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New Special Instructions
author Klaus Thoden <kthoden@mpiwg-berlin.mpg.de>
date Wed, 30 Jul 2014 15:58:21 +0200
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1 <?xml version="1.0" encoding="utf-8" ?>
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5 <html>
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7 <head><title></title>
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9 <meta name="generator" content="TeX4ht (http://www.cse.ohio-state.edu/~gurari/TeX4ht/mn.html)" />
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10 <meta name="originator" content="TeX4ht (http://www.cse.ohio-state.edu/~gurari/TeX4ht/mn.html)" />
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12 <meta name="src" content="Einst_Antwo_de_1915.tex" />
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13 <meta name="date" content="2005-03-01 19:47:00" />
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14 <link rel="stylesheet" type="text/css" href="Einst_Antwo_de_1915.css" />
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15 </head><body >
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17 <!--l. 13--><p class="noindent"><pb/></p>
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18 <div class="center" >
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20 <!--l. 14--><p class="noindent">
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21 </p><!--l. 15--><p class="noindent"><span
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22 class="cmr-12">2. </span><span
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23 class="cmbxti-10x-x-120">Antwort auf eine Abhandlung M. v. Laues </span> <br/><span
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24 class="cmbxti-10x-x-120">,,Ein Satz</span>
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25 <span
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26 class="cmbxti-10x-x-120">der Wahrscheinlichlceitsrechnung und </span> <br/><span
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27 class="cmbxti-10x-x-120">seine Anwendung</span>
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28 <span
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29 class="cmbxti-10x-x-120">auf die Strahlungstheorie&#8220;; </span> <br/><span
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30 class="cmbxti-10x-x-120">von A. Einstein.</span></p></div>
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31 <div class="center" >
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33 <!--l. 21--><p class="noindent">
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34 </p><!--l. 22--><p class="noindent">--------</p></div>
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35 <!--l. 25--><p class="indent"> In der zitierten Arbeit bringt Laue die mathematische <br/>Grundlage der Statistik der Strahlung in eine Form, die an <br/>Prägnanz und Schönheit nichts zu wünschen übrigläßt. Was <br/>aber die Anwendung jener Grundlage auf die Strahlungs-<br/>theorie anbelangt, so scheint er mir einem bedenklichen Irr-<br/>tume zum Opfer gefallen zu sein, der dringend Berichtigung <br/>fordert. Wenn Laues Behauptung, daß die Koeffizienten <br/>der Fourierentwicklung der bei natürlicher Strahlung auf-<br/>tretenden örtlichen Schwingung nicht voneinander statistisch <br/>unabhängig zu sein brauchten, berechtigt wäre, böte sich <br/>wirklich ein höchst aussichtsreicher Weg zur Überwindung <br/>der Schwierigkeiten dar, welche in der theoretischen Un-<br/>verdaulichkeit aller Gesetze besteht, in denen das Planck-<br/>sche ,,<span
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36 class="cmmi-10">h</span>&#8220; eine Rolle spielt. Dies war eben der Grund, der <br/>mich vor fünf Jahren veranlaßte, in einer mit L. Hopf zu-<br/>sammen publizierten Arbeit diese Frage näher zu prüfen.
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37 </p><!--l. 45--><p class="indent"> Das Resultat jener in ihrer Durchführung nicht ganz <br/>einwandfreien Arbeit, wird von Laue als richtige Konsequenz <br/>der zugrunde gelegten Voraussetzungen anerkannt. Aber <br/>Laue bestreitet die Zulässigkeit der Grundvoraussetzung, <br/>die sich so formulieren läßt:
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38 </p><!--l. 52--><p class="indent"> Wenn ich dadurch eine vollkommen ungeordnete Strah-<br/>lung (statistisch unabhängige Fourierkoeffizienten) erhalte <br/>daß ich unendlich viele vollkommen gegebene, ganz mitein-<br/>ander übereinstimmende Strahlungen derart superponiere, daß <br/>bei dieser Superposition die Gesamtphasen dieser superponierten <br/>Strahlungen zufällig gewählt werden, so muß die natürliche <br/>Strahlung erst recht statistisch ungeordnet sein.
