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update week 36
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mhjensen committed Sep 1, 2024
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77 changes: 77 additions & 0 deletions doc/pub/week36/html/week36-bs.html
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None,
'two-body-operators-in-second-quantization'),
('More operations', 2, None, 'more-operations'),
('Summing over all terms', 2, None, 'summing-over-all-terms'),
('Introducing second quantization',
2,
None,
'introducing-second-quantization'),
('Inserting back', 2, None, 'inserting-back'),
('Removing restrictions', 2, None, 'removing-restrictions'),
('Changing summation indices',
2,
None,
'changing-summation-indices'),
('Using the new operator expressions',
2,
None,
'using-the-new-operator-expressions'),
('Two-body state', 2, None, 'two-body-state'),
('Expectation value', 2, None, 'expectation-value'),
('Final expression', 2, None, 'final-expression'),
('Rewriting the two-body operator',
2,
None,
'rewriting-the-two-body-operator'),
('Antisymmetrized matrix elements',
2,
None,
'antisymmetrized-matrix-elements'),
("Wick's theorem", 2, None, 'wick-s-theorem')]}
end of tocinfo -->

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<!-- navigation toc: --> <li><a href="#explicit-results" style="font-size: 80%;">Explicit results</a></li>
<!-- navigation toc: --> <li><a href="#two-body-operators-in-second-quantization" style="font-size: 80%;">Two-body operators in second quantization</a></li>
<!-- navigation toc: --> <li><a href="#more-operations" style="font-size: 80%;">More operations</a></li>
<!-- navigation toc: --> <li><a href="#summing-over-all-terms" style="font-size: 80%;">Summing over all terms</a></li>
<!-- navigation toc: --> <li><a href="#introducing-second-quantization" style="font-size: 80%;">Introducing second quantization</a></li>
<!-- navigation toc: --> <li><a href="#inserting-back" style="font-size: 80%;">Inserting back</a></li>
<!-- navigation toc: --> <li><a href="#removing-restrictions" style="font-size: 80%;">Removing restrictions</a></li>
<!-- navigation toc: --> <li><a href="#changing-summation-indices" style="font-size: 80%;">Changing summation indices</a></li>
<!-- navigation toc: --> <li><a href="#using-the-new-operator-expressions" style="font-size: 80%;">Using the new operator expressions</a></li>
<!-- navigation toc: --> <li><a href="#two-body-state" style="font-size: 80%;">Two-body state</a></li>
<!-- navigation toc: --> <li><a href="#expectation-value" style="font-size: 80%;">Expectation value</a></li>
<!-- navigation toc: --> <li><a href="#final-expression" style="font-size: 80%;">Final expression</a></li>
<!-- navigation toc: --> <li><a href="#rewriting-the-two-body-operator" style="font-size: 80%;">Rewriting the two-body operator</a></li>
<!-- navigation toc: --> <li><a href="#antisymmetrized-matrix-elements" style="font-size: 80%;">Antisymmetrized matrix elements</a></li>
<!-- navigation toc: --> <li><a href="#wick-s-theorem" style="font-size: 80%;">Wick's theorem</a></li>

</ul>
Expand Down Expand Up @@ -521,6 +558,8 @@ <h2 id="more-operations" class="anchor">More operations </h2>

<p>where on the rhs we have a term for each distinct pairs. </p>

<!-- !split -->
<h2 id="summing-over-all-terms" class="anchor">Summing over all terms </h2>
<p>For the other terms on the rhs we obtain similar expressions and summing over all terms we obtain</p>
$$
\begin{align}
Expand All @@ -536,6 +575,10 @@ <h2 id="more-operations" class="anchor">More operations </h2>
\end{align}
$$


<!-- !split -->
<h2 id="introducing-second-quantization" class="anchor">Introducing second quantization </h2>

<p>We introduce second quantization via the relation</p>
$$
\begin{align}
Expand All @@ -549,6 +592,10 @@ <h2 id="more-operations" class="anchor">More operations </h2>
\end{align}
$$


