1
CH4 Preparatory Concepts. Function Spaces and Hermitian Operators
Particle in a one dimensional Box
A point mass m constrained to move on an infinitely thin, frictionless
wire which is strung tightly between two impenetrable walls.This
configuration is
CH.3
POSTULATES OF QUANTUM MECHANICS.
OPERATORS, EIGENFUNCTIONS AND EIGENVALUES
Postulate #1)The state function: The state of a system at any instant of time may be represented by a
state function or a wave function. All the information regarding the stat
1
CH.2
HISTORICAL REVIEW: ERXPERIMENTS AND THEORIES
ORIGIN OF QUANTUM THEORY
At the end of the 19th century, physics consisted essentially of
Newtons Classical Mechanics: Classical mechanics was used to predict the dynamics of
material bodies, and its pre
Chapter 6
Measurements and two-level
quantum systems
In the previous chapter we saw how the quantum state, represented by the
wave function of a quantum system, evolves in space and time as governed
by the Schr
odinger equation. We discussed the interpret
Chapter 5
Schr
odinger equation
Figure 5.1: Erwin Schrodinger
In the autumn of 1925 Erwin Schrodinger was invited by Professor Peter
Debye to give a talk at a seminar in Zurich on de Broglies thesis. During
the discussion that followed, Professor Debye co
Phys 300 - Homework II
Assigned: March 19, 2013, Tuesday.
Due: March 25, 2013, Monday, at 15:30 pm.
Notes:
(i) You can discuss and solve the problems together with your friends. However, you
should write your own solution on your homework. Directly copyin
Chapter 1
Historical overview of the
developments of quantum
mechanics
1.1
Quantum Ideas Course Overview
Course synopsis: The overall purpose of this course is to introduce you all
to the core concepts that underlie quantum physics, the key experimental
a
PHYS-300: QUANTUM PHYSICS
HOMEWORK I
(Due March 25, 2014)
Important Note: You must do the home work on your own. Late
deliveries are not accepted. For a very nice account of early beginnings of
quantum mechanics, read the following paper
http:/physics.cla
Phys 300 - Homework III
Assigned: May 8, 2013, Wednesday.
Due: Will not be collected.
1. Consider a particle in 1D with the following wavefunction.
cfw_
N ex/a for x 0
(x) = N exp( |x| /a) =
N ex/a for x 0
(x)
where a is a positive number with the dimensi
PHYS-300: QUANTUM PHYSICS
HOMEWORK II
(Due May 14, 2014)
Read the following article about the origins of the quantum theory
http:/www.slac.stanford.edu/pubs/beamline/30/2/30-2-carson.pdf
Q1: The wave function of a free particle that lives in one-dimension
Chapter 2
Particle-like behavior of light
2.1
Photoelectric effect
(a) Heinrich Rudolf Hertz
(b) Albert Einstein
(c) Robert Andrews Millikan
When a metal surface is illuminated by light, electrons can be emitted
from the surface. This phenomenon is known
Chapter 7
Two-particle systems and
entanglement
7.1
Introduction
In the previous chapter we discussed the standard formalism associated with
quantum measurement, and the measurement-induced collapse of the wave
function. The difficulties that arise from t
Phys 300 - Homework I
Assigned: March 12, 2013, Tuesday.
Due: March 18, 2013, Monday, at 15:30 pm.
Notes:
(i) You can discuss and solve the problems together with your friends. However, you
should write your own solution on your homework. Directly copying
Massachusetts Institute of Technology
Spring 2006 Exam I 1. a) Physics 8.04 Vuletic
page 1 of 4
i) The ratio is proportional to the Rutherford scattering rate and ring area.
G
Figure I: Ring area =p 2 sin d 4 4 R2/4 sin sin 2 0.383 = = = 2.95 102 4 sin R
Chapter 7: Angular Momentum
The applications of quantum mechanics to three-dimensional problems begin with a
description of angular momentum. Angular momentum in quantum mechanics is a more
general concept than its classical counterpart.
Classically, ang
Chapter 8: The Schrodinger Equation in Three Dimensions and the
Hydrogen Atom
Time-independent Schrodinger equation for a nonrelativistic motion of a particle in
three dimensions is
r
r h2 r 2
r
+ V(r) ( r ) = E ( r )
H ( r ) =
2
where we use for the
Chapter 16:
The Interaction of Charged Particles with the Electromagnetic Field
The classical electromagnetic field in vacuum is described by electric and magnetic
rr
rr
field vectors E( r , t) and B( r , t) which satisfy Maxwell equations (in Gauss unit
Orbital angular momentum
Consider a particle described by the Cartesian coordinates
conjugate momenta
and their
. The classical definition of the orbital angular
momentum of such a particle about the origin is
, giving
(297)
(298)
(299)
Let us assume that
Zeeman Effect
Hydrogen Atom: 3D Spherical Coordinates
= (spherical harmonics)(radial) and probability density P
E, L2, Lz operators and resulting eigenvalues
Angular momenta: Orbital L and Spin S
Addition of angular momenta
Magnetic moments and Zeem
Outline of section 5
Angular momentum in quantum mechanics
Classical definition of angular momentum
Linear Hermitian Operators for angular momentum
Commutation relations
Physical consequences
Simultaneous eigenfunctions of total angular
momentum and th
- CHAPTER 1. -
Chapter One
Section 1.1 1.
For C "& , the slopes are negative, and hence the solutions decrease. For C "& , the slopes are positive, and hence the solutions increase. The equilibrium solution appears to be Ca>b oe "& , to which all other so
- CHAPTER 2. -
Chapter Two
Section 2.1 1a+b
a,b Based on the direction field, all solutions seem to converge to a specific increasing function. a- b The integrating factor is .a>b oe /$> , and hence Ca>b oe >$ "* /#> - /$> It follows that all solutions co
- CHAPTER 4. -
Chapter Four
Section 4.1 1. The differential equation is in standard form. Its coefficients, as well as the function 1a>b oe > , are continuous everywhere. Hence solutions are valid on the entire real line. 3. Writing the equation in standa
- CHAPTER 3. -
Chapter Three
Section 3.1 1. Let C oe /<> , so that C w oe < /<> and C ww oe < /<> . Direct substitution into the differential equation yields a<# #< $b/<> oe ! . Canceling the exponential, the characteristic equation is <# #< $ oe ! The ro
- CHAPTER 5. -
Chapter Five
Section 5.1 1. Apply the ratio test : lim aB $b8" k a B $b 8 k
Hence the series converges absolutely for kB $k " . The radius of convergence is 3 oe " . The series diverges for B oe # and B oe % , since the n-th term does not a
- CHAPTER 6. -
Chapter Six
Section 6.1 3.
The function 0 a>b is continuous. 4.
The function 0 a>b has a jump discontinuity at > oe " . 7. Integration is a linear operation. It follows that (
E !
-9=2 ,> /=> .> oe
" E ,> => " E ,> => ( / / .> ( / / .> # !
- CHAPTER 8. -
Chapter Eight
Section 8.1 2. The Euler formula for this problem is C8" oe C8 2^& >8 $C8 , C8" oe C8 &82# $2 C8 ,
in which >8 oe >! 82 Since >! oe ! , we can also write
a+b. Euler method with 2 oe !& >8 C8 8oe# !" "&*)! 8oe% !# "#*#) 8oe' !$
- CHAPTER 9. -
Chapter Nine
Section 9.1 2a+b Setting x oe 0 /<> results in the algebraic equations OE &< $
For a nonzero solution, we must have ./>aA < Ib oe <# ' < ) oe ! . The roots of the characteristic equation are <" oe # and <# oe % . For < oe #, th