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39 </p><!--l. 61--><p class="indent"> Diese Grundvoraussetzung schien mir damals evident. <br/>Der Umstand aber, daß sie von einem so erfahrenen Fachmann, <br/><pb/>
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40 </p><!--l. 67--><p class="indent">
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42 </p><!--l. 68--><p class="noindent">wie Laue, nicht geteilt wird, beweist das Gegenteil. Ich will <br/>deshalb im folgenden einen Beweis geben, der von einer der-<br/>artigen Voraussetzung frei ist und -- wie ich hoffe -- un-<br/>widerleglich dartut, daß unsere Undulationstheorie die sta-<br/>tistische Unabhängigkeit der Fourierkoeffizienten unbedingt <br/>fordert. Bevor ich diesen Beweis beginne, will ich aber zeigen, <br/>warum die in den Teilen II und III der Laueschen Abhand-<br/>lung gegebene Betrachtung nach meiner Ansicht nicht be-<br/>weisend ist.
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43 </p><!--l. 79--><p class="indent"> Laue betrachtet eine Strahlung, die durch eine große <br/>Anzahl unregelmäßig über eine Schicht von der Dicke <span
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44 class="cmmi-10">c <img
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45 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Antwo_de_1915/fulltext/img/cmmi10-1c.png" alt="t" class="10x-x-1c" /> </span>ver-<br/>teilter Resonatoren senkrecht zu dieser Schicht emittiert wird. <br/>Im Teile II seiner Abhandlung nimmt er an, daß alle diese
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46 <br/>Resonatoren gleichzeitig und nach demselben Gesetze schwin-<br/>gen; im Teile III, daß die Schwingungen aller Resonatoren <br/>durch dasselbe, als gegeben zu denkende statistische Gesetz <br/>beherrscht seien. In beiden Fällen ergibt sich nicht die sta-<br/>tistische Unabhängigkeit der Fourierkoeffizienten der Ent-<br/>wicklung für die resultierende Strahlung. Hieraus darf aber <br/>nach meiner Meinung keineswegs die Zulässigkeit der Hypo-<br/>these gefolgert werden, daß auch <span
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47 class="cmti-10">bei der nat</span><span
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48 class="cmti-10">ürlichen Strahlung</span>
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49 <br/>jene Unabhängigkeit nicht vorhanden sei. Denn es ist doch <br/>gar nicht gesagt, daß der Grad von Unordnung, welchen jene <br/>ungeordnete Verteilung der Resonatoren über die Schicht <br/>von der Dicke <span
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50 class="cmmi-10">c <img
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51 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Antwo_de_1915/fulltext/img/cmmi10-1c.png" alt="t" class="10x-x-1c" /> </span>mit sich bringt, derselbe sei wie bei der <br/>natürlichen Strahlung.
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52 </p><!--l. 92--><p class="indent"> Dieser Verdacht erhebt sich um so dringender, als nach <br/>Laues rechnerischen Ergebnissen der Grad der statistischen <br/>Abhängigkeit zweier durch die Indizes <span
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53 class="cmmi-10">p </span>und
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54 <span
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55 class="cmmi-10">p</span><span
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56 class="cmsy-10">' </span>charakteri-<br/>sierten Glieder der Entwicklung für die resultierende Strah-<br/>lung wesentlich durch den Wert
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57 </p>
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58 <center class="par-math-display" >
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59 <img
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60 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Antwo_de_1915/fulltext/img/Einst_Antwo_de_19150x.png" alt="p(p---p')t T " class="par-math-display" /></center>
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61 <!--l. 102--><p class="nopar">
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62 </p><!--l. 106--><p class="noindent">bedingt werde, d. h. <span
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63 class="cmti-10">durch eine von der Schichtdicke abh</span><span
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64 class="cmti-10">ängige </span> <br/><span
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65 class="cmti-10">Gr</span><span
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66 class="cmti-10">öße</span>, während doch eine derartige statistische Abhängigkeit <br/>bei der natürlichen Strahlung -- falls eine solche vorhanden <br/>wäre -- nichts zu tun haben dürfte mit der besonderen Er-<br/>zeugungsart der betrachteten Strahlung.
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67 </p><!--l. 113--><p class="indent"> Nach meiner Ansicht ist daher keiner der von Laue <br/>betrachteten Fälle der natürlichen Strahlung bezüglichen <br/><pb/>
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68 </p><!--l. 119--><p class="indent">
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69
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70 </p><!--l. 120--><p class="noindent">Unordnung äquivalent, so daß aus seinen Ergebnissen über <br/>die natürliche Strahlung nichts gefolgert werden kann. Ich <br/>halte vielmehr meine frühere Behauptung aufrecht und <br/>suche dieselbe im folgenden durch einen neuen Beweis zu <br/>stützen, indem ich mich der von Laue in seiner Arbeit dar-<br/>gelegten Sätze aus der Wahrscheinlichkeitsrechnung bediene.
