Physlib.QuantumMechanics.HilbertSpaces.SpaceD.PositionStates

Position states

i. Overview

Informally, the position "state" at `x : Space d` has a non-normalizable wavefunction which is a Dirac-delta function centered at `x`. More precisely, the position "state" lives in the _rigged_ Hilbert space `𝓢(Space d, ℂ) < SpaceDHilbertSpace d μ < StrongDual ℂ 𝓢(Space d, ℂ)` as the element of the dual of `𝓢(Space d, ℂ)` defined by evaluation at `x`.

ii. Key results

iii. Table of contents

iv. References

https://en.wikipedia.org/wiki/Rigged_Hilbert_space

2 declarations

definition

Position state as a Dirac delta distribution $\delta_x$

#positionState

For a point $x$ in the $d$ -dimensional space $\text{Space } d$ , the position state is defined as the tempered distribution $\delta_x$ belonging to the strong dual $\mathcal{S}'(\text{Space } d, \mathbb{C})$ of the Schwartz space $\mathcal{S}(\text{Space } d, \mathbb{C})$ . This distribution acts on a test function $f \in \mathcal{S}(\text{Space } d, \mathbb{C})$ via evaluation at $x$ , such that $\langle \delta_x, f \rangle = f(x)$ . In the context of rigged Hilbert spaces, this represents a non-normalizable wavefunction given by the Dirac delta function $\psi(y) = \delta^{(d)}(y - x)$ .

theorem

The action of the position state $\delta_x$ on a test function $f$ is $f(x)$

#positionState_apply

For any point $x$ in the $d$ -dimensional space $\text{Space } d$ and any test function $f$ in the Schwartz space $\mathcal{S}(\text{Space } d, \mathbb{C})$ , the position state at $x$ (viewed as a tempered distribution $\delta_x$ ) applied to $f$ is equal to the value of the function at $x$ . That is, $\delta_x(f) = f(x)$ where $\delta_x$ is the position state defined in the dual of the Schwartz space.