---
title: "Selection rules"
firstLetter: "S"
publishDate: 2021-06-02
categories:
- Physics
- Quantum mechanics

date: 2021-05-29T14:42:08+02:00
draft: false
markup: pandoc
---

# Selection rules

In quantum mechanics, it is often necessary to evaluate
matrix elements of the following form,
where $\ell$ and $m$ respectively represent
the total angular momentum and its $z$-component:

$$\begin{aligned}
    \matrixel{f}{\hat{O}}{i}
    = \matrixel{n_f \ell_f m_f}{\hat{O}}{n_i \ell_i m_i}
\end{aligned}$$

Where $\hat{O}$ is an operator, $\ket{i}$ is an initial state, and
$\ket{f}$ is a final state (usually at least; $\ket{i}$ and $\ket{f}$
can be any states). **Selection rules** are requirements on the relations
between $\ell_i$, $\ell_f$, $m_i$ and $m_f$, which, if not met,
guarantee that the above matrix element is zero.
Note that $n_f$ and $n_i$ typically do not matter in this context,
so they will be omitted from now on.


## Parity rules

Let $\hat{O}$ denote any operator which is odd under spatial inversion
(parity):

$$\begin{aligned}
    \hat{\Pi}^\dagger \hat{O} \hat{\Pi} = - \hat{O}
\end{aligned}$$

Where $\hat{\Pi}$ is the parity operator.
We wrap this property of $\hat{O}$
in the states $\ket{\ell_f m_f}$ and $\ket{\ell_i m_i}$:

$$\begin{aligned}
    \matrixel{\ell_f m_f}{\hat{O}}{\ell_i m_i}
    &= - \matrixel{\ell_f m_f}{\hat{\Pi}^\dagger \hat{O} \hat{\Pi}}{\ell_i m_i}
    \\
    &= - \matrixel{\ell_f m_f}{(-1)^{\ell_f} \hat{O} (-1)^{\ell_i}}{\ell_i m_i}
    \\
    &= (-1)^{\ell_f + \ell_i + 1} \matrixel{\ell_f m_f}{\hat{O}}{\ell_i m_i}
\end{aligned}$$

Which clearly can only be true if the exponent is even,
so $\Delta \ell \equiv \ell_f - \ell_i$ must be odd.
This leads to the following selection rule,
often referred to as **Laporte's rule**:

$$\begin{aligned}
    \boxed{
        \Delta \ell \:\:\text{is odd}
    }
\end{aligned}$$

If this is not the case,
then the only possible way that the above equation can be satisfied
is if the matrix element vanishes $\matrixel{\ell_f m_f}{\hat{O}}{\ell_i m_i} = 0$.
We can derive an analogous rule for
any operator $\hat{E}$ which is even under parity:

$$\begin{aligned}
    \hat{\Pi}^\dagger \hat{E} \hat{\Pi} = \hat{E}
    \quad \implies \quad
    \boxed{
        \Delta \ell \:\:\text{is even}
    }
\end{aligned}$$


## Dipole rules

Arguably the most common operator found in such matrix elements
is a position vector operator, like $\vu{r}$ or $\hat{x}$,
and the associated selection rules are known as **dipole rules**.

For the $z$-component of angular momentum $m$ we have the following:

$$\begin{aligned}
    \boxed{
        \Delta m = 0 \:\:\mathrm{or}\: \pm 1
    }
\end{aligned}$$

<div class="accordion">
<input type="checkbox" id="proof-dipole-m"/>
<label for="proof-dipole-m">Proof</label>
<div class="hidden">
<label for="proof-dipole-m">Proof.</label>
We know that the angular momentum $z$-component operator $\hat{L}_z$ satisfies:

$$\begin{aligned}
    \comm*{\hat{L}_z}{\hat{x}} = i \hbar \hat{y}
    \qquad
    \comm*{\hat{L}_z}{\hat{y}} = - i \hbar \hat{x}
    \qquad
    \comm*{\hat{L}_z}{\hat{z}} = 0
\end{aligned}$$