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71 </p>
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72 <div class="center" >
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74 <!--l. 129--><p class="noindent">
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75 </p><!--l. 132--><p class="noindent"><span
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76 class="cmbsy-10">§</span><span
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77 class="cmbx-10">1. Statistische Eigenschaften einer Strahlung, die durch</span>
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78 <br/><span
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79 class="cmbx-10">Superposition unendlich vieler, voneinander unabh</span><span
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80 class="cmbx-10">ängig erzeugter</span>
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81 <br/><span
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82 class="cmbx-10">Strahlungen entstanden ist.</span></p></div>
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83 <!--l. 135--><p class="indent"> Jede der betrachteten Teilstrahlungen sei durch eine <br/>Fouriersche Entwicklung von der Form
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84 </p>
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85 <table width="100%"
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86 class="equation"><tr><td><a
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87 id="x1-2r1"></a>
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88 <center class="math-display" >
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89 <img
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90 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Antwo_de_1915/fulltext/img/Einst_Antwo_de_19151x.png" alt=" sum t t an(n)cos 2pn --+ bn(n)sin 2pn -- n T T " class="math-display" /></center></td><td width="5%">(1)</td></tr></table>
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91 <!--l. 141--><p class="nopar">
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92 </p><!--l. 145--><p class="noindent">für das Zeitintervall 0 bis <span
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93 class="cmmi-10">T </span>dargestellt, wobei die Koeffizienten <br/>dem Wahrscheinlichkeitsgesetz
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94 </p>
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95 <table width="100%"
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96 class="equation"><tr><td><a
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97 id="x1-3r2"></a>
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98 <center class="math-display" >
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99 <img
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100 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Antwo_de_1915/fulltext/img/Einst_Antwo_de_19152x.png" alt=" (n) (n) (n) (n) (n) (n) (n) dW = f (a1 ...az ..., b1 ...bz ) da1 ... dbz ... " class="math-display" /></center></td><td width="5%">(2)</td></tr></table>
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101
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102 <!--l. 154--><p class="nopar">
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103 </p><!--l. 158--><p class="noindent">genügen sollen, welches Gesetz für jedes (<span
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104 class="cmmi-10"><img
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105 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Antwo_de_1915/fulltext/img/cmmi10-17.png" alt="n" class="10x-x-17" /></span>), d. h. für jede der <br/>betrachteten Teilstrahlungen ein besonderes sein kann. Das <br/>Gesetz sei ferner ein solches, daß
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106 </p>
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107 <table width="100%"
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108 class="equation"><tr><td><a
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109 id="x1-4r3"></a>
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110 <center class="math-display" >
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111 <img
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112 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Antwo_de_1915/fulltext/img/Einst_Antwo_de_19153x.png" alt=" integral
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113 --n n (n) (n) (n) { an = an f d a1 ... d bz = 0 --n integral n (n) (n) (n) bn = bn f d a1 ... dbz = 0 " class="math-display" /></center></td><td width="5%">(3)</td></tr></table>
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114 <!--l. 171--><p class="nopar">
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115 </p><!--l. 175--><p class="indent"> Die resultierende Strahlung ist für das Zeitintervall 0 bis <span
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116 class="cmmi-10">T </span> <br/>durch die Entwicklung
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117 </p>
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118 <table width="100%"
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119 class="equation"><tr><td><a
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120 id="x1-5r4"></a>
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121 <center class="math-display" >
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122 <img
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123 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Antwo_de_1915/fulltext/img/Einst_Antwo_de_19154x.png" alt=" sum A cos 2pn t-+B sin 2pn t- n n T n T { sum sum ( ) = an(n) cos 2pn t-+bn(n) sin 2pn t n n T T " class="math-display" /></center></td><td width="5%">(4)</td></tr></table>
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124 <!--l. 186--><p class="nopar">
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125
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126 </p><!--l. 190--><p class="noindent">gegeben, woraus die Gültigkeit der Beziehungen
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127 </p>
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128 <table width="100%"
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129 class="equation"><tr><td><a
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130 id="x1-6r5"></a>
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131 <center class="math-display" >
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132 <img
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133 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Antwo_de_1915/fulltext/img/Einst_Antwo_de_19155x.png" alt=" An = sum an(n) { n sum (n) Bn = bn n " class="math-display" /></center></td><td width="5%">(5)</td></tr></table>
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134 <!--l. 199--><p class="nopar">
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135 </p><!--l. 203--><p class="noindent">hervorgeht. Welches statistische Gesetz folgt für die Fourier-<br/>koeffizienten <span
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136 class="cmmi-10">A</span><sub ><span
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137 class="cmr-7">1</span></sub><span
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138 class="cmmi-10">...</span><span
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139 class="cmmi-10">B</span><sub ><span
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140 class="cmmi-7">z</span></sub>?
