coordinates: F x = md 2 x/dt 2 F y = md 2 y/dt 2 F z = md 2 z/dt 2

Examples of position or velocity dependent forces: Gravitational force: F = Gm 1 m 2 /r 2 (=mg, on Earth surface) Electrostatic force: F = kq 1 q 2 /r 2 Charge moving in Magnetic field: F = qvxB Other forces (not “fundamental”) Harmonic force: F = -kr

Coordinate transformations Polar coordinates: x=rsin  ; y=rcos  Spherical coordinates: x=rsin  cos  ; y=rsin  sin  ; z=rcos  Cylindrical coordinates: x=  cos  ; y=  sin  ; z=z It is harder to do a vector transformation such as from a Cartesian coordinate system to other coordinate systems. But it is easier to transform scalar such as

Inclass I-1. a) Write down Newton equation of motion in Cartesian coordinates for an object moving under the influence of a two-dimensional central force of the form F=k/r 2, where k is a constant. b) What difficulty you will encounter if you would like to derive the Newton equations of motion in polar coordinates? y F x 0 y F x 0 r 

Lagrangian approach: Instead of force, one uses potential to construct equations of motion---Much easier. Also, potential is more fundamental: sometimes there is no force in a system but still has a potential that can affect the motion. Use generalized coordinates: (x,y,z), (r, ,  ), …..In general: (q 1,q 2,q 3 ….) Define Lagrangian: Equations of motion becomes:

Inclass I-2. Write down the Lagrangian in polar coordinates for an object moving under the influence of a two-dimensional central potential of the form V(r)=k/r, where k is a constant. Derive the equations of motion using Lagrangian approach. Compare this result with that obtained in Inclass I-1. y x 0 r  V(r)=k/r

Hamiltonian Definition of generalized momenta: If L  L(q j ), then p j =constant, “cyclic” in q j. Definition of Hamiltonian: What are the differences between L and H ?

Inclass I-3. An object is moving under the influence of a two-dimensional central potential of the form V(r)=k/r, where k is a constant. Determine the Hamiltonian in a) the Cartesian coordinate system; b) in polar coordinate system. (Hint: determine the generalized momenta first before you determine the Hamiltonian.)

(Inclass) I-4. An electron is placed in between two electrostatic plates separated by d. The potential difference between the plates is  o. a) Derive the equations of motion using Lagrangian method (3-dimensional motion) in Cartesian coordinate system. b) Determine the Hamiltonian using Cartesian coordinate system. c) Determine the Hamiltonian using cylindrical coordinate system. z e-e- d

Introduction to Quantum Mechanics Homework 1: Due:Jan 20, 12.00pm (Will not accept late homework) Inclass I-1 to I-4. Problems: 1.5, 1.7, 1.11, 1.12