# 1.25: The Dirac Delta Function


The Dirac delta function expressed in Dirac notation is: $$\Delta(x - x_1) = \langle x | x_1 \rangle$$. The $$\langle x | x_1 \rangle$$ bracket is evaluated using the momentum completeness condition. See the Mathematical Appendix for definitions of the required Dirac brackets and other mathematical tools used in the analysis that follows.

$\langle x | x_{1}\rangle=\int_{-\infty}^{\infty}\langle x | p\rangle\langle p | x_{1}\rangle d p=\frac{1}{2 \pi} \int_{-\infty}^{\infty} \exp (i p x) \exp \left(-i p x_{1}\right) d p=\frac{1}{2 \pi} \int_{-\infty}^{\infty} \exp \left[i p\left(x-x_{1}\right)\right] d p \nonumber$

Evaluation of this integral over a finite range of momentum values shows that the delta function is small except in the immediate neighborhood of x1. Integrating from -20 to 20 to reduce computational time shows that < x | x1 = 2> is small except in the area x = 2.

$\mathrm{x}_{1} =2 \quad \mathrm{x} =0,0 .01 \ldots 4 \quad \operatorname{Dirac}\left(\mathrm{x}, \mathrm{x}_{1}\right) =\frac{1}{2 \cdot \pi} \cdot \int_{-20}^{20} \exp \left[\mathrm{i} \cdot \mathrm{p} \cdot\left(\mathrm{x}-\mathrm{x}_{1}\right)\right] \mathrm{d} \mathrm{p} \nonumber$

The Fourier transform of the Dirac delta function into the momentum representation yields the following result.

$\int_{-\infty}^{\infty}\langle p | x\rangle\langle x | x_{1}\rangle d x=\frac{1}{\sqrt{2 \pi}} \exp \left(-i p x_{1}\right)=\langle p | x_{1}\rangle \nonumber$

The normalization constant is omitted for clarity of expression and the previous value of x1 is cleared to allow symbolic calculation.

$\mathrm{x}_{1} =\mathrm{x}_{1} \qquad \int_{-\infty}^{\infty} \exp (-\mathrm{i} \cdot \mathrm{p} \cdot \mathrm{x}) \cdot \Delta\left(\mathrm{x}-\mathrm{x}_{1}\right) \mathrm{d} \mathrm{x} \text { simplify } \rightarrow \; \mathrm{e}^{-\mathrm{p} \cdot \mathrm{x}_{1} \cdot \mathrm{i}} \nonumber$

## Mathematical Appendix

The position and momentum completeness conditions:

$\int | x \rangle\langle x|d x=1 \qquad \int| p\rangle\langle p|d p=1 \nonumber$

The momentum eigenstate in the coordinate representation:

$\langle x | p\rangle=\frac{1}{\sqrt{2 \pi}} \exp (i p x) \nonumber$

The position eigenstate in the momentum representation:

$\langle p | x\rangle=\frac{1}{\sqrt{2 \pi}} \exp (-i p x) \nonumber$

This page titled 1.25: The Dirac Delta Function is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Frank Rioux via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.