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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 ~ 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=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


r and r
from a Cartesian coordinate system
to other coordinate systems.
•But it is easier to transform scalar such as



  
x, y, z, and x, y, z .
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 is
a constant.
b) What difficulty you will encounter if you
would like to derive the Newton equations
of motion in polar coordinates?
y
y
F
F
r
0

x
0
x
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:
.
L  L(qj , qj )  Kinetic Energy  Potential Energy
.
 T (qj , qj )  V (qj )
Equations of motion
becomes:
d L
L

0

dt  q
q j
j
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
V(r)=k/r
r

0
x
Hamiltonian
•Definition of generalized momenta:
pj 
L

qj
d
L

pj 
0
dt
q j
•If L L(qj), then pj=constant, “cyclic” in qj.
•Definition of Hamiltonian:

H  H (q j , p j )  T (q j , q j )  V (q j )
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
d
e-
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