Physics 751 Final
18 December 2007
Only hand in answers to five questions!
Possibly useful info:
Simple Harmonic Oscillator:
a=
m
2
p
x+i
, a | n = n + 1 | n + 1
m
Angular Momentum Operators:
[ J i , J j ] = i ijk J k , J 2 | jm =
2
j ( j + 1) | jm
J =
Orbital Angular Momentum in Three Dimensions
Michael Fowler 11/05/07
The Angular Momentum Operators in Spherical Polar Coordinates
The angular momentum operator L = r p = i r .
In spherical polar coordinates,
x = r sin cos
y = r sin sin
z = r cos
ds 2
Main
An L = 3 Resonance in a Radial Square Well
The square well (red line) plus the L = 3 angular momentum barrier combine to give an effective potential (green line)
which amounts to a well between two radii with barriers on both sides.
See Sheet 3 for h
Main
A Resonance inside a Radial Square Barrier
Click the "Using this Spreadsheet" Tab at the bottom
for some ideas on exploring wave functions.
Radial Square Barrier Wavefunct ion
15
Use the slider (or click and hold on its ends) to see how
the phase cha
Quantizing Radiation
Michael Fowler, 5/4/06
Introduction
In analyzing the photoelectric effect in hydrogen, we derived the rate of ionization of a hydrogen
atom in a monochromatic electromagnetic wave of given strength, and the result we derived is in
goo
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Photoelectric Effect in Hydrogen
Michael Fowler 3/26/10
Introduction
In the photoelectric effect, incoming light causes an atom to eject an electron. We consider the
simplest possible scenario: that the atom is hydrogen in its ground s
Finding the Prefactor in the Simple Harmonic Oscillator
Propagator
Michael Fowler 10/24/07
Hand-waving Argument
Recall that the free particle propagator has the form
im ( x x )2
m
U ( x, T ; x, 0 ) =
exp
.
2 iT
2T
From a classical mechanical evaluatio
Path Integrals in Quantum Mechanics
Michael Fowler 10/24/07
Huygens Picture of Wave Propagation
If a point source of light is switched on, the wavefront is an expanding sphere centered at the
source. Huygens suggested that this could be understood if at a
Charged Particle in a Magnetic Field
Michael Fowler 1/16/08
Introduction
Classically, the force on a charged particle in electric and magnetic fields is given by the Lorentz
force law:
vB
F = q E +
c
This velocity-dependent force is quite different from t
Time-Independent Perturbation Theory
Michael Fowler 2/16/06
Introduction
If an atom (not necessarily in its ground state) is placed in an external electric field, the energy
levels shift, and the wave functions are distorted. This is called the Stark effe
Div, Grad and Curl in Orthogonal Curvilinear Coordinates
The treatment here is standard, following that in Abraham and Becker, Classical Theory of
Electricity and Magnetism.
Problems with a particular symmetry, such as cylindrical or spherical, are best a
Time-Dependent Perturbation Theory
Michael Fowler 7/6/07
Introduction: General Formalism
We look at a Hamiltonian H = H 0 + V ( t ) , with V ( t ) some time-dependent perturbation, so now
the wave function will have perturbation-induced time dependence.
O
Orbital Angular Momentum Eigenfunctions
Michael Fowler 1/11/08
Introduction
In the last lecture , we established that the operators J 2 , J z have a common set of eigenkets
j, m , J 2 j , m = j ( j + 1)
2
j, m , J z j, m = m
j , m where j, m are integers
Main
Schrdingers Equation for the Finite Square Well
This spreadsheet integrates f"(x) = (V(x)-E)f(x) with V(x) a finite-depth square well.
We take V(x) = 0 inside the well, V(x) = D outside the well.
The "look for bound state" button activates a macro,
w
Schrdingers Equation in 1-D: Some Examples
Michael Fowler, UVa. 9/24/07
Curvature of Wave Functions
Schrdingers equation in the form
d 2 ( x) 2m(V ( x) E )
=
( x)
2
dx 2
can be interpreted by saying that the left-hand side, the rate of change of slope, i
Identical Particles Revisited
Michael Fowler
Introduction
For two identical particles confined to a one-dimensional box, we established earlier that the
normalized two-particle wavefunction ( x1 , x2 ) , which gives the probability of finding
simultaneous
Spin
Michael Fowler 11/26/06
Introduction
The Stern Gerlach experiment for the simplest possible atom, hydrogen in its ground state,
demonstrated unambiguously that the component of the magnetic moment of the atom along the
z-axis could only have two valu
The Hydrogen Atom
Michael Fowler 11/22/06
Factoring Out the Center of Mass Motion
The hydrogen atom consists of two particles, the proton and the electron, interacting via the
Coulomb potential V ( r1 r2 ) = e 2 / r , where as usual r = r1 r2 . Writing th
Higher Order Perturbation Theory
Michael Fowler 03/07/06
The Interaction Representation
Recall that in the first part of this course sequence, we discussed the Schrdinger and Heisenberg
representations of quantum mechanics here. In the Schrdinger represen
Main
Schrdingers Equation for the Simple Harmonic Oscillator
Michael Fowler
This spreadsheet integrates f"(x) = (V(x)-E)f(x) with V(x) a simple harmonic oscillator.
The table of values and calculation are on Sheet 2.
The "Eigenvalue" button activates the
Scattering Theory
Michael Fowler 1/16/08
References:
Baym, Lectures on Quantum Mechanics, Chapter 9.
Sakurai, Modern Quantum Mechanics, Chapter 7.
Shankar, Principles of Quantum Mechanics, Chapter 19.
Introduction
Almost everything we know about nuclei an
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The Energy-Time Uncertainty Principle: Decaying States and
Resonances
Michael Fowler 10/1/08
Model of a Decaying State
The momentum-position uncertainty principle p x has an energy-time analog,
E t . Evidently, though, this must be a d
Relating Scattering Amplitudes to Bound States
Michael Fowler, UVa. 1/17/08
Low Energy Approximations for the S Matrix
In this section, we examine the properties of the partial-wave scattering matrix
Sl ( k ) = 1 + 2ikfl ( k )
for complex values of the mo
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Electron in a Box
Michael Fowler, University of Virginia 9/1/08
Plane Wave Solutions
The best way to gain understanding of Schrdingers equation is to solve it for various
potentials. The simplest is a one-dimensional particle in a box
More Scattering: the Partial Wave Expansion
Michael Fowler 1/17/08
Plane Waves and Partial Waves
We are considering the solution to Schrdingers equation for scattering of an incoming plane
wave in the z-direction by a potential localized in a region near
The Density Matrix
Michael Fowler 11/19/07
Pure States and Mixed States
Our treatment here more or less follows that of Sakurai, beginning with two imagined SternGerlach experiments. In that experiment, a stream of (non-ionized) silver atoms from an oven