From 9b12e1072d4662d3aaf4c3f8e0f0272c3c1a6ec8 Mon Sep 17 00:00:00 2001 From: Prefetch Date: Wed, 2 Jun 2021 20:14:06 +0200 Subject: Expand knowledge base --- content/know/concept/selection-rules/index.pdc | 546 +++++++++++++++++++++++++ 1 file changed, 546 insertions(+) create mode 100644 content/know/concept/selection-rules/index.pdc (limited to 'content/know/concept/selection-rules') diff --git a/content/know/concept/selection-rules/index.pdc b/content/know/concept/selection-rules/index.pdc new file mode 100644 index 0000000..5da97e7 --- /dev/null +++ b/content/know/concept/selection-rules/index.pdc @@ -0,0 +1,546 @@ +--- +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: + +$$\begin{aligned} + \boxed{ + \Delta m = 0 \:\:\mathrm{or}\: \pm 1 + } +\end{aligned}$$ + +
+ + + +
+ +Meanwhile, for the total angular momentum $\ell$ we have the following: + +$$\begin{aligned} + \boxed{ + \Delta \ell = \pm 1 + } +\end{aligned}$$ + +
+ + + +
+ + +## 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. -- cgit v1.2.3