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Diffstat (limited to 'source/know/concept/selection-rules/index.md')
| -rw-r--r-- | source/know/concept/selection-rules/index.md | 46 |
1 files changed, 18 insertions, 28 deletions
diff --git a/source/know/concept/selection-rules/index.md b/source/know/concept/selection-rules/index.md index 373486e..620e345 100644 --- a/source/know/concept/selection-rules/index.md +++ b/source/know/concept/selection-rules/index.md @@ -25,6 +25,7 @@ between $$\ell_i$$, $$\ell_f$$, $$m_i$$ and $$m_f$$, which, if not met, guarantee that the above matrix element is zero. + ## Parity rules Let $$\hat{O}$$ denote any operator which is odd under spatial inversion @@ -73,6 +74,7 @@ $$\begin{aligned} \end{aligned}$$ + ## Dipole rules Arguably the most common operator found in such matrix elements @@ -87,11 +89,8 @@ $$\begin{aligned} } \end{aligned}$$ -<div class="accordion"> -<input type="checkbox" id="proof-dipole-m"/> -<label for="proof-dipole-m">Proof</label> -<div class="hidden" markdown="1"> -<label for="proof-dipole-m">Proof.</label> + +{% include proof/start.html id="proof-dipole-m" -%} We know that the angular momentum $$z$$-component operator $$\hat{L}_z$$ satisfies: $$\begin{aligned} @@ -166,8 +165,8 @@ whenever $$\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}$$ are nonzero. -</div> -</div> +{% include proof/end.html id="proof-dipole-m" %} + Meanwhile, for the total angular momentum $$\ell$$ we have the following: @@ -177,11 +176,8 @@ $$\begin{aligned} } \end{aligned}$$ -<div class="accordion"> -<input type="checkbox" id="proof-dipole-l"/> -<label for="proof-dipole-l">Proof</label> -<div class="hidden" markdown="1"> -<label for="proof-dipole-l">Proof.</label> + +{% include proof/start.html id="proof-dipole-l" -%} We start from the following relation (which is already quite a chore to prove): @@ -190,11 +186,8 @@ $$\begin{aligned} = 2 \hbar^2 (\vu{r} \hat{L}^2 + \hat{L}^2 \vu{r}) \end{aligned}$$ -<div class="accordion"> -<input type="checkbox" id="proof-dipole-l-comm"/> -<label for="proof-dipole-l-comm">Proof</label> -<div class="hidden" markdown="1"> -<label for="proof-dipole-l-comm">Proof.</label> + +{% include proof/start.html id="proof-dipole-l-commutator" -%} To begin with, we want to find the commutator of $$\hat{L}^2$$ and $$\hat{x}$$: $$\begin{aligned} @@ -364,8 +357,8 @@ $$\begin{aligned} At last, this brings us to the desired equation for $$\comm{\hat{L}^2}{\comm{\hat{L}^2}{\vu{r}}}$$, with $$\vu{r} = (\hat{x}, \hat{y}, \hat{z})$$. -</div> -</div> +{% include proof/end.html id="proof-dipole-l-commutator" %} + We then multiply this relation by $$\Bra{f} = \Bra{\ell_f m_f}$$ on the left and $$\Ket{i} = \Ket{\ell_i m_i}$$ on the right, @@ -458,9 +451,8 @@ $$\begin{aligned} (\ell_f - \ell_i)^2 = 1 \end{aligned}$$ +{% include proof/end.html id="proof-dipole-l" %} -</div> -</div> ## Rotational rules @@ -502,11 +494,8 @@ $$\begin{aligned} } \end{aligned}$$ -<div class="accordion"> -<input type="checkbox" id="proof-rot-scalar"/> -<label for="proof-rot-scalar">Proof</label> -<div class="hidden" markdown="1"> -<label for="proof-rot-scalar">Proof.</label> + +{% include proof/start.html id="proof-rotation-scalar" -%} Firstly, we look at the commutator of $$\hat{s}$$ with the $$z$$-component $$\hat{L}_z$$: $$\begin{aligned} 0 @@ -578,8 +567,8 @@ $$\begin{aligned} Which means that the value of the matrix element does not depend on $$m_i$$ (or $$m_f$$) at all. -</div> -</div> +{% include proof/end.html id="proof-rotation-scalar" %} + Similarly, given a general (pseudo)vector operator $$\vu{V}$$, which, by nature, must satisfy the following commutation relations, @@ -631,6 +620,7 @@ $$\begin{gathered} \end{gathered}$$ + ## Superselection rule Selection rules are not always about atomic electron transitions, or angular momenta even. |