Similar to the Kronecker Delta, but for continuous values

δ(x)≜{1x=00x≠0x∈R\delta (x)\triangleq\begin{cases}1&x=0\\0&x\neq 0\end{cases}\quad x\in\mathbb{R}

Properties

  1. Note that
⟨x∣x′⟩=δ(x−x′)\braket{x|x'}=\delta(x-x')
  1. Note that
δ(x)=0ifx≠0\delta (x)=0\quad if\quad x\neq 0
  1. Note that
∫−∞∞δ(x)dx=1\int_{-\infty}^\infty \delta(x)dx=1
  1. Note that
∫−∞∞δ(x−x′)f(x′)dx′=f(x)\int_{-\infty}^\infty\delta(x-x')f(x')dx'=f(x)
  1. Note the Fourier Transform representation of the Dirac Delta.
δ(z)=12π∫−∞∞eiyzdy\delta(z)=\frac{1}{2\pi}\int_{-\infty}^\infty e^{iyz}dy