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141 <pb/>
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142 </p><!--l. 210--><p class="indent">
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143
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144 </p><!--l. 211--><p class="indent"> Aus einer Betrachtung, die der im Teile I der Laueschen <br/>Arbeit durchgeführten ganz analog ist, findet man, daß das <br/>gesuchte statistische Gesetz das folgende ist:
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145 </p>
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146 <table width="100%"
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147 class="equation"><tr><td><a
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148 id="x1-7r6"></a>
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149 <center class="math-display" >
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150 <img
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151 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Antwo_de_1915/fulltext/img/Einst_Antwo_de_19156x.png" alt=" sum
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152 -mn(amnAmAn+bm nBm Bn+2gmnAm Bn) dW = konst.e dA1 ...dBz. " class="math-display" /></center></td><td width="5%">(6)</td></tr></table>
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153 <!--l. 218--><p class="nopar">
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154 </p><!--l. 220--><p class="noindent">Hieraus ersieht man, daß durch Superposition unendlich vieler <br/>Teilstrahlungen die statistische Unabhängigkeit der Fourier-<br/>Koeffizienten noch keineswegs garantiert wird. Wohl aber <br/>gestattet das Gesetz (6) die Frage nach der statistischen Un-<br/>abhängigkeit der Fourierkoeffizienten auf eine einfachere Frage <br/>zu reduzieren. Jene statistische Unabhängigkeit wird nämlich <br/>dann und nur dann erfüllt sein, wenn im Exponenten der <br/>Exponentialfunktion nur die Quadrate der
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155 <span
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156 class="cmmi-10">A</span><sub ><span
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157 class="cmmi-7">m</span></sub> und <span
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158 class="cmmi-10">B</span><sub ><span
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159 class="cmmi-7">m</span></sub>, aber <br/>keine Produkte dieser Größen auftreten; d. h. es muß sein:
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160 </p>
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161 <table width="100%"
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162 class="equation"><tr><td><a
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163 id="x1-8r7"></a>
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164 <center class="math-display" >
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165 <img
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166 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Antwo_de_1915/fulltext/img/Einst_Antwo_de_19157x.png" alt="{ amn = bmn = 0 f¨ur m /= n gmn = 0. " class="math-display" /></center></td><td width="5%">(7)</td></tr></table>
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167 <!--l. 235--><p class="nopar">
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168 </p><!--l. 239--><p class="indent"> Es ist ferner wegen (3) und (5) klar, daß im Falle sta-<br/>tistischer Unabhängigkeit die Beziehungen
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169 </p>
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170
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171 <table width="100%"
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172 class="equation"><tr><td><a
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173 id="x1-9r8"></a>
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174 <center class="math-display" >
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175 <img
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176 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Antwo_de_1915/fulltext/img/Einst_Antwo_de_19158x.png" alt=" ------ ------ { AmAn = BmBn = 0 f¨ur m /= n ------ AmBn = 0 " class="math-display" /></center></td><td width="5%">(7a)</td></tr></table>
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177 <!--l. 248--><p class="nopar">
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178 </p><!--l. 252--><p class="noindent">bestehen müssen. Da die Zahl der Bedingungen (7a) gleich <br/>ist der Zahl der Bedingungen (7), und alle Bedingungen (7a) <br/>voneinander unabhängig sind, so folgt, daß im Falle der Gültig-<br/>keit von (6) die Bedingungen (7a) <span
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179 class="cmti-10">hinreichend </span>sind für die
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180 <br/>statistische Unabhängigkeit der Fourierkoeffizienten.
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181 </p><!--l. 260--><p class="indent"> Wir gelangen daher zu folgendem vorläufigen Ergebnis: <br/>Da wir von der natürlichen Strahlung annehmen müssen, daß <br/>ihre statistischen Eigenschaften durch Superposition von in-<br/>kohärenten Teilstrahlungen nicht geändert werden, so sind <br/>die Gleichungen (7a) bei der natürlichen Strahlung hinreichende <br/>Bedingungen für die statistische Unabhängigkeit der Fourier-<br/>koeffizienten.
