From 29b49508a751649310173e592b63415dbf563a2a Mon Sep 17 00:00:00 2001
From: Prefetch
Date: Thu, 3 Sep 2026 15:05:35 +0200
Subject: Improve knowledge base

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+---
+title: "Discrete spectrum summation"
+sort_title: "Discrete spectrum summation"
+date: 2026-09-02
+categories:
+- Physics
+- Quantum mechanics
+layout: "concept"
+---
+
+This article is about a trick used in many calculations,
+especially in condensed matter physics and advanced quantum mechanics,
+which, as far as I know, does not have a specific name
+(this is the best I could come up with),
+but is so common and useful that it deserves attention.
+
+Often, we find ourselves doing calculations
+about a $$D$$-dimensional system with periodic boundary conditions.
+Generally, there are two sources of such boundary conditions:
+an inherent periodicity of the system (e.g. crystals),
+and/or a need to chop up an infinite system into finite pieces
+to prevent mathematical problems (e.g. divergences).
+
+In the second case, if studying the whole infinity directly is not possible,
+we restrict ourselves to a hypercube with side $$L$$
+and $$D$$-dimensional volume $$V = L^D$$,
+with the intention to let $$L \to \infty$$ at the end.
+We then often impose periodic boundary conditions on the hypercube,
+in order to be able to use [Fourier transforms](/know/concept/fourier-transform/)
+on such a finite domain, and/or to study transport phenomena.
+This idea is trivial to generalize to "hyperrectangles"
+with different side lengths $$L_x$$, $$L_y$$, etc.
+
+In both cases, we might end up expanding functions
+from a [Hilbert space](/know/concept/hilbert-space/) defined on the hypercube
+in a basis of plane waves $$\ket{\psi_\vb{k}}$$ with wavevectors $$\vb{k}$$,
+often as the result of a Fourier transform.
+But due to the hypercube's finite size and its boundary conditions,
+those plane waves occupy a discrete set of allowed $$\vb{k}$$-values
+(whereas in an infinite system, $$\vb{k}$$ would be a continuous variable).
+
+Hence, those normalized basis waves $$\ket{\psi_\vb{k}}$$ are as follows
+in $$\vb{r}$$-space (modulo a constant phase):
+
+$$\begin{aligned}
+    \inprod{\vb{r}}{\psi_{\vb{k}}}
+    = \psi_{\vb{k}}(\vb{r})
+    = \frac{1}{\sqrt{L^D}} \exp(i \vb{k} \cdot \vb{r})
+    \qquad \qquad
+    \vb{k} = \frac{2 \pi}{L} (n_1, ..., n_D)
+\end{aligned}$$
+
+Where $$n_1, ..., n_D \in \mathbb{Z}$$.
+The discreteness is typically an artifact of our mathematical setup,
+and then disappears into the true continuous spectrum for $$L \to \infty$$.
+Until then, every plane wave occupies a nonzero volume
+$$(2 \pi)^D / L^D$$ in $$\vb{k}$$-space.
+
+Here is the key: as $$L$$ increases, the allowed $$\vb{k}$$-values become denser,
+until any sum over those $$\vb{k}$$ turns into a Riemann integral:
+
+$$\begin{aligned}
+    \lim_{L \to \infty} \frac{(2 \pi)^D}{L^D} \sum_{\vb{k}} f(\vb{k})
+    = \int_{-\infty}^\infty f(\vb{k}) \dd{\vb{k}}
+\end{aligned}$$
+
+Where $$(2 \pi) / L$$ is the spacing between $$\vb{k}$$-values.
+This trick to convert nasty sums to easier integrals
+is used all over physics because it is so powerful.
+We can even get away with postponing taking the limit,
+and doing the conversion as an exact equality in the middle of our calculation,
+on the condition that we take $$L \to \infty$$ at the end.
+
+Actually, this trick is not exclusive to periodic boundary conditions,
+but is also valid for Dirichlet ("particle in a box") boundaries,
+in which case the wavevector spectrum is discrete too,
+also with constant spacing between allowed $$\vb{k}$$-values.
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