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 MechanicsGiven force F, determine position of an object at anytime:

F ~ d2r/dt2

Proportionality constant = m, property of the object.

Integration of eq. (1) gives r=r(t) ---the solution: prediction of the motion.

In Cartesian coordinates:

Fx = md2x/dt2

Fy = md2y/dt2

Fz = md2z/dt2

Examples of position or velocity dependent forces:

•Gravitational force: F = Gm1m2/r2

(=mg, on Earth surface)

•Electrostatic force: F = kq1q2/r2

•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=rsincos; y=rsinsin; 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

rr and

.,,and,,,

zyxzyx

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/r2, where k isa constant.

b) What difficulty you will encounter if youwould like to derive the Newton equations of motion in polar coordinates?

yF

x0

yF

x0

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: (q1,q2,q3….)

Define Lagrangian:

)(),(

),(

jqVT

EnergyPotentialEnergyKineticLL

.

.

jj

jj

qq

qq

Equations of motion becomes:

0

jj

q

L

q

L

dt

d

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

x0

r

V(r)=k/r

Hamiltonian

•Definition of generalized momenta:

j

j

q

Lp

•If L L(qj), then pj=constant, “cyclic” in qj.

•Definition of Hamiltonian:

)(),(),( jjjjj qVqqTpqHH

What are the differences between L and H ?

0

j

j q

Lp

dt

d

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-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