postulates

Physical states are represented as "rays" in a Hilbert State $H$ .
- Vectors are called ket
- A Hilbert Space is a normed vector space with a inner product
- Vector are represented using $| ⟩$
- there exists a dual space $⟨ |$ (bra space)
- Inner products are defined as $⟨ ψ | ψ ⟩$ (note that $⟨ ψ | ψ ⟩ \geq 0$ )
- the space is called linear if $⟨ ψ | a ϕ_{1} + b ϕ_{2} ⟩ = a ⟨ ψ | ϕ_{1} ⟩ + b ⟨ ψ | ϕ_{2} ⟩$
- $⟨ ψ | ϕ ⟩^{*} = ⟨ ϕ | ψ ⟩$
- $|\langle \psi | \phi \rangle| {#2} \leq \langle \psi | \psi \rangle \langle \phi | \phi \rangle$ (Schwarz Inequality)
- In the real world there is no difference between $c | ψ ⟩$ and $| ψ ⟩$ where $c \in C$
- A vector space is said to be spanned by ${| e_{n} ⟩}$ i.e. $| ψ ⟩ = \sum_{n} a_{n} | e_{n} ⟩$ . In particular these bases can be chosen to be orthonormal $⟨ e_{m} | e_{n} ⟩ = δ_{m n}$ and $a_{n} = ⟨ e_{n} | ψ ⟩$
- The state may be represented by a column vector. In $n$ dimensions, one can represent a state by a set of $n$ complex numbers $| ψ ⟩ = (c_{1} \dots c_{n})^{T}$ . The corresponding element in the dual space (bra) can be written by a row vector $⟨ ψ | = (c_{1}^{*} \dots c_{n}^{*})$
- The inner product between two vectors is given by $⟨ ψ | ϕ ⟩ = \sum_{n} ψ_{n}^{*} ϕ_{n}$

Copenhagen Interpretation: State $| ψ ⟩$ has a probabilistic interpretation with $| α_{n} ⟩^{2}$ as the probability with which the system would be found in the state $| e_{n} ⟩$

Observables (position, momentum, energy, etc.) are represented by linear self-adjoint operators in the Hilbert Space $H$
An arbitrary vector can be written as a linear superposition of a complete set of eigenstates of any operator in that space $$| \psi \rangle = \sum_n c_n | \lambda_n \rangle$$If a measurement of the observable is made in such a state, the result will be the eigenvalue $λ_{n}$ with a probability $| c_{n} |^{2} = ⟨ ψ | P_{n} | ψ ⟩$ with $P_{n} = | λ_{n} ⟩ ⟨ λ_{n} |$ as the projection operator.
Time Evolution: (Schrodinger's Evolution) $$\begin{align} i \hbar \frac{\partial}{\partial t} | \psi \rangle &= H | \psi \rangle \ |\psi\rangle (t) &= U(t) | \psi(0) \rangle \ U &= e^{-i Ht/\hbar}\end{align}$$