• Everything starts with the Stern-Gerlach experiment . A silver atom shoots through a magnet and it either lands on one side or the other side of glass pane. This single experiment brings up the concept of quantum spin and is, in my opinion, the single most important experiment that contributes to the weirdness of Quantum Mechanics. If this experiment didn't exist, Schrodinger's cat, quantum entanglement, and quantum computers wouldn't exist.
• A quantum state is represented by this thing called a ket. A lot of the times, kets can only either be an up ket or a down ket. Kets by themselves don't really have any physical meaning, but their meaning can be understood by transforming it with these tools called operators: position operators, momentum operators, and probability operators.
• Operators are essentially transformers for functions like f(x) = sin(x). Operators can be really simple and turn a function into f(x) = 0 for example. However, they can also be really complicated and turn a function into f(x) = p^2 * sin(x) / 2m, just to give an example.
• Oftentimes, because operators are so complicated when applied to functions, physicists will forget the fact that functions are functions and just call them vectors. A common example of this is calling f(x) = sin(x) [1, 0] and calling g(x) = cos(x) [0, 1]. This has the same meaning, because these two functions' relationsips have the same relationships as these two vectors. They are both perpendicular to each other at all points. And now that we can reduce complicated functions to simple vectors, we can reduce the complicated operators into simple matrices.
• Sometimes, not all the time, when you apply a transformation to something, you can get the same thing, but just stretched or shrunk. In linear algebra, this is called eigenvector. In quantum mechanics, this is called eigenkets.
• Kets are often represented as column vectors, this is so that we can conveniently have row vectors that can be multiplie with these column vectors. These row vectors go by the name of bras. Interestingly enough, if you put these words together, you get the name bra-ket. BRA-KET. Bracket... Clever, clever physicists..
• Some operators have certain relationships with each other. Specifically, if two operators can be applied one after the other without completely destroying the ket, then the operators are considered compatible. However, if one operator gets applied to a ket, and then another operator, resulting in the ket getting destroyed. Then the operators are incompatible.
• If a matrix acts on a vector and it doesn't fundamentally change, it just gets stretched or shrunk, the vector is called the eigenvector of the matrix. If a quantum matrix acts on a ket and it doesn't fundamentally change, it just gets stretched or shrunk. the ket is called an eigenket. Similarly, if an operator acts on a function and it just stretches or shrinks, it is called an eigenfunction of the operator. Often, these are just the same names for the same idea. When you apply a transformation, you don't get anything too different. In these rare situations, you can receive interesting information from the amount that it stretches or shrinks.