1. Introduction to the course
1.1
1.2
Why study Quantum Mechanics?
Striking Characteristics of QM
1.1 Why study Quantum Mechanics?
Quantum mechanics (QM) is a fundamental and general theory that applies on a very wide range of scale, from
subatomic system

2. Mathematical Formalism of Quantum Mechanics
2.1
2.2
Linear vectors and Hilbert space
Operators
2.2.1 Hermitian operators
2.2.2 Operators and their properties
2.2.3 Functions of operators
Quantum mechanics is a linear theory, and so it is natural that v

3. Axioms of Quantum Mechanics
3.1
3.2
Introduction
The axioms of quantum mechanics
3.2.1 Observables and State Space
3.2.2
Quantum measurement
3.2.3 Law of motion
3.3 Strong measurements
3.3.1 Expectation values
3.3.2 Uncertainty relationships
3.3.3 Repe

4. Two-level systems
4.1
4.2
Generalities
Rotations and angular momentum
4.2.1 Classical rotations
4.2.2 QM angular momentum as generator of rotations
4.2.3 Example of Two-Level System: Neutron Interferometry
4.2.4 Spinor behavior
4.2.5 The SU(2) and SO(3

5. Time evolution
5.1
5.2
5.3
5.4
5.5
5.6
The Schrodinger and Heisenberg pictures
Interaction Picture
5.2.1 Dyson Time-ordering operator
5.2.2 Some useful approximate formulas
Spin- 1 precession
2
Examples: Resonance of a Two-Level System
5.4.1 Dressed st

22.51 Quantum Theory of Radiation Interactions
Final Exam - Solutions
Tuesday December 15, 2009
Problem 1
Harmonic oscillator
20 points
Consider an harmonic oscillator described by the Hamiltonian H = l (N + 1 ). Calculate the evolution of the expectation

22.51 Quantum Theory of Radiation Interactions
Final Exam
December 14, 2010
Problem 1:
Name: . . . . . . . . . . . . . . . . . .
Electric Field Evolution
20 points
Consider a single mode electromagnetic eld in a volume V = L3 . Calculate the evolution of

22.51 Quantum Theory of Radiation Interactions
Final Exam
December 19, 2011
Problem 1:
Solution
Rutherford Scattering
35 points
100 years ago, in 1911, Rutherford explained the observed surprising behavior of alpha particles scattering from a gold
foil, l

22.51 Quantum Theory of Radiation Interactions
Final Exam
December 17, 2012
Problem 1:
Name: . . . . . . . . . . . . . . . . . .
30 points
Quantized LC circuit
(2
)
2
A) The energy of an LC circuit is given by E = 1 Q + , where Q is the charge stored in t

22.51 Quantum Theory of Radiation Interactions
Mid-Term Exam
Thursday Oct. 29, 2009
Problem 1
Simple two-level system
20 points
The operator x (y ) corresponds to the component of the spin of an electron along the x-axis (y-axis), in units of l/2. In the

22.51 Quantum Theory of Radiation Interactions
Mid-Term Exam
October 27, 2010
Problem 1:
Solution
Electron Spin: Magnetization
20 points
1
Consider an isolated electron with spin- 2 , placed in a large magnetic f eld B = Bz z at zero temperature. The spin

22.51 Quantum Theory of Radiation Interactions
Mid-Term Exam
November 2, 2011
Solution
For most of this midterm exam we are interested in describing the quantum behavior of a particle such as an atom or
a ion which is conned in a double well potential. Th

22.51 Quantum Theory of Radiation Interactions
Mid-Term Exam
October 31, 2012
Name: . . . . . . . . . . . . . . . . . .
In this mid-term we will study the dynamics of an atomic clock,
which is one of the applications of David Winelands research (No
bel pr