Tutorial 6
PH 101: Physics I (2014)
1. A uniform thin rod of length L and mass M is pivoted at one end. The pivot is attached to
the top of a car accelerating at rate A, as shown in Fig.1. , (a) What is the equilibrium value
of the angle between the rod a

The Quantum Harmonic Oscillator
Consider the familiar problem of a mass tied to a spring. In classical physics we
know that the position and momentum of the particle at time t is given by,
x(t) = A cos(t + )
and momentum is,
p(t) = mx(t)
= Am sin(t + )
is

Prof. Girish Setlur
Department of Physics
Room No. 302
Email: [email protected]
Tel: 2715
(Lecture hall 1)
Dr. Ashwini Kumar Sharma
Department of Physics
Room No. 107
Email: [email protected]
Tel: 2724
(Lecture hall 2)
Lecture 14
Rigid Body Motio

Lecture-XIII
Noninertial systems
Apparent Force in Rotating Coordinates
The force in the rotating system is
where
The first term is called the Coriolis force, a velocity dependent force and the second
term, radially outward from the axis of rotation, is c

Lecture-XI
Angular momentum and Fixed axis
rotation
General rigid body motion
The general motion of a rigid body of mass m consists of a translation of the center of
mass with velocity
and a rotation about the center of mass with all elements of the
rigid

Lecture-XII
Noninertial systems
Accelerating frame
Newton's second law F = ma holds true only in inertial coordinate systems.
However, there are many noninertial (that is, accelerating) frames that one needs to
consider, such as elevators, merry-go-rounds

Lecture-IX
Angular momentum and Fixed axis
rotation
How to study rotational motion of an object?
We need to develop techniques to handle the rotational motion of solid bodies.
For example, consider the common Yo-Yo running up and down its
string as the sp

Lecture-VIII
Work and Energy
The Physical Meaning of the Gradient
To see the relation between U and contour lines of constant potential energy,
consider the change in U due to a displacement dr along a contour.
2 > 1
since on a contour line, U is constant

Lecture-VII
Work and Energy
The problem:
As first glance there seems to be no problem in finding the motion of a particle if we
know the force; starting with Newton's second law, we obtain the acceleration, and by
integrating we can find first the velocit

Lecture-VI
Momentum and variable mass
Momentum and the Flow of Mass
External force causes the momentum of a system to change as
One needs to apply this generalized version of Newtons Second Law to systems in
which mass flows between constituent objects.
A

Lecture-V
Momentum
Generalization of Newtons laws:
Two major assumptions:
1. Point mass
What about large bodies with elaborate structure?
Such as landing of a spacecraft on the moon.
2. Constant mass
What about bodies with variable mass?
Such as rain drop

Lecture-IV
Position, velocity and time
dependent forces
Solving differential equations:
Consider the force as a function of time, position, or velocity,
and solve the
d 2x
m 2 = F ( x, t , v )
dt
differential equation to find the position, x(t), as a func

Lecture-III
Newtons laws of motion
A recapitulation
Newtons first law of motion:
Every body continues in its state of rest, or of uniform motion in a straight line,
unless it is compelled to change that state by applying external forces upon it.
What does

Lecture-2
Kinematics in polar coordinates
Velocity and Acceleration in polar coordinate:
r
The position vector r in polar coordinate is given by :
r
r = rr
j
j
And the unit vectors are: r = cos i + sin & = sin i + cos
Since the unit vectors are not const

Lecture-1
Newtonian Mechanics
Domain of Newtonian Mechanics
Particle: A point mass object. A body having some mass has negligible dimension in
comparison to other length scales present in the system.
Study of motion of a particle: Kinematics and Kinetics

Lecture-VIII
Work and Energy
The problem:
As first glance there seems to be no problem in finding the motion of a particle if we
know the force; starting with Newton's second law, we obtain the acceleration, and by
integrating we can find first the veloci

Lecture-XVIII
Rigid body motion
Equation of motion
In the body frame: L = I
dL
dL
= + L,
dt fix dt body
dL
= + L
dt body
Therefore,
If the body frame coincides with the principal axes of the system, the moment of
inertia tensor will be gi

Lecture 15
Rigid Body Motion II
29.09.2015
Velocity of a small piece of any rigid body
Rate of change of any vector of the rigid body
It tells us that a vector A(t) associated with a rigid body changes
with time simply on account of the spinning motion of

Lecture 16
Rigid Body Motion III
05.10.2015
The Gyroscope
One of the most useful application of spinning rigid bodies is the gyroscope.
The gyroscope is a heavy, rapidly spinning disk or wheel whose axis of
rotation is free to change. Here is a picture of

The story so far .
Einstein and de Broglie together wrote down the mathematical
formulae that explained the meaning of wave particle duality
=
=
T=
Schrodinger used this to write down the expression for the
probability amplitude of a matter wave with

Wave Mechanics
A Mathematical Description
of Matter Waves
Schrodingers Equation
In January 1926, Schrdinger
published in Annalen der
Physik
the
paper
"Quantisierung
als
Eigenwertproblem"
[tr.
Quantization as an Eigenvalue
Problem] on wave mechanics
and pr

PH204: Quantum Mechanics (3-1-0-8)
Review of wave mechanics: Youngs double slit, de Broglie relation, wave packets, Schrodinger
equation; Observable, Eigen values and Eigen functions; Simple applications: particle in a box;
potential well; potential barri

HYDROGEN ATOM IN QUANTUM MECHANICS
An electron bound to the nucleus is described by an elliptical or circular trajectory in classical physics. In quantum
physics we know that there is no trajectory rather a probability amplitude that describes the likelih

Generator of translations and rotations
1
Generator of Translations
Consider a function (x). I want to find an operator that takes this and maps it to
(x + a). Thus I want to find Ta such that for any (x),
Ta (x) = (x + a)
Imagine I expand the right hand

Heisenberg Uncertainty Relation
In the following we provide a general proof of Heisenbergs uncertainty relation
h
2
This is a special case of the uncertainty relation of any two self adjoint operators A and B so that,
x
A
p
< [A; B] >
2i
B
(1)
where the

Basic Operator Theory
Here we want to understand the properties of operators such as momentum operp2
h2
ator px = ih x
or in 3D p = i
h, kinetic energy operator K = 2m
= 2m
2 ,
angular momentum operator L = r p = ih r e.t.c. Specifically, we want to
relat

ANGULAR MOMENTUM IN QUANTUM MECHANICS
We saw that L = r p is a self adjoint operator. So it is perfectly acceptable to choose this to represent angular
momentum in quantum mechanics as it were the case in classical physics. In classical physics it is poss

Fourier Analysis
The main goal of Fourier analysis is to express any complicated function f (x) as some combination of the extremely
simple function eikx . This function eikx is infinitely many times differentiable and integrable and each time such an
ope

Lecture 23
Quantum Mechanics III
26.10.2015
Announcement
Quiz II
Date
: 29.10.2015
Duration : 45 minutes
Syllabus : Lectures 14 to 19
Mode
: Same as that of Midsem exam.
Write only the final answer in the question paper.
Only question paper having final a

Lecture 22
Quantum Mechanics II
23.10.2015
Schrodingers Equation
In January 1926, Schrdinger
published in Annalen der
Physik
the
paper
"Quantisierung
als
Eigenwertproblem"
[tr.
Quantization as an Eigenvalue
Problem] on wave mechanics
and presented what is