Physlib.QuantumMechanics.Operators.OneDimension.Momentum

Momentum operator

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

We show that plane waves are generalized eigenvectors of the momentum operator.

The momentum operator on functions `ℝ → ℂ`

The momentum operator on Schwartz maps

Generalized eigenvectors of the momentum operator

The momentum operator is self adjoint

1 declaration

theorem

The momentum operator $\hat{p}$ is symmetric

#momentumOperatorUnbounded_isSymmetric

The momentum operator $\hat{p}$ on the Hilbert space $L^2(\mathbb{R}, \mathbb{C})$ , defined with the space of Schwartz functions $\mathcal{S}(\mathbb{R}, \mathbb{C})$ as its domain such that $\hat{p}\psi = -i \hbar \frac{d\psi}{dx}$ , is a symmetric operator.