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182 </p>
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183 <div class="center" >
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184
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185 <!--l. 269--><p class="noindent">
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186 </p><!--l. 272--><p class="noindent"><span
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187 class="cmbsy-10">§</span><span
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188 class="cmbx-10">2. Nachweis der statistischen Unabh</span><span
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189 class="cmbx-10">ängigkeit der Fourier-</span>
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190 <br/><span
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191 class="cmbx-10">koeffizienten bei der nat</span><span
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192 class="cmbx-10">ürlichen Strahlung.</span></p></div>
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193 <!--l. 275--><p class="indent"> Es sei <span
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194 class="cmmi-10">F</span>(<span
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195 class="cmmi-10">t</span>) eine Komponente des Strahlungsvektors sta-<br/>tionärer natürlicher Strahlung, gegeben für unendlich lange <br/>Zeit. <span
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196 class="cmmi-10">T </span>sei eine gegen die Schwingungsdauer der langwelligsten <br/><pb/>
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197 </p><!--l. 282--><p class="indent">
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198
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199 </p><!--l. 283--><p class="noindent">in der Strahlung auftretenden Lichtart große Zeitdauer. Zwischen <br/>den Zeiten <span
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200 class="cmmi-10">t</span><sub ><span
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201 class="cmr-7">0</span></sub> und
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202 <span
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203 class="cmmi-10">t</span><sub ><span
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204 class="cmr-7">0</span></sub> + <span
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205 class="cmmi-10">T </span>sei <span
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206 class="cmmi-10">F</span>(<span
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207 class="cmmi-10">t</span>) dargestellt durch die Fourier-<br/>reihe
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208 </p>
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209 <table width="100%"
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210 class="equation"><tr><td><a
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211 id="x1-10r8"></a>
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212 <center class="math-display" >
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213 <img
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214 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Antwo_de_1915/fulltext/img/Einst_Antwo_de_19159x.png" alt=" ( ) sum t--t0 t--t0 An cos 2p n T + Bn sin 2p n T . n " class="math-display" /></center></td><td width="5%">(4a)</td></tr></table>
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215 <!--l. 291--><p class="nopar">
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216 </p><!--l. 295--><p class="indent"> Es ist klar, daß die zu <span
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217 class="cmmi-10">F</span>(<span
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218 class="cmmi-10">t</span>) gehörigen Fourierkoeffizienten <br/><span
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219 class="cmmi-10">A</span><sub ><span
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220 class="cmmi-7">n</span></sub>, <span
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221 class="cmmi-10">B</span><sub ><span
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222 class="cmmi-7">n</span></sub> von der Wahl der Epoche <span
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223 class="cmmi-10">t</span><sub ><span
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224 class="cmr-7">0</span></sub> abhängen werden. Indem <br/>wir die Entwicklung für sehr viele, zufällig gewählte <span
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225 class="cmmi-10">t</span><sub ><span
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226 class="cmr-7">0</span></sub> aus-<br/>geführt denken, erlangen wir ein statistisches Material zur <br/>Ableitung statistischer Eigenschaften der Koeffizienten <span
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227 class="cmmi-10">A</span><sub ><span
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228 class="cmmi-7">n</span></sub>, <span
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229 class="cmmi-10">B</span><sub ><span
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230 class="cmmi-7">n</span></sub>, <br/>welche wir bei der natürlichen Strahlung notwendig fordern <br/>müssen.