We take the first relation,
and wrap it in $\bra{\ell_f m_f}$ and $\ket{\ell_i m_i}$, giving:

$$\begin{aligned}
    i \hbar \matrixel{\ell_f m_f}{\hat{y}}{\ell_i m_i}
    &= \matrixel{\ell_f m_f}{\hat{L}_z \hat{x}}{\ell_i m_i} - \matrixel{\ell_f m_f}{\hat{x} \hat{L}_z}{\ell_i m_i}
    \\
    &= \hbar m_f \matrixel{\ell_f m_f}{\hat{x}}{\ell_i m_i} - \hbar m_i \matrixel{\ell_f m_f}{\hat{x}}{\ell_i m_i}
    \\
    &= \hbar (m_f - m_i) \matrixel{\ell_f m_f}{\hat{x}}{\ell_i m_i}
\end{aligned}$$

Next, we do the same thing with the second relation, for $[\hat{L}_z, \hat{y}]$, giving:

$$\begin{aligned}
    - i \hbar \matrixel{\ell_f m_f}{\hat{x}}{\ell_i m_i}
    &= \matrixel{\ell_f m_f}{\hat{L}_z \hat{y}}{\ell_i m_i} - \matrixel{\ell_f m_f}{\hat{y} \hat{L}_z}{\ell_i m_i}
    \\
    &= \hbar m_f \matrixel{\ell_f m_f}{\hat{y}}{\ell_i m_i} - \hbar m_i \matrixel{\ell_f m_f}{\hat{y}}{\ell_i m_i}
    \\
    &= \hbar (m_f - m_i) \matrixel{\ell_f m_f}{\hat{y}}{\ell_i m_i}
\end{aligned}$$

Respectively isolating the two above results for $\hat{x}$ and $\hat{y}$,
we arrive at these equations:

$$\begin{aligned}
    \matrixel{\ell_f m_f}{\hat{x}}{\ell_i m_i}
    &= i (m_f - m_i) \matrixel{\ell_f m_f}{\hat{y}}{\ell_i m_i}
    \\
    \matrixel{\ell_f m_f}{\hat{y}}{\ell_i m_i}
    &= - i (m_f - m_i) \matrixel{\ell_f m_f}{\hat{x}}{\ell_i m_i}
\end{aligned}$$

By inserting the first into the second,
we find (part of) the selection rule:

$$\begin{aligned}
    \matrixel{\ell_f m_f}{\hat{y}}{\ell_i m_i}
    &= (m_f - m_i)^2 \matrixel{\ell_f m_f}{\hat{y}}{\ell_i m_i}
\end{aligned}$$

This can only be true if $\Delta m = \pm 1$,
unless the inner products of $\hat{x}$ and $\hat{y}$ are zero,
in which case we cannot say anything about $\Delta m$ yet.
Assuming the latter, we take the inner product of
the commutator $\comm*{\hat{L}_z}{\hat{z}} = 0$, and find:

$$\begin{aligned}
    0
    &= \matrixel{\ell_f m_f}{\hat{L}_z \hat{z}}{\ell_i m_i} - \matrixel{\ell_f m_f}{\hat{z} \hat{L}_z}{\ell_i m_i}
    \\
    &= \hbar m_f \matrixel{\ell_f m_f}{\hat{z}}{\ell_i m_i} - \hbar m_i \matrixel{\ell_f m_f}{\hat{z}}{\ell_i m_i}
    \\
    &= \hbar (m_f - m_i) \matrixel{\ell_f m_f}{\hat{z}}{\ell_i m_i}
\end{aligned}$$

If $\matrixel{f}{\hat{z}}{i} \neq 0$, we require $\Delta m = 0$.
The previous requirement was $\Delta m = \pm 1$,
implying that $\matrixel{f}{\hat{x}}{i} = \matrixel{f}{\hat{y}}{i} = 0$
whenever $\matrixel{f}{\hat{z}}{i} \neq 0$.
Only if $\matrixel{f}{\hat{z}}{i} = 0$
does the previous rule $\Delta m = \pm 1$ hold,
in which case the inner products of $\hat{x}$ and $\hat{y}$ are nonzero.
</div>
</div>

Meanwhile, for the total angular momentum $\ell$ we have the following:

$$\begin{aligned}
    \boxed{
        \Delta \ell = \pm 1
    }
\end{aligned}$$

<div class="accordion">
<input type="checkbox" id="proof-dipole-l"/>
<label for="proof-dipole-l">Proof</label>
<div class="hidden">
<label for="proof-dipole-l">Proof.</label>
We start from the following relation
(which is already quite a chore to prove):

$$\begin{aligned}
    \comm{\hat{L}^2}{\comm*{\hat{L}^2}{\vu{r}}}
    = 2 \hbar^2 (\vu{r} \hat{L}^2 + \hat{L}^2 \vu{r})
\end{aligned}$$

