In the classical picture, light waves with general polarizations are superpositions of waves with orthogonal linear polarizations. To quantitatively explain the polarizer experiments based on photons, we will argue that the individual photons with general polarizations should still be viewed mathematically as superpositions of the two special photon states, but one in which the overall amplitude has no physical meaning. Thus, we'll arrive at one of the most basic rules for quantum systems: for each possible result of a given measurement, there are special states, called "eigenstates", for which that result will definitely be obtained. More general states (for which the result is not predetermined) are superpositions of these eigenstates, and the amount of each eigenstate in the superposition determines the probability of obtaining the corresponding outcome. Wave properties of particles With the understanding that classical electromagnetic waves are comprised of photon particles, one might wonder whether other kinds of particles give rise to wavelike phenomena. While we don't see any classical electron waves (this has to do with the Pauli exclusion principle) it turns out that a beam of electrons at some fixed momentum does exhibit diffraction phenomena, with a wavelength inversely proportional to momentum, just as for photons. By discussing a very simple diffraction experiment, known as the "double-slit experiment", we'll argue that states of individual electrons with a given momentum do not have well-defined positions, and propose (motivated by our discussion of polarization experiments) that these and more general states of the electrons are superpositions of eigenstates where the electrons do have definite positions. The information about the amount of each position eigenstate in a given superposition is known as the "wavefunction", and this information determines the probabilities of the possible outcomes when the position is measured. The wavefunction gives a complete description of the state of a particle at a given time and replaces the classical description in terms of instantaneous position and velocity. The description of general electron states as superpositions of states with definite positions provides another example of our general rules for quantum mechanics. For any given question that we might ask about a particle (in this case "what is the position?" or "what is the momentum?"), there are some states for which the answer is predetermined, while general states are superpositions of these states. An important point is that an eigenstate for one physical quantity (e.g. an electron with some definite position) is usually not an eigenstate for another physical quantity. For example, there are no states that are eigenstates of both position and momentum, and we'll see that the more certain we are about the position of a given particle, the more uncertainty there is in the momentum. This is known as the "Heisenberg Uncertainty Principle". The Schrödinger Equation Since the classical description of a particle in terms of position and velocity have been replaced by the idea of a quantum state described by a wavefunction, we'll need to understand what replaces