Skip to main content Contents Index Prev Up Next \(\newcommand{\vf}[1]{\mathbf{\boldsymbol{\vec{#1}}}}
\renewcommand{\Hat}[1]{\mathbf{\boldsymbol{\hat{#1}}}}
\let\VF=\vf
\let\HAT=\Hat
\newcommand{\Prime}{{}\kern0.5pt'}
\newcommand{\PARTIAL}[2]{{\partial^2#1\over\partial#2^2}}
\newcommand{\Partial}[2]{{\partial#1\over\partial#2}}
\newcommand{\tr}{{\mathrm tr}}
\newcommand{\CC}{{\mathbb C}}
\newcommand{\HH}{{\mathbb H}}
\newcommand{\KK}{{\mathbb K}}
\newcommand{\RR}{{\mathbb R}}
\newcommand{\HR}{{}^*{\mathbb R}}
\renewcommand{\AA}{\vf{A}}
\newcommand{\BB}{\vf{B}}
\newcommand{\CCv}{\vf{C}}
\newcommand{\EE}{\vf{E}}
\newcommand{\FF}{\vf{F}}
\newcommand{\GG}{\vf{G}}
\newcommand{\HHv}{\vf{H}}
\newcommand{\II}{\vf{I}}
\newcommand{\JJ}{\vf{J}}
\newcommand{\KKv}{\vf{Kv}}
\renewcommand{\SS}{\vf{S}}
\renewcommand{\aa}{\VF{a}}
\newcommand{\bb}{\VF{b}}
\newcommand{\ee}{\VF{e}}
\newcommand{\gv}{\VF{g}}
\newcommand{\iv}{\vf{imath}}
\newcommand{\rr}{\VF{r}}
\newcommand{\rrp}{\rr\Prime}
\newcommand{\uu}{\VF{u}}
\newcommand{\vv}{\VF{v}}
\newcommand{\ww}{\VF{w}}
\newcommand{\grad}{\vf{\nabla}}
\newcommand{\zero}{\vf{0}}
\newcommand{\Ihat}{\Hat I}
\newcommand{\Jhat}{\Hat J}
\newcommand{\nn}{\Hat n}
\newcommand{\NN}{\Hat N}
\newcommand{\TT}{\Hat T}
\newcommand{\ihat}{\Hat\imath}
\newcommand{\jhat}{\Hat\jmath}
\newcommand{\khat}{\Hat k}
\newcommand{\nhat}{\Hat n}
\newcommand{\rhat}{\HAT r}
\newcommand{\shat}{\HAT s}
\newcommand{\xhat}{\Hat x}
\newcommand{\yhat}{\Hat y}
\newcommand{\zhat}{\Hat z}
\newcommand{\that}{\Hat\theta}
\newcommand{\phat}{\Hat\phi}
\newcommand{\LL}{\mathcal{L}}
\newcommand{\DD}[1]{D_{\textrm{$#1$}}}
\newcommand{\bra}[1]{\langle#1|}
\newcommand{\ket}[1]{|#1/rangle}
\newcommand{\braket}[2]{\langle#1|#2\rangle}
\newcommand{\LargeMath}[1]{\hbox{\large$#1$}}
\newcommand{\INT}{\LargeMath{\int}}
\newcommand{\OINT}{\LargeMath{\oint}}
\newcommand{\LINT}{\mathop{\INT}\limits_C}
\newcommand{\Int}{\int\limits}
\newcommand{\dint}{\mathchoice{\int\!\!\!\int}{\int\!\!\int}{}{}}
\newcommand{\tint}{\int\!\!\!\int\!\!\!\int}
\newcommand{\DInt}[1]{\int\!\!\!\!\int\limits_{#1~~}}
\newcommand{\TInt}[1]{\int\!\!\!\int\limits_{#1}\!\!\!\int}
\newcommand{\Bint}{\TInt{B}}
\newcommand{\Dint}{\DInt{D}}
\newcommand{\Eint}{\TInt{E}}
\newcommand{\Lint}{\int\limits_C}
\newcommand{\Oint}{\oint\limits_C}
\newcommand{\Rint}{\DInt{R}}
\newcommand{\Sint}{\int\limits_S}
\newcommand{\Item}{\smallskip\item{$\bullet$}}
\newcommand{\LeftB}{\vector(-1,-2){25}}
\newcommand{\RightB}{\vector(1,-2){25}}
\newcommand{\DownB}{\vector(0,-1){60}}
\newcommand{\DLeft}{\vector(-1,-1){60}}
\newcommand{\DRight}{\vector(1,-1){60}}
\newcommand{\Left}{\vector(-1,-1){50}}
