Elements of Classical and Quantum Physics pp 43-50 | Cite as

# Dirac’s Delta

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## Abstract

Let us start with the Heavyside (Oliver Heaviside (1850–1925) was probably the first to use the \(\delta \) before Dirac, and the work of George Green also implies the concept. Often the names are not historically fair) \(\theta \) discontinuous function, also known as the step function, defined by With it we can define a rectangular-shaped peak function, of width \(2\alpha \), such that

$$\begin{aligned} \theta (x) = \left\{ \begin{array}{ll} 1 &{} \text {se } x>0, \\ {1 \over 2} &{} \text {if } x=0, \\ 0 &{} \text {se } x<0, \end{array} \right. \end{aligned}$$

$$\begin{aligned} \delta _{\alpha }(x) ={\theta (\alpha ^{2} - x^{2}) \over 2\alpha }, \end{aligned}$$

$$\begin{aligned} \int _{-\infty }^{\infty }\delta _{\alpha }(x) d x=1. \end{aligned}$$

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