Hilbert Space

Axiom 1 : A state is a ray in the Hilbert space. A ray is a general case of a vector that need not to have an origin and need not to be represented by basis vectors.

Axiom 2 : An observable is a self-adjoint Hermitian operator on the Hilbert space.

* Recall that an operator is defined as a thing that linearly transforming some state $\ket{\psi}$ where linear here means:

    $$ A(a\ket{\psi} + b\ket{\varphi}) = aA\ket{\psi} + bA\ket{\varphi}$$

* Self-adjoint means that the operator acted upon a braket can be either done by first acting on the bra or firt acting on the ket,

    $$ \bra{\varphi}A\ket{\psi} = A\bra{\varphi}\ket{\psi}$$

Property: Real Eigenvalues
: Recall that if,

    $$ A\ket{\psi} = a\ket{\psi} $$

    then we are sure that $a$ is real and $\ket{\psi}$ is a vector that can from the complete basis of the Hilbert space.

Property: Spectral Representatoin
: Given $N$ eigenstates denoted by the ket vector $\ket{n}$, we may write the operator as a linear combination of each $n$th eigenstate's projection vector. This is called the **spectral representation** of the operator.

    The projection vector is defined as,

    $$ P_n \equiv \ket{n}\bra{n} $$

    We form the spectral representation,

    $$ A = \sum_{n}^N{a_nP_n} $$

Axiom 3: Unitarity (Dynamics) : A unitary operator is where its hermitian conjugate equals its inverse,

$$ \begin{align}
    U^\dagger \equiv (U^*)^T \\
    \boxed{U^{\dagger} = U^{-1}} \tag{Unitary}
\end{align}$$

The time evolution is unitary and described by a self adjoin operator called the Hamiltonian. Recall the Schrodinger Equation,

$$ \frac{\d}{\d t}\ket{\Psi} = -iH\ket{\Psi} \tag{$\hbar = 1$}$$

For instance we can represent an infinitesimal time step as,

$$ \ket{\Psi(t + \d t)} = \ket{\Psi(t)} - iH\ket{\Psi(t)} \d t = (1 - iH\d t)\ket{\Psi(t)}$$

In general this is,

$$
\ket{\psi(t)} = U(t-t_0)\ket{\Psi(t_0)} \\
U(t-t_0) = \exp\left[-i\int{H\d t}\right]
$$

The unitary operator preserve orthonomality $\bra{\varphi}\ket{\psi}$ given that the same unitary operator applies to both states (interpret as if the Hamiltonian and time elapsed is the same for the two states),

$$ \bra{u\varphi}\ket{u\psi} = \bra{\varphi}\ket{\psi} $$