Bra-Ket Notation

Notation

$| ψ ⟩$ - ket vector $\in E$ .

$⟨ ψ |$ - bra vector $\in E^{*}$ dual space.

$⟨ ψ | ψ ⟩$ - bra-ket.

$ψ (\vec{r}, t) = ⟨ \vec{r}, t | ψ ⟩$ .

$ψ (\vec{p}, t) = ⟨ \vec{p}, t | ψ ⟩$ .

$⟨ φ | ψ ⟩ = (φ, ψ)$ is the notation for an inner product.

For position/ momentum space representation:

$⟨ φ | ψ ⟩ = \int φ^{*} (\vec{r}, t) ψ (\vec{r}, t) d^{3} \vec{r} = \int φ_{P}^{*} (\vec{p}, t) ψ_{P} (\vec{p}, t) d^{3} \vec{p}$ .

$(| ψ ⟩)^{*} = ⟨ ψ |$
$(a | ψ ⟩)^{*} = a ⟨ ψ |$
$| a ψ ⟩ = a | ψ ⟩$
$⟨ a ψ | = a^{*} ⟨ ψ |$
$⟨ φ | ψ = \overset{―}{⟨ ψ | φ ⟩}$ since the complex conjugate transposes the swapped bra-kets
$⟨ ψ | ψ ⟩$ is real, positive
Normalization of Wavefunction
Schwarz Inequality
Triangle Inequality
Orthogonality of States
Orthonormality of States
$| φ ⟩ | ψ ⟩$ is allowed (and is shorthand for a tensor product of the spaces) only if $| φ ⟩$ and $| ψ ⟩$ are from different spaces (e.g. Positon vector and spin vector)
$⟨ φ | ψ ⟩$ is a projection, or checks the overlap, of $ψ$ on $φ$