If ⟨φ|Ψ⟩ = 0, then the states φ and Ψ are called orthogonal, and are as different as can be. Inner products between vectors are expressed like ⟨φ|Ψ⟩, and represent the “closeness” between these states. These conjugated vectors are written like ⟨φ|. Similarly, any operator A has a conjugate operator A*. By analogy with complex conjugation of numbers, you can also conjugate vectors.
The notation used for a state vector is |Ψ⟩, and is read as “the state vector psi”. These vectors are all unit length, and encode all of the observable information about the system. In quantum mechanics, the state of a system is described by a vector in a complex vector space. If you want to know more QM, I highly highly recommend Leonard Susskind’s online video lectures. I’m going to assume a lot of background knowledge of quantum mechanics for the purposes of this post, so as to keep it from getting too long. I want to present it here because I like it a lot. This video contains a really short and sweet derivation of the form of the Schrodinger equation from some fundamental principles.