<div class="accordion">
<input type="checkbox" id="proof-dipole-l-comm"/>
<label for="proof-dipole-l-comm">Proof</label>
<div class="hidden">
<label for="proof-dipole-l-comm">Proof.</label>
To begin with, we want to find the commutator of $\hat{L}^2$ and $\hat{x}$:

$$\begin{aligned}
    \comm*{\hat{L}^2}{\hat{x}}
    &= \comm*{\hat{L}_x^2}{\hat{x}} + \comm*{\hat{L}_y^2}{\hat{x}} + \comm*{\hat{L}_z^2}{\hat{x}}
    = \comm*{\hat{L}_y^2}{\hat{x}} + \comm*{\hat{L}_z^2}{\hat{x}}
    \\
    &= \hat{L}_y \comm*{\hat{L}_y}{\hat{x}} + \comm*{\hat{L}_y}{\hat{x}} \hat{L}_y
    + \hat{L}_z \comm*{\hat{L}_z}{\hat{x}} + \comm*{\hat{L}_z}{\hat{x}} \hat{L}_z
\end{aligned}$$

Evaluating these commutators gives us:

$$\begin{aligned}
    \comm*{\hat{L}_y}{\hat{x}}
    &= \comm*{\hat{z} \hat{p}_x}{\hat{x}} - \comm*{\hat{x} \hat{p}_z}{\hat{x}}
    = \hat{z} \comm*{\hat{p}_x}{\hat{x}} + \comm*{\hat{z}}{\hat{x}} \hat{p}_x
    - \hat{x} \comm*{\hat{p}_z}{\hat{x}} - \comm*{\hat{x}}{\hat{x}} \hat{p}_z
    = - i \hbar \hat{z}
    \\
    \comm*{\hat{L}_z}{\hat{x}}
    &= \comm*{\hat{x} \hat{p}_y}{\hat{x}} - \comm*{\hat{y} \hat{p}_x}{\hat{x}}
    = \hat{x} \comm*{\hat{p}_y}{\hat{x}} + \comm*{\hat{x}}{\hat{x}} \hat{p}_y
    - \hat{y} \comm*{\hat{p}_x}{\hat{x}} - \comm*{\hat{y}}{\hat{x}} \hat{p}_x
    = i \hbar \hat{y}
\end{aligned}$$

Which we then insert back into the original equation, yielding:

$$\begin{aligned}
    \comm*{\hat{L}^2}{\hat{x}}
    &= i \hbar (- \hat{L}_y \hat{z} - \hat{z} \hat{L}_y + \hat{L}_z \hat{y} + \hat{y} \hat{L}_z)
\end{aligned}$$

This can be simplified by introducing some more commutators:

$$\begin{aligned}
    \comm*{\hat{L}^2}{\hat{x}}
    &= i \hbar \big( \!-\! ( \comm*{\hat{L}_y}{\hat{z}} + \hat{z} \hat{L}_y ) - \hat{z} \hat{L}_y
    + ( \comm*{\hat{L}_z}{\hat{y}} + \hat{y} \hat{L}_z ) + \hat{y} \hat{L}_z \big)
\end{aligned}$$

Evaluating these commutators gives us:

$$\begin{aligned}
    \comm*{\hat{L}_y}{\hat{z}}
    &= \comm*{\hat{z} \hat{p}_x}{\hat{z}} - \comm*{\hat{x} \hat{p}_z}{\hat{z}}
    = \hat{z} \comm*{\hat{p}_x}{\hat{z}} + \comm*{\hat{z}}{\hat{z}} \hat{p}_x
    - \hat{x} \comm*{\hat{p}_z}{\hat{z}} - \comm*{\hat{x}}{\hat{z}} \hat{p}_z
    = i \hbar \hat{x}
    \\
    \comm*{\hat{L}_z}{\hat{y}}
    &= \comm*{\hat{x} \hat{p}_y}{\hat{y}} - \comm*{\hat{y} \hat{p}_x}{\hat{y}}
    = \hat{x} \comm*{\hat{p}_y}{\hat{y}} + \comm*{\hat{x}}{\hat{y}} \hat{p}_y
    - \hat{y} \comm*{\hat{p}_x}{\hat{y}} - \comm*{\hat{y}}{\hat{y}} \hat{p}_x
    = - i \hbar \hat{x}
\end{aligned}$$