<!-- !split -->
<h2 id="inserting-back" class="anchor">Inserting back </h2>

<p>Inserting this in \eqref{eq:2-34} gives</p>
$$
\begin{align}
Expand All @@ -571,6 +618,10 @@ <h2 id="more-operations" class="anchor">More operations </h2>
\end{align}
$$


<!-- !split -->
<h2 id="removing-restrictions" class="anchor">Removing restrictions </h2>

<p>Here we let \( \sum' \) indicate that the sums running over \( \alpha \) and \( \beta \) run over all
single-particle states, while the summations \( \gamma \) and \( \delta \)
run over all pairs of single-particle states. We wish to remove this restriction and since
Expand All @@ -594,6 +645,9 @@ <h2 id="more-operations" class="anchor">More operations </h2>

<p>where we have used the anti-commutation rules.</p>

<!-- !split -->
<h2 id="changing-summation-indices" class="anchor">Changing summation indices </h2>

<p>Changing the summation indices
\( \alpha \) and \( \beta \) in \eqref{eq:2-38b} we obtain
</p>
Expand All @@ -617,6 +671,9 @@ <h2 id="more-operations" class="anchor">More operations </h2>
\( \beta \), \( \gamma \) og \( \delta \).
</p>

<!-- !split -->
<h2 id="using-the-new-operator-expressions" class="anchor">Using the new operator expressions </h2>

<p>With this expression we can now verify that the second quantization form of \( \hat{H}_I \) in Eq.&nbsp;\eqref{eq:2-39}
results in the same matrix between two anti-symmetrized two-particle states as its corresponding coordinate
space representation. We have
Expand All @@ -631,6 +688,10 @@ <h2 id="more-operations" class="anchor">More operations </h2>
\end{equation}
$$


<!-- !split -->
<h2 id="two-body-state" class="anchor">Two-body state </h2>

<p>Using the commutation relations we get </p>
$$
\begin{align}
Expand All @@ -650,6 +711,10 @@ <h2 id="more-operations" class="anchor">More operations </h2>
\end{align}
$$


<!-- !split -->
<h2 id="expectation-value" class="anchor">Expectation value </h2>

<p>The vacuum expectation value of this product of operators becomes</p>
$$
\begin{align}
Expand All @@ -663,6 +728,10 @@ <h2 id="more-operations" class="anchor">More operations </h2>
\end{align}
$$


<!-- !split -->
<h2 id="final-expression" class="anchor">Final expression </h2>

<p>Insertion of
Eq.&nbsp;\eqref{eq:2-42b} in Eq.&nbsp;\eqref{eq:2-40} results in
</p>
Expand All @@ -676,6 +745,10 @@ <h2 id="more-operations" class="anchor">More operations </h2>
\end{align}
$$


<!-- !split -->
<h2 id="rewriting-the-two-body-operator" class="anchor">Rewriting the two-body operator </h2>

<p>The two-body operator can also be expressed in terms of the anti-symmetrized matrix elements we discussed previously as</p>
$$
\begin{align}
Expand All @@ -689,6 +762,10 @@ <h2 id="more-operations" class="anchor">More operations </h2>
\end{align}
$$


<!-- !split -->
<h2 id="antisymmetrized-matrix-elements" class="anchor">Antisymmetrized matrix elements </h2>

<p>The factors in front of the operator, either \( \frac{1}{4} \) or
\( \frac{1}{2} \) tells whether we use antisymmetrized matrix elements or not.
</p>
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43 changes: 43 additions & 0 deletions doc/pub/week36/html/week36-reveal.html
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Expand Up @@ -607,7 +607,10 @@ <h2 id="more-operations">More operations </h2>
<p>&nbsp;<br>

<p>where on the rhs we have a term for each distinct pairs. </p>
</section>

<section>
<h2 id="summing-over-all-terms">Summing over all terms </h2>
<p>For the other terms on the rhs we obtain similar expressions and summing over all terms we obtain</p>
<p>&nbsp;<br>
$$
Expand All @@ -624,6 +627,10 @@ <h2 id="more-operations">More operations </h2>
\end{align}
$$
<p>&nbsp;<br>
</section>

