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Contents Index Dark Mode Prev Up Next \(\newcommand{\vf}[1]{\mathbf{\boldsymbol{\vec{#1}}}}
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\)
Section 22.6 Normalization of States on a Ring
In
Section 22.5 , 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{22.6.1}
\end{equation}
where
\begin{equation}
m\in\{0,\pm1,\pm2,...\}\tag{22.6.2}
\end{equation}
and \(N\) is a normalization constant.
As usual, we choose the normalization
\(N\) in
(22.5.2) 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{22.6.3}\\
\amp = \int_0^{2\pi} N^* e^{-im\phi}\, N e^{im\phi}\, r_0 d\phi\tag{22.6.4}\\
\amp = 2\pi r_0 \vert N\vert^2\tag{22.6.5}\\
\amp\qquad \Rightarrow\quad N=\frac{1}{\sqrt{2\pi r_0}}\tag{22.6.6}
\end{align}
where we have chosen the arbitrary phase in \(N\) to be one.