Substituting these then leads us to the first milestone of this proof:

$$\begin{aligned}
    \comm*{\hat{L}^2}{\hat{x}}
    &= i \hbar \big( \!-\! i \hbar \hat{x} - \hat{z} \hat{L}_y - \hat{z} \hat{L}_y
    - i \hbar \hat{x} + \hat{y} \hat{L}_z + \hat{y} \hat{L}_z \big)
    \\
    &= 2 i \hbar (\hat{y} \hat{L}_z - \hat{z} \hat{L}_y - i \hbar \hat{x})
\end{aligned}$$

Repeating this process for $\comm*{\hat{L}^2}{\hat{y}}$ and $\comm*{\hat{L}^2}{\hat{z}}$,
we find analogous expressions:

$$\begin{aligned}
    \comm*{\hat{L}^2}{\hat{y}}
    &= 2 i \hbar (\hat{z} \hat{L}_x - \hat{x} \hat{L}_z - i \hbar \hat{y})
    \\
    \comm*{\hat{L}^2}{\hat{z}}
    &= 2 i \hbar (\hat{x} \hat{L}_y - \hat{y} \hat{L}_x - i \hbar \hat{z})
\end{aligned}$$

Next, we take the commutator with $\hat{L}^2$ of the commutator we just found:

$$\begin{aligned}
    \comm*{\hat{L}^2}{\comm*{\hat{L}^2}{\hat{x}}}
    &= 2 i \hbar \big(\comm*{\hat{L}^2}{\hat{y} \hat{L}_z} - \comm*{\hat{L}^2}{\hat{z} \hat{L}_y} - i \hbar \comm*{\hat{L}^2}{\hat{x}}\big)
    \\
    &= 2 i \hbar \big( \hat{y} \comm*{\hat{L}^2}{\hat{L}_z} + \comm*{\hat{L}^2}{\hat{y}} \hat{L}_z
    - \hat{z} \comm*{\hat{L}^2}{\hat{L}_y} - \comm*{\hat{L}^2}{\hat{z}} \hat{L}_y
    - i \hbar \comm*{\hat{L}^2}{\hat{x}} \big)
\end{aligned}$$

Where we used that $\comm*{\hat{L}^2}{\hat{L}_y} = \comm*{\hat{L}^2}{\hat{L}_z} = 0$.
The other commutators look familiar:

$$\begin{aligned}
    \comm*{\hat{L}^2}{\comm*{\hat{L}^2}{\hat{x}}}
    &= 2 i \hbar \big( \comm*{\hat{L}^2}{\hat{y}} \hat{L}_z
    - \comm*{\hat{L}^2}{\hat{z}} \hat{L}_y
    - i \hbar \comm*{\hat{L}^2}{\hat{x}} \big)
\end{aligned}$$

By inserting the expressions we found earlier for these commutators, we get:

$$\begin{aligned}
    \comm*{\hat{L}^2}{\comm*{\hat{L}^2}{\hat{x}}}
    %&= - 2 \hbar^2 \big( 2 (\hat{z} \hat{L}_x - \hat{x} \hat{L}_z - i \hbar \hat{y}) \hat{L}_z
    %- 2 (\hat{x} \hat{L}_y - \hat{y} \hat{L}_x - i \hbar \hat{z}) \hat{L}_y
    %- (\hat{L}^2 \hat{x} - \hat{x} \hat{L}^2) \big)
    %\\
    &= - 4 \hbar^2 \big( \hat{z} \hat{L}_x \hat{L}_z - \hat{x} \hat{L}_z^2  - i \hbar \hat{y} \hat{L}_z
    + \hat{y} \hat{L}_x \hat{L}_y - \hat{x} \hat{L}_y^2 + i \hbar \hat{z} \hat{L}_y \big) \\
    &\qquad\qquad + 2 \hbar^2 \big( \hat{L}^2 \hat{x} - \hat{x} \hat{L}^2 \big)
\end{aligned}$$