\newcommand{\Down}{\vector(0,-1){50}}
\newcommand{\Right}{\vector(1,-1){50}}
\newcommand{\ILeft}{\vector(1,1){50}}
\newcommand{\IRight}{\vector(-1,1){50}}
\newcommand{\Partials}[3]
{\displaystyle{\partial^2#1\over\partial#2\,\partial#3}}
\newcommand{\Jacobian}[4]{\frac{\partial(#1,#2)}{\partial(#3,#4)}}
\newcommand{\JACOBIAN}[6]{\frac{\partial(#1,#2,#3)}{\partial(#4,#5,#6)}}
\newcommand{\LLv}{\vf{L}}
\newcommand{\OOb}{\boldsymbol{O}}
\newcommand{\PPv}{\vf{P}_\text{cm}}
\newcommand{\RRv}{\vf{R}_\text{cm}}
\newcommand{\ff}{\vf{f}}
\newcommand{\pp}{\vf{p}}
\newcommand{\tauv}{\vf{\tau}}
\newcommand{\Lap}{\nabla^2}
\newcommand{\Hop}{H_\text{op}}
\newcommand{\Lop}{L_\text{op}}
\newcommand{\Hhat}{\hat{H}}
\newcommand{\Lhat}{\hat{L}}
\newcommand{\defeq}{\overset{\rm def}{=}}
\newcommand{\absm}{\vert m\vert}
\newcommand{\ii}{\ihat}
\newcommand{\jj}{\jhat}
\newcommand{\kk}{\khat}
\newcommand{\dS}{dS}
\newcommand{\dA}{dA}
\newcommand{\dV}{d\tau}
\renewcommand{\ii}{\xhat}
\renewcommand{\jj}{\yhat}
\renewcommand{\kk}{\zhat}
\newcommand{\lt}{<}
\newcommand{\gt}{>}
\newcommand{\amp}{&}
\definecolor{fillinmathshade}{gray}{0.9}
\newcommand{\fillinmath}[1]{\mathchoice{\colorbox{fillinmathshade}{$\displaystyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\textstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptscriptstyle\phantom{\,#1\,}$}}}
\)
Section 8.4 The Dirac Delta Function
Definition 8.4.1 . The Dirac Delta Function.
The
Dirac delta function \(\delta(x)\) has has the following defining properties:
\begin{equation}
\delta(x)
= \begin{cases}
0\quad \amp x\not= 0\\
\infty\quad\amp x=0
\end{cases}\tag{8.4.1}
\end{equation}
\begin{equation}
\int_b^c \delta(x)\, dx = 1 \qquad\qquad b\lt 0\lt c\tag{8.4.2}
\end{equation}
\begin{equation}
x\,\delta(x) \equiv 0\text{.}\tag{8.4.3}
\end{equation}
Using the Dirac Delta Function in an Integral.
The properties of the delta function allow us to compute
\begin{align*}
\int_{-\infty}^{\infty} f(x)\,\delta(x) \,dx
\amp = \int_{-\infty}^{\infty} f(0)\,\delta(x) \,dx\\
\amp = f(0) \int_{-\infty}^{\infty} \delta(x) \,dx\\
\amp = f(0)
\end{align*}
In the first equality, it is safe to replace the function \(f(x)\) with its value \(f(0)\) at \(x=0\) since everywhere else the integrand is zero, due to the delta function.
We can shift the “spike” in the delta function as usual, obtaining \(\delta(x-a)\text{.}\) This shifted delta function satisfies
\begin{equation}
\int_{-\infty}^{\infty} f(x)\,\delta(x-a) \,dx
= f(a)\tag{8.4.4}
\end{equation}
To Remember. The Dirac delta function can be used inside an integral to pick out the value of a function at any desired point.
