Showcases in Mathematical Physics

Notes, discussions, everything about Mathematical Physics.

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Many texts in physics naively define the Dirac delta function $\delta:\mathbb{R}\to\mathbb{R}$ as $$ \delta(x)=\begin{cases} \infty,\text{ if }x=0\\ 0,\text{ if }x\neq0 \end{cases}. $$ However, $\infty\notin\mathbb{R}$, so this rule does not defines a function properly. To remedy this, some references, such as Butkov, states that the Dirac delta must be thought as symbol, that only makes sense inside an integral, for which $$ \int_\mathbb{R}\delta(x)\:dx = 1, $$ and, for any continuous function $f:\mathbb{R}\to\mathbb{R}$, $$ \int_\mathbb{R}f(x)\delta(x)\:dx = f(0). $$ We may also interpret this symbol as a limit of a sequence of functions. Such sequences are called as delta sequences. For instance, consider the sequence $(f_n)$ of gaussian functions $f_n:\mathbb{R}\to\mathbb{R}$ given by $$ f_n(x)=\frac{n}{\sqrt{\pi}}e^{-n^2x^2}. $$ However, this approach need a more careful treatment. For instance, we need to establish the topology in which the limit is taken.