Substituting the well-known commutators
$i \hbar \hat{L}_y = \comm*{\hat{L}_z}{\hat{L}_x}$ and
$i \hbar \hat{L}_z = \comm*{\hat{L}_x}{\hat{L}_y}$:

$$\begin{aligned}
    \comm*{\hat{L}^2}{\comm*{\hat{L}^2}{\hat{x}}}
    &= - 4 \hbar^2 \big( \hat{z} \hat{L}_x \hat{L}_z + \hat{y} \hat{L}_x \hat{L}_y
    - \hat{x} \hat{L}_y^2 - \hat{x} \hat{L}_z^2
    + \hat{z} \comm*{\hat{L}_z}{\hat{L}_x} - \hat{y} \comm*{\hat{L}_x}{\hat{L}_y} \big) \\
    &\qquad\qquad + 2 \hbar^2 \big( \hat{L}^2 \hat{x} - \hat{x} \hat{L}^2 \big)
    \\
    &= - 4 \hbar^2 \big( \hat{z} \hat{L}_z \hat{L}_x + \hat{y} \hat{L}_y \hat{L}_x
    - \hat{x} \hat{L}_y^2 - \hat{x} \hat{L}_z^2 \big)
    + 2 \hbar^2 \big( \hat{L}^2 \hat{x} - \hat{x} \hat{L}^2 \big)
\end{aligned}$$

By definition, $\hat{L}_x^2 + \hat{L}_y^2 + \hat{L}_z^2 = \hat{L}^2$,
which we use to arrive at:

$$\begin{aligned}
    \comm*{\hat{L}^2}{\comm*{\hat{L}^2}{\hat{x}}}
    %&= - 4 \hbar^2 \big( \hat{z} \hat{L}_z \hat{L}_x + \hat{y} \hat{L}_y \hat{L}_x + \hat{x} \hat{L}_x^2
    %- \hat{x} \hat{L}_x^2 - \hat{x} \hat{L}_y^2 - \hat{x} \hat{L}_z^2 \big)
    %+ 2 \hbar^2 \big( \hat{L}^2 \hat{x} - \hat{x} \hat{L}^2 \big)
    %\\
    &= - 4 \hbar^2 \big( \hat{z} \hat{L}_z \hat{L}_x + \hat{y} \hat{L}_y \hat{L}_x + \hat{x} \hat{L}_x^2 - \hat{x} \hat{L}^2 \big)
    + 2 \hbar^2 \big( \hat{L}^2 \hat{x} - \hat{x} \hat{L}^2 \big)
    \\
    &= - 4 \hbar^2 \big( \hat{z} \hat{L}_z \hat{L}_x + \hat{y} \hat{L}_y \hat{L}_x + \hat{x} \hat{L}_x^2 \big)
    + 2 \hbar^2 \big( \hat{L}^2 \hat{x} + \hat{x} \hat{L}^2 \big)
\end{aligned}$$

The second term is what we want to prove,
so the first term must vanish:

$$\begin{aligned}
    \hat{z} \hat{L}_z \hat{L}_x + \hat{y} \hat{L}_y \hat{L}_x + \hat{x} \hat{L}_x^2
    = (\vu{r} \cdot \vu{L}) \hat{L}_x
    = (\vu{r} \cdot (\vu{r} \cross \vu{p})) \hat{L}_x
    = (\vu{p} \cdot (\vu{r} \cross \vu{r})) \hat{L}_x
    = 0
\end{aligned}$$

Where $\vu{L} = \vu{r} \cross \vu{p}$ by definition,
and the cross product of a vector with itself is zero.

This process can be repeated for
$\comm*{\hat{L}^2}{\comm*{\hat{L}^2}{\hat{y}}}$ and
$\comm*{\hat{L}^2}{\comm*{\hat{L}^2}{\hat{z}}}$,
leading us to:

$$\begin{aligned}
    \comm*{\hat{L}^2}{\comm*{\hat{L}^2}{\hat{x}}}
    &= 2 \hbar^2 (\hat{x} \hat{L}^2 + \hat{L}^2 \hat{x})
    \\
    \comm*{\hat{L}^2}{\comm*{\hat{L}^2}{\hat{y}}}
    &= 2 \hbar^2 (\hat{y} \hat{L}^2 + \hat{L}^2 \hat{y})
    \\
    \comm*{\hat{L}^2}{\comm*{\hat{L}^2}{\hat{z}}}
    &= 2 \hbar^2 (\hat{z} \hat{L}^2 + \hat{L}^2 \hat{z})
\end{aligned}$$

