Dirac Notation

Also called “Bracket/Bra-ket” Notation

$$⟨g(x)⟩=\int {\psi }^{\ast }(x)g(x)\psi (x)dx=⟨\psi |g|\psi ⟩$$

The entire expression $⟨\psi |g|\psi ⟩$ is called the “bracket” $⟨\psi |$ is called the “bra” $|\psi ⟩$ is called the “ket”

• Use kets to represent the probabilisitc state of a particle

• For example, the expected position $x$ is $⟨\psi |x|\psi ⟩$

• To include time dependence: $⟨g(x,t)⟩=⟨\Psi |g(x,t)|\Psi ⟩$

• The normalization condition is $⟨\Psi |\Psi ⟩=⟨\psi |\psi ⟩=1$

• The dipole moment: $⟨\vec{d}⟩=⟨{\psi }_f|e\vec{r}|{\psi }_i⟩$

H Atom State

$${\psi }_{nl{m}_l{m}_s}=|nl{m}_l{m}_s⟩$$

See Hydrogen Radial Wave Functions and Schrödinger Wave Equation for Hydrogen Atom