Position Operator

It is a Continuous Operator that represents the position of a particle In the position basis:

(x^ψ)(x)→xψ(x)(\hat{x}\psi)(x)\rightarrow x\psi(x)

Note that

x^=∫x∣x⟩⟨x∣dx\hat{x}=\int x\ket{x}\bra{x}dx

check:

x^∣x⟩=∫x′ ∣x′⟩⟨x′∣x⟩ dx′\hat{x}\ket{x} = \int x'\,\ket{x'}\braket{x'|x}\, dx'

This uses the principle in the first note of Dirac Delta

=∫x′ ∣x′⟩ δ(x′−x) dx′= \int x'\,\ket{x'}\,\delta(x'-x)\, dx'

Because this only fires at x′=xx'=x, we're only going to evaluate x′∣x′⟩x'\ket{x'} at xx. Hence

=x∣x⟩=x\ket{x}

Properties

  1. Note
f(x^)∣x⟩=f(x)∣x⟩f(\hat{x})\ket{x} = f(x)\ket{x}
  1. Note
⟨x∣f(x^)=f(x)⟨x∣\bra{x}f(\hat{x}) = f(x)\bra{x}
  1. Note
⟨ψ∣f(x^)∣ψ⟩=∫f(x) ∣ψ(x)∣2 dx=∫f(x) Pψ(x) dx\bra{\psi}f(\hat{x})\ket{\psi} = \int f(x)\,|\psi(x)|^2\, dx = \int f(x)\,P_\psi(x)\, dx
  1. Note
⟨x∣x^∣ψ⟩=x⟨x∣ψ⟩=x ψ(x)\bra{x}\hat{x}\ket{\psi} = x\braket{x|\psi} = x\,\psi(x)