At last, this brings us to the desired equation for $\comm*{\hat{L}^2}{\comm*{\hat{L}^2}{\vu{r}}}$,
with $\vu{r} = (\hat{x}, \hat{y}, \hat{z})$.
</div>
</div>

We then multiply this relation by $\bra{f} = \bra{\ell_f m_f}$ on the left
and $\ket{i} = \ket{\ell_i m_i}$ on the right,
so the right-hand side becomes:

$$\begin{aligned}
    2 \hbar^2 \matrixel{f}{\vu{r} \hat{L}^2 \!\!+\! \hat{L}^2 \!\vu{r}}{i}
    %\matrixel{\ell_f m_f}{\comm*{\hat{L}^2}{\comm*{\hat{L}^2}{\vu{r}}}}{\ell_i m_i}
    &= 2 \hbar^2 \big( \matrixel{f}{\vu{r} \hat{L}^2}{i} + \matrixel{f}{\hat{L}^2 \vu{r}}{i} \big)
    \\
    &= 2 \hbar^2 \big( \hbar^2 \ell_i (\ell_i \!+\! 1) \matrixel{f}{\vu{r}}{i}
    + \hbar^2 \ell_f (\ell_f \!+\! 1) \matrixel{f}{\vu{r}}{i} \big)
    \\
    &= 2 \hbar^4 \big(\ell_f (\ell_f \!+\! 1) + \ell_i (\ell_i \!+\! 1)\big) \matrixel{f}{\vu{r}}{i}
\end{aligned}$$

And, likewise, the left-hand side becomes:

$$\begin{aligned}
    \matrixel{f}{\comm*{\hat{L}^2}{\comm*{\hat{L}^2}{\vu{r}}}}{i}
    &= \matrixel{f}{\hat{L}^2 \comm*{\hat{L}^2}{\vu{r}}}{i}
    - \matrixel{f}{\comm*{\hat{L}^2}{\vu{r}} \hat{L}^2}{i}
    \\
    &= \hbar^2 \ell_f (\ell_f \!+\! 1) \matrixel{f}{\comm*{\hat{L}^2}{\vu{r}}}{i}
    - \hbar^2 \ell_i (\ell_i \!+\! 1) \matrixel{f}{\comm*{\hat{L}^2}{\vu{r}}}{i}
    \\
    &= \hbar^2 \big( \ell_f (\ell_f \!+\! 1) - \ell_i (\ell_i \!+\! 1) \big) \matrixel{f}{\comm*{\hat{L}^2}{\vu{r}}}{i}
    \\
    &= \hbar^2 \big( \ell_f (\ell_f \!+\! 1) - \ell_i (\ell_i \!+\! 1) \big)
    \big( \matrixel{f}{\hat{L}^2 \vu{r}}{i} - \matrixel{f}{\vu{r} \hat{L}^2}{i} \big)
    \\
    &= \hbar^4 \big( \ell_f (\ell_f \!+\! 1) - \ell_i (\ell_i \!+\! 1) \big)^2 \matrixel{f}{\vu{r}}{i}
\end{aligned}$$

Obviously, both sides are equal to each other,
leading to the following equation:

$$\begin{aligned}
    2 \ell_f (\ell_f \!+\! 1) + 2 \ell_i (\ell_i \!+\! 1)
    &= \big( \ell_f (\ell_f \!+\! 1) - \ell_i (\ell_i \!+\! 1) \big)^2
\end{aligned}$$

To proceed, we rewrite the right-hand side like so:

$$\begin{aligned}
    \big( \ell_f (\ell_f \!+\! 1) - \ell_i (\ell_i \!+\! 1) \big)^2
    &= \big( \ell_f^2 - \ell_i^2 + \ell_f - \ell_i \big)^2
    \\
    &= \big( (\ell_f + \ell_i) (\ell_f - \ell_i) + (\ell_f - \ell_i) \big)^2
    \\
    &= (\ell_f + \ell_i)^2 (\ell_f - \ell_i)^2 + 2 (\ell_f + \ell_i) (\ell_f - \ell_i)^2 + (\ell_f - \ell_i)^2
    \\
    &= \big( (\ell_f + \ell_i)^2 + 2 (\ell_f + \ell_i) + 1 \big) (\ell_f - \ell_i)^2
    \\
    &= (\ell_f + \ell_i + 1)^2 (\ell_f - \ell_i)^2
\end{aligned}$$

