For discrete basis:

I=∑n∣n⟩⟨n∣I=\sum_n\ket{n}\bra{n}

Think about how this just creates a n×nn\times n diagonal matrix with 1's in it.

For a continuous basis:

I=∫∣x⟩⟨x∣ dxI=\int\ket{x}\bra{x}~dx

Note:

  1. note
I∣x⟩=∫−∞∞∣x′⟩⟨x′∣x⟩dx′=∫−∞∞∣x′⟩δ(x−x′)dx′=∣x⟩I\ket{x} = \int_{-\infty}^\infty \ket{x'}\braket{x'|x}dx'=\int_{-\infty}^\infty \ket{x'}\delta(x-x')dx'=\ket{x}
  1. Note
⟨x∣I∣ψ⟩=∫−∞∞⟨x∣x′⟩⟨x′∣ψ⟩dx\bra{x}I\ket{\psi}=\int_{-\infty}^\infty \braket{x|x'}\braket{x'|\psi}dx =∫−∞∞δ(x−x′)ψ(x′)dx′=\int_{-\infty}^\infty \delta (x-x')\psi(x')dx' =ψ(x)=⟨x∣ψ⟩=\psi(x)=\braket{x|\psi}
  1. Note
⟨ϕ∣ψ⟩=∫⟨ϕ∣x⟩⟨x∣ψ⟩dx\braket{\phi|\psi}=\int\braket{\phi|x}\braket{x|\psi}dx =∫ϕ∗(x)ψ(x) (dx)=\int\phi^*(x)\psi (x)~(dx)