# 3: The Tools of Quantum Mechanics

• Bra-Ket Notation
Bra-ket notation is a standard notation for describing quantum states in the theory of quantum mechanics composed of angle brackets and vertical bars. It can also be used to denote abstract vectors and linear functionals in mathematics. It is so called because the inner product (or dot product) of two states is denoted by a bracket, <Φ|Ψ>, consisting of a left part, <Φ|, called the bra, and a right part, |Ψ>, called the ket.
• Chapter 4. Principles of Quantum Mechanics
Here we will continue to develop the mathematical formalism of quantum mechanics, using heuristic arguments as necessary. This will lead to a system of postulates which will be the basis of our subsequent applications of quantum mechanics.
• Collapsing Wavefunctions
One of the postulates is that all measurable quantities in a quantum system are represented mathematically by so called observables. An observable is thus a mathematical object, more specifically a real linear operator whose 'eigenstates' form a complete set. This essentially means that any quantum state can be expressed as a linear combination of these eigenstates of the observable.
• Expectation Values (Averages)
The expected value (or expectation, mathematical expectation, mean, or first moment) refers to the value of a variable one would "expect" to find if one could repeat the random variable process an infinite number of times and take the average of the values obtained. More formally, the expected value is a weighted average of all possible values.
• Matrix Representation of Operators and Wavefunctions
• Properties of the solutions to the Schrödinger equation
Beginning in the early 20th century, physicists began to acknowledge that matter--much like electromagnetic radiation--possessed wave-like behaviors. While electromagnetic radiation were well understood to obey Maxwell's Equations, matter obeyed no known equations.