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Physlib.QuantumMechanics.OneDimension.Operators.Position

7 declarations

definition

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

#positionOperator

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)

#positionOperatorSchwartz

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)

#positionOperatorSchwartz_apply_fun

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)

#positionOperatorSchwartz_apply

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})

#positionOperatorUnbounded

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

#positionStates_generalized_eigenvector_positionOperatorUnbounded

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

#positionOperatorUnbounded_isSelfAdjoint

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.