<section>
<h2 id="introducing-second-quantization">Introducing second quantization </h2>

<p>We introduce second quantization via the relation</p>
<p>&nbsp;<br>
Expand All @@ -639,6 +646,10 @@ <h2 id="more-operations">More operations </h2>
\end{align}
$$
<p>&nbsp;<br>
</section>

<section>
<h2 id="inserting-back">Inserting back </h2>

<p>Inserting this in <a href="#mjx-eqn-20">(20)</a> gives</p>
<p>&nbsp;<br>
Expand All @@ -663,6 +674,10 @@ <h2 id="more-operations">More operations </h2>
\end{align}
$$
<p>&nbsp;<br>
</section>

<section>
<h2 id="removing-restrictions">Removing restrictions </h2>

<p>Here we let \( \sum' \) indicate that the sums running over \( \alpha \) and \( \beta \) run over all
single-particle states, while the summations \( \gamma \) and \( \delta \)
Expand Down Expand Up @@ -690,6 +705,10 @@ <h2 id="more-operations">More operations </h2>
<p>&nbsp;<br>

<p>where we have used the anti-commutation rules.</p>
</section>

<section>
<h2 id="changing-summation-indices">Changing summation indices </h2>

<p>Changing the summation indices
\( \alpha \) and \( \beta \) in <a href="#mjx-eqn-25">(25)</a> we obtain
Expand Down Expand Up @@ -717,6 +736,10 @@ <h2 id="more-operations">More operations </h2>
<p>where we sum freely over all single-particle states \( \alpha \),
\( \beta \), \( \gamma \) og \( \delta \).
</p>
</section>

<section>
<h2 id="using-the-new-operator-expressions">Using the new operator expressions </h2>

<p>With this expression we can now verify that the second quantization form of \( \hat{H}_I \) in Eq.&nbsp;<a href="#mjx-eqn-27">(27)</a>
results in the same matrix between two anti-symmetrized two-particle states as its corresponding coordinate
Expand All @@ -733,6 +756,10 @@ <h2 id="more-operations">More operations </h2>
\end{equation}
$$
<p>&nbsp;<br>
</section>

<section>
<h2 id="two-body-state">Two-body state </h2>

<p>Using the commutation relations we get </p>
<p>&nbsp;<br>
Expand All @@ -754,6 +781,10 @@ <h2 id="more-operations">More operations </h2>
\end{align}
$$
<p>&nbsp;<br>
</section>

<section>
<h2 id="expectation-value">Expectation value </h2>

<p>The vacuum expectation value of this product of operators becomes</p>
<p>&nbsp;<br>
Expand All @@ -769,6 +800,10 @@ <h2 id="more-operations">More operations </h2>
\end{align}
$$
<p>&nbsp;<br>
</section>

<section>
<h2 id="final-expression">Final expression </h2>

<p>Insertion of
Eq.&nbsp;<a href="#mjx-eqn-30">(30)</a> in Eq.&nbsp;<a href="#mjx-eqn-28">(28)</a> results in
Expand All @@ -784,6 +819,10 @@ <h2 id="more-operations">More operations </h2>
\end{align}
$$
<p>&nbsp;<br>
</section>

<section>
<h2 id="rewriting-the-two-body-operator">Rewriting the two-body operator </h2>

<p>The two-body operator can also be expressed in terms of the anti-symmetrized matrix elements we discussed previously as</p>
<p>&nbsp;<br>
Expand All @@ -799,6 +838,10 @@ <h2 id="more-operations">More operations </h2>
\end{align}
$$
<p>&nbsp;<br>
</section>

<section>
<h2 id="antisymmetrized-matrix-elements">Antisymmetrized matrix elements </h2>

<p>The factors in front of the operator, either \( \frac{1}{4} \) or
\( \frac{1}{2} \) tells whether we use antisymmetrized matrix elements or not.
Expand Down
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