There are number of ideas that are reviewed as a preparation to the study of QM. I recommend that the student focus at the beginning on building the mathematical skills so as to feel comfortable with the development.

Critical	Important or Interesting
Partial derivatives	Hamiltonian formulation of mechanics
Complex (Imaginary) numbers [ z]	Photoelectric effect
Complex conjugate of complex number [z^*]	Black Body radiation
Representation of z in terms of r,θ	Bohr atom
Differential equations	Hidden Variables
Energy	EPR: Einstein, Podolsky, Rosen
Potentials
Momentum Angular, Linear
Coordinates Cartesian, Spherical, Cylindrical
Orthogonal Functions
Interference
Vectors
Basis vectors
Amplitudes, Probability
Hamiltonian
Force, Torque
Conserved Quantities (Constants of Motion)
State of a System
Good Quantum Numbers
Time evolution of a system
Representations
Pauli Exclusion Principle
Particles: e, p, γ [electron, proton, photon]
Photon:
electron:
Uncertainty principle
Classical particle vs wave
Wavelength, frequency QM: ; ; ;
Dispersion relationship
Wavefunction

State of a System

The refinement of what we mean by the state of the system will be central to our understanding of QM. In classical mechanics this is an implied and obvious concept that so little formal attention is spent clarifying what is meant by the state of the system. In QM what one knows about a system is far less obvious. The refinement of this concept then becomes very critical. Some crtical questions:

· What do you know about a particular system?

· What can you know?

· How can you label a definite state for a system?

Good Quantum Numbers

A set of mutually measurable quantities that characterize a quantum system

Representations

Since there are mutually exclusive measurable quantities in QM, [i.e. x,p], one can characterize systems in terms of different quantum numbers. A choice of specific quantum numbers will be considered one representation of the states of the system. The relationship between different representations plays a major role in QM. It is possible to write down a specific state in terms of any one of the sets of good quantum numbers that are available. A state of definite momentum can be expressed in terms of the measurable locations for that particle.

Potentials and Forces

These are ways to characterize an interaction. The notion of particle interactions is important. What do we mean by an isolated system? How do we add interactions between isolated systems?

Pauli Exclusion Principle

Two identical particles may not occupy the same state. Not all particles follow the PEP. Particles that do follow the PEP are called Fermions. No two Fermions can occupy the same state. Other particles, referred to as Bosons, can be in same state.

electrons, protons è Fermions

photonè Boson

Uncertainty Principle

Waves vs Particles

A very good review of the differences and similarities is contained in the two-slit problem.

waves: add amplitudes and square the result to find intensity. The result is interference.

particles: add intensities. No interference.

Dispersion relationship

For a particle characterized by an energy and momentum and therefore c. The relationship between is described as a dispersion relationship. For mass less particles . Classically, one can examine systems in which the speed of wave propagation (phase velocity) is dependent on the frequency. These systems are called dispersive because the shape of a pulse moving through the medium changes as it propagates due to the variations of speed.

; ; phase velocity;

for light so the phase velocity is the same as the classical speed and is a constant.

particles with mass è

Wavefunction

The wavefunction is a critical element in the formulation of QM. To grasp the concept one needs to begin to consider how to define the wavefunction for a given system and how to interpret the wavefunction. This will be an ongoing development throughout the course.