Physlib

Physlib.QuantumMechanics.OneDimension.Operators.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

7 declarations

definition

Position operator (x^ψ)(x)=xψ(x)(\hat{x}\psi)(x) = x\psi(x)

The position operator is the C\mathbb{C}-linear map from the space of functions R→C\mathbb{R} \to \mathbb{C} to itself that maps a function ψ\psi to the function x↦xψ(x)x \mapsto x \psi(x).

definition

Position operator on Schwartz maps xψ(x)x \psi(x)

The position operator is a continuous linear map on the Schwartz space S(R,C)\mathcal{S}(\mathbb{R}, \mathbb{C}) that maps a function ψ\psi to the function defined by x↦xψ(x)x \mapsto x \psi(x).

theorem

Action of the position operator on S(R,C)\mathcal{S}(\mathbb{R}, \mathbb{C}) as x↦xψ(x)x \mapsto x \psi(x)

For any Schwartz function ψ∈S(R,C)\psi \in \mathcal{S}(\mathbb{R}, \mathbb{C}), the position operator applied to ψ\psi is the function mapping each x∈Rx \in \mathbb{R} to xψ(x)x \psi(x).

theorem

Pointwise Evaluation of Position Operator on Schwartz Maps (x^ψ)(x)=xψ(x)(\hat{x}\psi)(x) = x\psi(x)

For any Schwartz function ψ∈S(R,C)\psi \in \mathcal{S}(\mathbb{R}, \mathbb{C}) and any real number x∈Rx \in \mathbb{R}, the position operator applied to ψ\psi and evaluated at xx satisfies (x^ψ)(x)=xψ(x)(\hat{x}\psi)(x) = x \psi(x).

definition

Unbounded position operator X^\hat{X} on L2(R)L^2(\mathbb{R})

The unbounded position operator X^\hat{X} is an operator on the Hilbert space L2(R,C)L^2(\mathbb{R}, \mathbb{C}) with the Schwartz space S(R,C)\mathcal{S}(\mathbb{R}, \mathbb{C}) as its domain. It is defined by the action (Xψ)(x)=xψ(x)(X \psi)(x) = x \psi(x) for any Schwartz function ψ\psi.

theorem

Position states are generalized eigenvectors of the position operator X^\hat{X} with eigenvalue xx

For any real number x∈Rx \in \mathbb{R}, the position state (denoted by `positionState x`) is a generalized eigenvector of the unbounded position operator X^\hat{X} with eigenvalue xx.

theorem

The position operator X^\hat{X} is self-adjoint

The unbounded position operator X^\hat{X} on the Hilbert space L2(R,C)L^2(\mathbb{R}, \mathbb{C}) with the Schwartz space S(R,C)\mathcal{S}(\mathbb{R}, \mathbb{C}) as its domain, defined by the action (X^ψ)(x)=xψ(x)(\hat{X} \psi)(x) = x \psi(x), is self-adjoint.