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
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
- CHAPTER 11. -
Chapter Eleven
Section 11.1 1. Since the right hand sides of the ODE and the boundary conditions are all zero, the boundary value problem is homogeneous. 3. The right hand side of the ODE is nonzero. Therefore the boundary value problem is