1 <?xml version="1.0" encoding="UTF-8"?>
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4 <article xml:lang="it_IT" xmlns="http://docbook.org/ns/docbook" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:m="http://www.w3.org/1998/Math/MathML" xmlns:xi="http://www.w3.org/2001/XInclude" version="5.2">
6 <figure role='theorem'>
7 <title>Teorema 1.</title>
8 <para>For electrons in a perfect crystal</para>
9 <para>there is a basis of wave functions with the following two properties:</para>
11 <figure role='theorem'>
12 <title>Teorema 2.</title>
13 <para>1) each of these wave functions is an energy eigenstate;</para>
14 <para>2) each of these wave functions is a Bloch state, meaning that this wave function <inlineequation>
15 <alt role='tex'>\psi</alt>
16 <m:math display="inline">
18 <m:mrow><m:mi>ψ</m:mi>
21 </inlineequation> can be written in the form <inlineequation>
22 <alt role='tex'>\psi(r)=u(\boldsymbol{r})e^{i\boldsymbol{k}\cdot\boldsymbol{r}}</alt>
23 <m:math display="inline">
26 <m:mrow><m:mi>ψ</m:mi>
28 <m:mo form='prefix' fence='true' stretchy='true' symmetric='true'>(</m:mo>
30 <m:mo form='postfix' fence='true' stretchy='true' symmetric='true'>)</m:mo>
35 <m:mo form='prefix' fence='true' stretchy='true' symmetric='true'>(</m:mo>
36 <m:mstyle mathvariant='bold'>
37 <m:mi>r</m:mi></m:mstyle>
38 <m:mo form='postfix' fence='true' stretchy='true' symmetric='true'>)</m:mo>
44 <m:mstyle mathvariant='bold'>
45 <m:mi>k</m:mi></m:mstyle><m:mo>⋅</m:mo>
46 <m:mstyle mathvariant='bold'>
47 <m:mi>r</m:mi></m:mstyle>
53 </inlineequation> where <inlineequation>
54 <alt role='tex'>u</alt>
55 <m:math display="inline">
61 </inlineequation> has the same periodicity as the atomic structure of the crystal: <inlineequation>
62 <alt role='tex'>u_{\boldsymbol{k}}(\boldsymbol{r})=u_{\boldsymbol{k}}(\boldsymbol{r}+\boldsymbol{n}\cdot\boldsymbol{a})</alt>
63 <m:math display="inline">
69 <m:mstyle mathvariant='bold'>
70 <m:mi>k</m:mi></m:mstyle>
73 <m:mo form='prefix' fence='true' stretchy='true' symmetric='true'>(</m:mo>
74 <m:mstyle mathvariant='bold'>
75 <m:mi>r</m:mi></m:mstyle>
76 <m:mo form='postfix' fence='true' stretchy='true' symmetric='true'>)</m:mo>
81 <m:mstyle mathvariant='bold'>
82 <m:mi>k</m:mi></m:mstyle>
85 <m:mo form='prefix' fence='true' stretchy='true' symmetric='true'>(</m:mo>
87 <m:mstyle mathvariant='bold'>
88 <m:mi>r</m:mi></m:mstyle>
90 <m:mstyle mathvariant='bold'>
91 <m:mi>n</m:mi></m:mstyle><m:mo>⋅</m:mo>
92 <m:mstyle mathvariant='bold'>
93 <m:mi>a</m:mi></m:mstyle>
95 <m:mo form='postfix' fence='true' stretchy='true' symmetric='true'>)</m:mo>
100 </inlineequation>.</para>