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102 lines
3.2 KiB
XML
102 lines
3.2 KiB
XML
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<?xml version="1.0" encoding="UTF-8"?>
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<!-- This DocBook file was created by LyX 2.4.0-beta2
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See https://www.lyx.org/ for more information -->
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<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">
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<title>Test</title>
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<figure role='theorem'>
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<title>Teorema 1.</title>
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<para>For electrons in a perfect crystal</para>
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<para>there is a basis of wave functions with the following two properties:</para>
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</figure>
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<figure role='theorem'>
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<title>Teorema 2.</title>
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<para>1) each of these wave functions is an energy eigenstate;</para>
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<para>2) each of these wave functions is a Bloch state, meaning that this wave function <inlineequation>
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<alt role='tex'>\psi</alt>
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<m:math display="inline">
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<m:mrow><m:mi>ψ</m:mi>
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</m:mrow>
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</m:math>
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</inlineequation> can be written in the form <inlineequation>
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<alt role='tex'>\psi(r)=u(\boldsymbol{r})e^{i\boldsymbol{k}\cdot\boldsymbol{r}}</alt>
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<m:math display="inline">
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<m:mrow>
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<m:mrow><m:mi>ψ</m:mi>
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<m:mrow>
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<m:mo form='prefix' fence='true' stretchy='true' symmetric='true'>(</m:mo>
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<m:mi>r</m:mi>
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<m:mo form='postfix' fence='true' stretchy='true' symmetric='true'>)</m:mo>
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</m:mrow>
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<m:mo>=</m:mo>
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<m:mi>u</m:mi>
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<m:mrow>
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<m:mo form='prefix' fence='true' stretchy='true' symmetric='true'>(</m:mo>
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<m:mstyle mathvariant='bold'>
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<m:mi>r</m:mi></m:mstyle>
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<m:mo form='postfix' fence='true' stretchy='true' symmetric='true'>)</m:mo>
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</m:mrow>
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<m:msup>
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<m:mi>e</m:mi>
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<m:mrow>
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<m:mi>i</m:mi>
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<m:mstyle mathvariant='bold'>
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<m:mi>k</m:mi></m:mstyle><m:mo>⋅</m:mo>
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<m:mstyle mathvariant='bold'>
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<m:mi>r</m:mi></m:mstyle>
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</m:mrow>
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</m:msup>
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</m:mrow>
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</m:mrow>
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</m:math>
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</inlineequation> where <inlineequation>
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<alt role='tex'>u</alt>
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<m:math display="inline">
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<m:mrow>
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<m:mi>u</m:mi>
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</m:mrow>
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</m:math>
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</inlineequation> has the same periodicity as the atomic structure of the crystal: <inlineequation>
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<alt role='tex'>u_{\boldsymbol{k}}(\boldsymbol{r})=u_{\boldsymbol{k}}(\boldsymbol{r}+\boldsymbol{n}\cdot\boldsymbol{a})</alt>
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<m:math display="inline">
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<m:mrow>
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<m:mrow>
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<m:msub>
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<m:mi>u</m:mi>
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<m:mstyle mathvariant='bold'>
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<m:mi>k</m:mi></m:mstyle>
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</m:msub>
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<m:mrow>
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<m:mo form='prefix' fence='true' stretchy='true' symmetric='true'>(</m:mo>
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<m:mstyle mathvariant='bold'>
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<m:mi>r</m:mi></m:mstyle>
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<m:mo form='postfix' fence='true' stretchy='true' symmetric='true'>)</m:mo>
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</m:mrow>
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<m:mo>=</m:mo>
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<m:msub>
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<m:mi>u</m:mi>
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<m:mstyle mathvariant='bold'>
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<m:mi>k</m:mi></m:mstyle>
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</m:msub>
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<m:mrow>
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<m:mo form='prefix' fence='true' stretchy='true' symmetric='true'>(</m:mo>
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<m:mrow>
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<m:mstyle mathvariant='bold'>
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<m:mi>r</m:mi></m:mstyle>
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<m:mo>+</m:mo>
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<m:mstyle mathvariant='bold'>
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<m:mi>n</m:mi></m:mstyle><m:mo>⋅</m:mo>
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<m:mstyle mathvariant='bold'>
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<m:mi>a</m:mi></m:mstyle>
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</m:mrow>
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<m:mo form='postfix' fence='true' stretchy='true' symmetric='true'>)</m:mo>
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</m:mrow>
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</m:mrow>
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</m:mrow>
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</m:math>
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</inlineequation>.</para>
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</figure>
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</article>
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