And then we do the same to the left-hand side, yielding:

$$\begin{aligned}
    2 (\ell_f^2 + \ell_i^2 + \ell_f + \ell_i)
    &= 2 \ell_f^2 + 2 \ell_i^2 + 2 \ell_f \ell_i - 2 \ell_f \ell_i + 2 \ell_f + 2 \ell_i + 1 - 1
    \\
    &= (\ell_f + \ell_i + 1)^2 + \ell_f^2 + \ell_i^2 - 2 \ell_f \ell_i - 1
    \\
    &= (\ell_f + \ell_i + 1)^2 + (\ell_f - \ell_i)^2 - 1
\end{aligned}$$

The equation above has thus been simplified to the following form:

$$\begin{aligned}
    (\ell_f + \ell_i + 1)^2 + (\ell_f - \ell_i)^2 - 1
    &= (\ell_f + \ell_i + 1)^2 (\ell_f - \ell_i)^2
\end{aligned}$$

Rearranging yields a product equal to zero,
so one or both of the factors must vanish:

$$\begin{aligned}
    0
    &= (\ell_f + \ell_i + 1)^2 (\ell_f - \ell_i)^2 - (\ell_f + \ell_i + 1)^2 - (\ell_f - \ell_i)^2 + 1
    \\
    &= \big( (\ell_f + \ell_i + 1)^2 - 1 \big) \big( (\ell_f - \ell_i)^2 - 1 \big)
\end{aligned}$$

The first factor is zero if $\ell_f = \ell_i = 0$,
in which case the matrix element $\matrixel{f}{\vu{r}}{i} = 0$ anyway.
The other, non-trivial option is therefore:

$$\begin{aligned}
    (\ell_f - \ell_i)^2
    = 1
\end{aligned}$$
</div>
</div>


## Superselection rule

Selection rules need not always be about atomic electron transitions.
According to the **principle of indistinguishability**,
permutating identical particles never leads to an observable difference.
In other words, the particles are fundamentally indistinguishable,
so for any observable $\hat{O}$ and multi-particle state $\ket{\Psi}$, we can say:

$$\begin{aligned}
    \matrixel{\Psi}{\hat{O}}{\Psi}
    = \matrixel*{\hat{P} \Psi}{\hat{O}}{\hat{P} \Psi}
\end{aligned}$$

Where $\hat{P}$ is an arbitrary permutation operator.
Indistinguishability implies that $\comm*{\hat{P}}{\hat{O}} = 0$
for all $\hat{O}$ and $\hat{P}$,
which lets us prove the above equation, using that $\hat{P}$ is unitary:

$$\begin{aligned}
    \matrixel*{\hat{P} \Psi}{\hat{O}}{\hat{P} \Psi}
    = \matrixel{\Psi}{\hat{P}^{-1} \hat{O} \hat{P}}{\Psi}
    = \matrixel{\Psi}{\hat{P}^{-1} \hat{P} \hat{O}}{\Psi}
    = \matrixel{\Psi}{\hat{O}}{\Psi}
\end{aligned}$$

Consider a symmetric state $\ket{s}$ and an antisymmetric state $\ket{a}$
(see [Pauli exclusion principle](/know/concept/pauli-exclusion-principle/)),
which obey the following for a permutation $\hat{P}$:

$$\begin{aligned}
    \hat{P} \ket{s}
    = \ket{s}
    \qquad
    \hat{P} \ket{a}
    = - \ket{a}
\end{aligned}$$

Any obervable $\hat{O}$ then satisfies the equation below,
again thanks to the fact that $\hat{P} = \hat{P}^{-1}$:

$$\begin{aligned}
    \matrixel{s}{\hat{O}}{a}
    = \matrixel*{\hat{P} s}{\hat{O}}{a}
    = \matrixel{s}{\hat{P}^{-1} \hat{O}}{a}
    = \matrixel{s}{\hat{O} \hat{P}}{a}
    = \matrixel*{s}{\hat{O}}{\hat{P} a}
    = - \matrixel{s}{\hat{O}}{a}
\end{aligned}$$

This leads us to the **superselection rule**,
which states that there can never be any interference
between states of different permutation symmetry:

$$\begin{aligned}
    \boxed{
        \matrixel{s}{\hat{O}}{a}
        = 0
    }
\end{aligned}$$



## References
1.  D.J. Griffiths, D.F. Schroeter,
    *Introduction to quantum mechanics*, 3rd edition,
    Cambridge.