1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
|
---
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.
|