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231 </p><!--l. 304--><p class="indent"> Um diese Eigenschaften abzuleiten, entwickeln wir <span
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232 class="cmmi-10">F</span>(<span
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233 class="cmmi-10">t</span>) <br/>in eine Fourierreihe zwischen den Zeiten 0 und <span
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234 class="cmmi-10"><img
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235 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Antwo_de_1915/fulltext/img/cmmi10-23.png" alt="h" class="10x-x-23" /></span>, wobei <span
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236 class="cmmi-10"><img
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237 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Antwo_de_1915/fulltext/img/cmmi10-23.png" alt="h" class="10x-x-23" /> </span> <br/>eine gegenüber <span
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238 class="cmmi-10">T </span>sehr große Zeitdauer sei. Für dies Zeit-<br/>intervall sei
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239 </p>
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240 <table width="100%"
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241 class="equation"><tr><td><a
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242 id="x1-11r8"></a>
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243 <center class="math-display" >
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244 <img
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245 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Antwo_de_1915/fulltext/img/Einst_Antwo_de_191510x.png" alt=" ( ) sum t- F (t) = n an cos 2p nh + fn . " class="math-display" /></center></td><td width="5%">(8)</td></tr></table>
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246 <!--l. 314--><p class="nopar">
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247 </p><!--l. 318--><p class="indent"> Wählen wir <span
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248 class="cmmi-10">t</span><sub ><span
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249 class="cmr-7">0</span></sub> zwischen <span
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250 class="cmmi-10">t </span>= 0 und <span
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251 class="cmmi-10">t </span>= <span
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252 class="cmmi-10"><img
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253 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Antwo_de_1915/fulltext/img/cmmi10-23.png" alt="h" class="10x-x-23" /></span><span
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254 class="cmsy-10">-</span><span
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255 class="cmmi-10">T</span>, so können <br/>die Koeffizienten <span
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256 class="cmmi-10">A</span><sub ><span
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257 class="cmmi-7">n</span></sub> und
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258 <span
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259 class="cmmi-10">B</span><sub ><span
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260 class="cmmi-7">n</span></sub> durch <span
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261 class="cmmi-10">t</span><sub ><span
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262 class="cmr-7">0</span></sub> und die Koeffizienten <br/><span
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263 class="cmmi-10"><img
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264 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Antwo_de_1915/fulltext/img/cmmi10-b.png" alt="a" class="10x-x-b" /></span><sub ><span
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265 class="cmmi-7"><img
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266 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Antwo_de_1915/fulltext/img/cmmi7-17.png" alt="n" class="7x-x-17" /></span></sub> und <span
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267 class="cmmi-10"><img
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268 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Antwo_de_1915/fulltext/img/cmmi10-27.png" alt="f" class="cmmi-10x-x-27" align="middle" /></span><sub ><span
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269 class="cmmi-7"><img
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270 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Antwo_de_1915/fulltext/img/cmmi7-17.png" alt="n" class="7x-x-17" /></span></sub> der Entwicklung (8) ausgedrückt werden; man er-<br/>hält zunächst
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271 </p>
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272 <table width="100%"
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273 class="equation"><tr><td><a
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274 id="x1-12r9"></a>
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275 <center class="math-display" >
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276 <img
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277 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Antwo_de_1915/fulltext/img/Einst_Antwo_de_191511x.png" alt=" sum { t0 integral +T ( ) ( ) } An = -2 an cos 2pn t-+fn cos 2pn t--t0- d t T n t h T { t00+T 2- sum { integral ( -t ) ( t--t0) } Bn = T an cos 2pnh + fn sin 2pn T dt . n t0 " class="math-display" /></center></td><td width="5%">(9)</td></tr></table>
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278 <!--l. 338--><p class="nopar">
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279 </p><!--l. 340--><p class="noindent">Führt man die Integration aus, so erhält man, indem man <br/>in bekannter Weise Glieder mit dem Faktor <img
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280 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Antwo_de_1915/fulltext/img/Einst_Antwo_de_191512x.png" alt="p(n/h1+p/T)" class="frac" align="middle" /> gegen <br/>solche mit dem Faktor <img
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281 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Antwo_de_1915/fulltext/img/Einst_Antwo_de_191513x.png" alt="p(n/1h-n/T-)" class="frac" align="middle" />
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282 vernachlässigt:
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283 </p>
