Chapter 1 Matrices, Vectors,
and Vector Calculus
The physical law and the physical results must be
independent of coordinate systems.
The use of vector gives us this independence, also
provides a compact method of expressing the
Chapter 7 Hamilton's Principle:
Lagrangian and Hamiltonian Dynamics
To circumvent some difficulties in using Newton's equations, alternate methods
have been developed.
All such approaches are in essence aposteriori, because we know before
Chapter 4 Nonlinear Oscillations and Chaos
The equation of motion for the damped & driven oscillator x + f ( x )+ g (x )= h ( x )
If f ( x ) or g ( x) contains powers of x or x, higher than linear, then the physical
system is nonlinear. C
Chapter 8 Central-Force Motion
8.2 Reduced Mass
Describing a system of 2 particles requires the
specification of 6 quantities; the 3 components of
each of the 2 vectors r1 & r2 for the particles, or the
3 components of the center-of-mass vector
R and the
Chapter 3 Oscillations
consider the restoring force F is a function only of the displacement: F=F(x)
and assume F(x) possesses continuous derivatives of all orders so that the
function can be expanded in a Taylor series:
x d F
Chapter 5 Gravitation
Newton's law of gravity:
Chapter 2 Newtonian Mechanics - Single Particle
Physicallaws must be based on experimental fact.
The postulate based on the experiments assumes the status of a physical law.
If some experiments disagree with the predictions of the law, the theory must
Chapter 6 Some Methods in the Calculus of Variations
Many problems in Newtonian mechanics are easily analyzed by means of
Lagrange's equation and
An example of the use of
the theory of variations is
Prof. Yen-Chieh Huang
Dept of Electrical Engineering
National Tsing-Hua University
EE Photonics I Autumn 2000/01
Oct. 18, 2000/10/18
EE5140 Photonics I
due in class, Tuesday Oct.
1. Prove Snells law by applying Fermats principle. (10%)
Pathlength = n1d1 sec 1 + n2 d 2 sec 2 (1)
Pathlength is a function of 1 and 2 , which are related by
d1 tan 1 + d 2 tan 2 = d .(2)
The pathlength is minimized when
( / 1 ) [ n1d1 sec 1 + n2 d
1. Give three ways for defining the polarization state.
(1) the polarization ellipse
(2) the Poincare sphere
(3) the Stokes vector
(4) the Jones vector
2. What is Poincare sphere?
Ans: The Poincare sphere is a geometrical construct in whi
1. Derive the intensity distribution on a plane for the interference between two oblique plane
U1 = I 0 e j k1 i r = I 0 e jkz
U 2 = I 0 e j k2 i r = I 0 e
j( kx x+ kz z )
= I 0 e j ( k sin x + k cos z )
= 2 1 = k sin x
at z=0 pl
1. Describe the physical meaning of optical axis for anisotropical
Ans: , k (),
na = nb ,(optical axis),
2. What is double refraction?
Ans: In anisotropic material, each incident wave has two refracted waves with two
1. Interpret the physical meaning of Poynting vector.
Ans: power flow per unit area:
2. Write down the four Maxwell equations in the complete form. Derive the
integral form of four Maxwell equations, and interpret their physical
1. Prove the Fourier transform of
( t m )
m = s
= [ ( t m )]exp ( j 2 t )dt =
m = s
m = s
( t m ) exp ( j 2 t )dt
exp j 2
1 exp ( j 2 M )
= exp ( j 2 m ) =
1 exp ( j 2 )
m = s
e j e j M e j M
1 e j 2
1. Why a spherical mirror can not have a rigorous focal point?
2. Describe how to use the ray transfer matrix.
Ans: An optical system is a set of optical components placed between two transverse planes at z1 and z2 ,
referred to as t
1. Define all the parameters which can characterize a Gaussian beam.
Ans: q-parameter z0 (Rayleigh range), Gaussian
q ( z ) = z + jz0
q( z ) R( z )
W 2 ( z)
Beam width: W ( z ) = W0 1 +
1. Describe Fermats principle in detail.
2. Describe the important characteristics of paraboloidal mirror.
Ans: It has the useful property of focusing all incident rays parallel to its axis to a single point called the
3. Define pa
Stimulated and Spontaneous Emission of
Radiation in a Single Mode for N-TLMs
Michael T. Tavis and Frederick W. Cummings
The general expressions for the time development of the ensemble averages of and + are found
for N two level molecules (TLMs) interact
1 A brief reminder of linear Algebra
1.1 Linear vector space . . . . . . . . . . . . . . . . . . . .
1.2 Linear operators and their corresponding matrices . . .
1.3 Function of an operator . . . . . . .
INSTITUTE OF PHYSICS PUBLISHING
EUROPEAN JOURNAL OF PHYSICS
Eur. J. Phys. 24 (2003) 519524
Vector potential of the Coulomb gauge
A M Stewart
Research School of Physical Sciences and Engineering, The Australian National Universit
Up until now, we have dealt with general features of quantum eld theories. For example, we
have seen how to calculate scattering amplitudes starting from a general Lagrangian
resonance at DPPH
Nguyn Vn ng
L Th Qu
Set up and Procedure
Results and discussion
Electron spin resonance is based on the
absorption of high-frequency radiation by
paramagnetic substances in