Physics 11513, Quiz #1, Fri. Oct. 23, 1998
NAME: cfw_4/2 SW
You have /2 hour. Please show all work. Total ssible score is 24
Here is some information that goal be useful:
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Physics 216
Solution Set 5
Spring 2012
1. Consider a two-level system with E1 < E2 . There is a time-dependent potential that
connects the two levels as follows:
V11 = V22 = 0,
V12 = eit ,
V21 = eit
( real) ,
(1)
where Vij = hi|V |ji. At time t = 0, it is
Physics 216
Final Exam Solutions
Spring 2012
1. Consider a positively-charged spin-1/2 particle in an external magnetic field, governed by
the Hamiltonian:
~ S
~,
H = H0 I B
~ is the vector of spin-1/2 spin matrices, and
where I is the identity operator i
Physics 216
Midterm Exam
Spring 2012
This is a take home exam. You may refer to the textbooks by Shankar and Baym, the
class handouts and any third quantum mechanics textbook of your choosing. (If you do
consult a third text, please indicate which one you
Solutions for R. Shankar, Principles of Quantum Mechanics, 2nd ed.
14.4.3.
Plugging in the rotating ket, |r (t)i = eitSz /h |(t)i, to find the modified Hamiltonian
gives:
d
d
|r (t)i = i
heitSz /h |(t)i + Sz eitSz /h |(t)i .
dt
dt
We can identify the orig
Solutions for R. Shankar, Principles of Quantum Mechanics, 2nd ed.
17.2.7.
The sought after operator must satisfy:
H 1 = [, H 0 ]
where H 1 = qf X and H 0 =
operator suggests itself.
h2
2
2m
+
m 2 X 2
.
2
Since H 1 X and H 0 X 2 , a differential
Trying =
Solutions for R. Shankar, Principles of Quantum Mechanics, 2nd ed.
14.4.1.
The time evolution of the expectation value is found via the Ehrenfest relation (it should
be noted that [B, L] = 0 is assumed):
dL
i
i
= [H, L] = [(L B)L + L(L B)].
dt
h
h
It is e
Solutions for R. Shankar, Principles of Quantum Mechanics, 2nd ed.
14.4.4.
It was shown in 14.4.1. that for position-independent B, the expectation value of torque is
given by the decoupled expression
d hLi
= hi B
dt
where L = as usual. The expectation va
INTRODUCTION TO
TOPOLOGY
PURE AND APPLIED
.
f
F
F-
I
I
F
I
tat
41
01iPP
Cohn's Dedication:
To Ahexa and Colton
Bob's Dedication:
Ahla Famighia
Contents
o
1
2
Preface
11
Introduction
0.1 What is Topology and How is it Applied?
0.2 A Glimpse at the History
The Economic Lives of the Poor
Abhijit V. Banerjee and Esther Duflo
October 2006
Abstract
This paper uses survey data from 13 countries to
document the economic lives of the poor (those living
on less than $2 dollar per day per capita at purchasing
power
260
Chapter 7 Lagranges Equations
1
i
i
but since L in (7 6]) is cl -
_ ' early Independent 0 n - .
generalized momentum 33/8517 is constant f (12 3 JUSt tells us that the
Mq'g +1118: + 4100501) = const (7 6%
. .l
a result you will recognize as conser
80
Mechanics, PHY 101
Date: 14 October 2016
Additional problems, Set 3
m
m
1. A ball is thrown vertically upwards. It rises up, falls down, hits the floor, rises back again
(to a smaller height), falls again, hits the floor, and so on until the ball comes
Mechanics, PHY 101
Date: 17 October 2016
Additional problems, Set 3
1. A ball is thrown vertically upwards. It rises up, falls down, hits the floor, rises back again
(to a smaller height), falls again, hits the floor, and so on until the ball comes to a r
secondn vector
n has the
0 s, what
he angle
n of the
a radar decoy (Fig. 4-33). The horizontal distance between the release point and the point where
PHY
theMechanics,
decoy strikes
the101
ground is d !
d
700 m. (a) How long is the decoy in
the air? (b) H
Mechanics, PHY 101
Date: 14 October 2016
Additional problems, Set 2
1. A boat is traveling upstream in the positive direction of an x axis at 14 km/h with respect
to the water of a river. The water is flowing at 9.0 km/h with respect to the ground.
(a) Wh
Tutorial 4 (Supplement)
PHY 101
(Newtonian) Mechanics
Junaid Aftab
The question(s) in this document are meant to supplement the the questions being discussed
in all the tutorial sessions running this week. The supplementary question(s) are meant to
be mor