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284 <table width="100%"
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285 class="equation"><tr><td><a
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286 id="x1-13r10"></a>
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287 <center class="math-display" >
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288 <img
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289 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Antwo_de_1915/fulltext/img/Einst_Antwo_de_191514x.png" alt=" ( ) ( ) sum sin-p-nTh--n(-cos-xn)n +-2pn-t0h- An = n an p nTh-- n { sum ( T- ) ( t0) Bn = - ansin p-nh---n(-sin--xn)n-+2pn-h- , n p nTh-- n " class="math-display" /></center></td><td width="5%">(10)</td></tr></table>
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290 <!--l. 359--><p class="nopar">
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291 <pb/>
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292 </p><!--l. 365--><p class="indent">
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293
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294 </p><!--l. 366--><p class="noindent">wobei
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295 </p>
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296 <center class="par-math-display" >
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297 <img
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298 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Antwo_de_1915/fulltext/img/Einst_Antwo_de_191515x.png" alt=" ( ) T- xnn = p nh + n + fn " class="par-math-display" /></center>
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299 <!--l. 372--><p class="nopar">
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300 </p><!--l. 376--><p class="noindent">gesetzt ist. Die Formeln (10) gelten nur für Werte von <span
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301 class="cmmi-10">t</span><sub ><span
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302 class="cmr-7">0</span></sub> <br/>zwischen <span
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303 class="cmmi-10">t</span><sub ><span
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304 class="cmr-7">0</span></sub> = 0 und
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305 <span
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306 class="cmmi-10">t</span><sub ><span
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307 class="cmr-7">0</span></sub> = <span
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308 class="cmmi-10"><img
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309 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Antwo_de_1915/fulltext/img/cmmi10-23.png" alt="h" class="10x-x-23" /> </span><span
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310 class="cmsy-10">- </span><span
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311 class="cmmi-10">T</span>, weil die Entwicklung ge-<br/>mäß (8) nur für das Zeitintervall 0 <span
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312 class="cmsy-10">- </span><span
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313 class="cmmi-10"><img
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314 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Antwo_de_1915/fulltext/img/cmmi10-23.png" alt="h" class="10x-x-23" /></span>
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315 gilt. Wir erlauben <br/>uns jedoch, die Formel (8) für das Intervall 0 <span
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316 class="cmsy-10">- </span>(<span
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317 class="cmmi-10"><img
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318 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Antwo_de_1915/fulltext/img/cmmi10-23.png" alt="h" class="10x-x-23" /> </span>+ <span
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319 class="cmmi-10">T</span>) an-<br/>zuwenden. Damit ersetzen wir zwischen den Zeitwerten <span
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320 class="cmmi-10"><img
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321 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Antwo_de_1915/fulltext/img/cmmi10-23.png" alt="h" class="10x-x-23" /> </span> <br/>und <span
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322 class="cmmi-10"><img
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323 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Antwo_de_1915/fulltext/img/cmmi10-23.png" alt="h" class="10x-x-23" /> </span>+ <span
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324 class="cmmi-10">T</span>
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325 die Funktion <span
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326 class="cmmi-10">F</span>(<span
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327 class="cmmi-10">t</span>) durch die Werte von <span
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328 class="cmmi-10">F</span>(<span
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329 class="cmmi-10">t</span>) zwischen <br/>den Zeiten 0 und <span
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330 class="cmmi-10">T</span>. Durch dieses Vorgehen werden im fol-<br/>genden unsere Mittelwertbetrachtungen gefälscht, aber nur <br/>relativ unendlich wenig, weil das Zeitintervall <span
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331 class="cmmi-10">T </span>gegen
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332 <span
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333 class="cmmi-10"><img
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334 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Antwo_de_1915/fulltext/img/cmmi10-23.png" alt="h" class="10x-x-23" /> </span>un-<br/>endlich klein ist. Von dieser Erwägung ausgehend, werden <br/>wir die Gleichungen (10) so anwenden, wie wenn sie im ganzen <br/>Intervall 0 <span
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335 class="cmmi-10">&lt; t</span><sub ><span
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336 class="cmr-7">0</span></sub> <span
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337 class="cmmi-10">&lt; <img
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338 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Antwo_de_1915/fulltext/img/cmmi10-23.png" alt="h" class="10x-x-23" /> </span>gelten würden.
