Physlib.QuantumMechanics.Operators.OneDimension.Position

Position operator

In this module we define: - The position operator on functions `ℝ → ℂ` - The position operator on Schwartz maps as an unbounded operator on the Hilbert space.

We show that position wavefunctions are generalized eigenvectors of the position operator.

The position operator on functions `ℝ → ℂ`

The position operator on Schwartz maps

Generalized eigenvectors of the momentum operator

Position operator is self adjoint

1 declaration

theorem

The Unbounded Position Operator $\hat{X}$ is Symmetric

#positionOperatorUnbounded_isSymmetric

The unbounded position operator $\hat{X}$ on the Hilbert space $L^2(\mathbb{R}, \mathbb{C})$ with the Schwartz space $\mathcal{S}(\mathbb{R}, \mathbb{C})$ as its domain, defined by $(\hat{X} \psi)(x) = x \psi(x)$ , is symmetric. That is, for all $\psi_1, \psi_2 \in \mathcal{S}(\mathbb{R}, \mathbb{C})$ , the $L^2$ inner product satisfies $\langle \hat{X} \psi_1, \psi_2 \rangle = \langle \psi_1, \hat{X} \psi_2 \rangle$ .