Skip to main content
Contents Index Dark Mode 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 24.6 Normalization of States on a Ring
In
Section 24.5 , we found the energy and angular momentum eigenstates for a quantum particle on a ring to be
\begin{equation}
\Phi_m(\phi)\defeq N\,e^{im\phi}\tag{24.6.1}
\end{equation}
where
\begin{equation}
m\in\{0,\pm1,\pm2,...\}\tag{24.6.2}
\end{equation}
and \(N\) is a normalization constant.
As usual, we choose the normalization
\(N\) in
(24.5.2) so that, if the particle is in an eigenstate, the probability of finding it somewhere on the ring is unity.
\begin{align}
1 \amp= \int_0^{2\pi} \Phi_m^*(\phi)\, \Phi_m(\phi)\, r_0\, d\phi\tag{24.6.3}\\
\amp = \int_0^{2\pi} N^* e^{-im\phi}\, N e^{im\phi}\, r_0 d\phi\tag{24.6.4}\\
\amp = 2\pi r_0 \vert N\vert^2\tag{24.6.5}\\
\amp\qquad \Rightarrow\quad N=\frac{1}{\sqrt{2\pi r_0}}\tag{24.6.6}
\end{align}
where we have chosen the arbitrary phase in \(N\) to be one.
