A completeness relation is a really fancy way of writing the identity operator. You can write a completeness relation for ANY quantum system in ANY complete basis.
For spin 1/2 systems in the \(z\)-basis, we have
\begin{align}
1&=\left|{+}\right\rangle \left\langle {+}\right|\, +\,\left|{-}\right\rangle \left\langle {-}\right|\\
&\doteq\begin{pmatrix}1\\0\end{pmatrix}\begin{pmatrix}1&0\end{pmatrix}
+\begin{pmatrix}0\\1\end{pmatrix}\begin{pmatrix}0&1\end{pmatrix}\\
&=\begin{pmatrix}1&0\\0&0\end{pmatrix}+\begin{pmatrix}0&0\\0&1\end{pmatrix}\\
&=\begin{pmatrix}1&0\\0&1\end{pmatrix}
\end{align}
The trick is to SUM the ket/bras over a complete set of basis states.
For the quantum ring in the energy basis, we have
\begin{align}
1&=\sum_{m=-\infty}^{\infty}\left|{m}\right\rangle \left\langle {m}\right|
\end{align}
or in the position basis, we have
\begin{align}
1&=\int_{0}^{2\pi}\left|{\phi}\right\rangle \left\langle {\phi}\right|\, r_0\, d\phi
\end{align}
- Write the completeness relations for the hydrogen atom in both the energy and the position bases.
- Briefly interpret the symbols in the following statements involving completeness relations for the hydrogen atom:
\begin{align}
\left|{\psi}\right\rangle &=\color{brown}{\left(\sum_{n=0}^{\infty}\sum_{\ell=0}^{n-1}\sum_{m=-\ell}^{\ell}
\left|{n\ell m}\right\rangle \left\langle {n\ell m}\right|\right)}\left|{\psi}\right\rangle \\
&=\sum_{n=0}^{\infty}\sum_{\ell=0}^{n-1}\sum_{m=-\ell}^{\ell}
\color{red}{\left|{n\ell m}\right\rangle c_{n\ell m}}\\
\left|{\psi}\right\rangle &=\color{blue}{\left(\int_{0}^{2\pi}\int_{0}^{\pi}\int_{0}^{\infty}
\left|{r\,\theta\,\phi}\right\rangle \left\langle {r\,\theta\,\phi}\right|\, r^2\sin\theta\,
dr\, d\theta\, d\phi\right)} \left|{\psi}\right\rangle \\
&=\int_{0}^{2\pi}\int_{0}^{\pi}\int_{0}^{\infty}
\left|{r\,\theta\,\phi}\right\rangle \left\langle {r\,\theta\,\phi}\middle|{\psi}\right\rangle \, r^2\sin\theta\,
dr\, d\theta\, d\phi\\
&=\int_{0}^{2\pi}\int_{0}^{\pi}\int_{0}^{\infty}
\color{red}{\left|{r\,\theta\,\phi}\right\rangle \psi(r, \theta, \phi)}\, r^2\sin\theta\,
dr\, d\theta\, d\phi\\
\end{align}
- Use the hydrogen atom completeness relations to find new formulas for the following expressions. When would you use these formulas?
\begin{align*}
c_{n\ell m}&=\left\langle {n\ell m}\middle|{\psi}\right\rangle \\
&=?\\
\psi(r, \theta, \phi)&=\left\langle {r\, \theta\, \phi}\middle|{\psi}\right\rangle \\
&=?
\end{align*}