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339 </p><!--l. 393--><p class="indent"> Wir bilden nun mit Hilfe von (10) den Mittelwert <span class="overline"><span
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340 class="cmmi-10">A</span><sub ><span
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341 class="cmmi-7">m</span></sub> <span
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342 class="cmmi-10">A</span><sub ><span
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343 class="cmmi-7">n</span></sub><span
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344 class="cmmi-10">,</span></span> <br/>d. h. die Größe
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345 </p>
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346 <center class="par-math-display" >
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347 <img
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348 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Antwo_de_1915/fulltext/img/Einst_Antwo_de_191516x.png" alt="------ 1 integral h Am An = h- Am An dt0. 0 " class="par-math-display" /></center>
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349 <!--l. 400--><p class="nopar">
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350 </p><!--l. 403--><p class="noindent">Dabei tritt das Integral
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351 </p>
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352 <center class="par-math-display" >
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353 <img
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354 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Antwo_de_1915/fulltext/img/Einst_Antwo_de_191517x.png" alt=" h ( ) ( ) integral t0- t0 cos xmm + 2pm h cos xnn + 2pnh dt0 0
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355 " class="par-math-display" /></center>
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356 <!--l. 411--><p class="nopar">
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357 </p><!--l. 415--><p class="noindent">auf. Dieses verschwindet wegen der Ganzzahligkeit von <span
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358 class="cmmi-10"><img
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359 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Antwo_de_1915/fulltext/img/cmmi10-16.png" alt="m" class="cmmi-10x-x-16" align="middle" /> </span>und <span
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360 class="cmmi-10"><img
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361 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Antwo_de_1915/fulltext/img/cmmi10-17.png" alt="n" class="10x-x-17" /></span>, <br/>wenn <span
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362 class="cmmi-10"><img
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363 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Antwo_de_1915/fulltext/img/cmmi10-16.png" alt="m" class="cmmi-10x-x-16" align="middle" /></span><img
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364 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Antwo_de_1915/fulltext/img/Einst_Antwo_de_191518x.png" alt="/=" class="neq" align="middle" /><span
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365 class="cmmi-10"><img
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366 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Antwo_de_1915/fulltext/img/cmmi10-17.png" alt="n" class="10x-x-17" /></span>, und hat für <span
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367 class="cmmi-10"><img
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368 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Antwo_de_1915/fulltext/img/cmmi10-16.png" alt="m" class="cmmi-10x-x-16" align="middle" /> </span>= <span
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369 class="cmmi-10"><img
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370 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Antwo_de_1915/fulltext/img/cmmi10-17.png" alt="n" class="10x-x-17" /> </span>den Wert <img
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371 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Antwo_de_1915/fulltext/img/Einst_Antwo_de_191519x.png" alt="h 2" class="frac" align="middle" />(<span
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372 class="cmsy-10">-</span>1)<sup ><span
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373 class="cmmi-7">m</span><span
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374 class="cmsy-7">-</span><span
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375 class="cmmi-7">n</span></sup>. <br/>Mit Rücksicht darauf ergibt die erste der Gleichungen (10)
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376 </p>
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377 <table width="100%"
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378 class="equation"><tr><td><a
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379 id="x1-14r11"></a>
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380
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381 <center class="math-display" >
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382 <img
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383 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Antwo_de_1915/fulltext/img/Einst_Antwo_de_191520x.png" alt=" ( ) ( ) ------ (- 1)m-n sum 2sin p n Th- m sin p nTh-- n Am An = --2- an ---p2(n-T--m)--(nT---n)---- n h h " class="math-display" /></center></td><td width="5%">(11)</td></tr></table>
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384 <!--l. 428--><p class="nopar">
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385 </p>
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386 <center class="par-math-display" >
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387 <img
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388 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Antwo_de_1915/fulltext/img/Einst_Antwo_de_191521x.png" alt=" 1 sum 2 sin2 pnTh- = 2 an p2(n-T--m)--(nT---n). h h " class="par-math-display" /></center>
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389 <!--l. 438--><p class="nopar">
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390 </p><!--l. 442--><p class="indent"> A priori ist klar, daß eine statistische Abhängigkeit nur <br/>zwischen Strahlungskomponenten von sehr nahe gleicher <br/>Frequenz zu erwarten ist. <span
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391 class="cmmi-10">m </span>und <span
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392 class="cmmi-10">n</span>
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393 gehören also demselben <br/>engen Spektralbereich an, ebenso jene Werte von <span
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394 class="cmmi-10"><img
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395 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Antwo_de_1915/fulltext/img/cmmi10-17.png" alt="n" class="10x-x-17" /></span>, welche
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396 <br/>zu unserer Summe merklich beitragen. <pb/>
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397 </p><!--l. 452--><p class="indent">
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398
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399 </p><!--l. 453--><p class="indent"> In (11) ist der Bruch auf der rechten Seite eine wegen <br/>der Kleinheit von <span
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400 class="cmti-10">T/</span><span
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401 class="cmmi-10"><img
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402 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Antwo_de_1915/fulltext/img/cmmi10-23.png" alt="h" class="10x-x-23" /> </span>mit
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403 <span
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404 class="cmmi-10"><img
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405 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Antwo_de_1915/fulltext/img/cmmi10-17.png" alt="n" class="10x-x-17" /> </span>langsam veränderliche Größe. <br/>Deshalb kann bezüglich der Größe <span
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406 class="cmmi-10"><img
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407 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Antwo_de_1915/fulltext/img/cmmi10-b.png" alt="a" class="10x-x-b" /></span><sub ><span
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408 class="cmmi-7"><img
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409 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Antwo_de_1915/fulltext/img/cmmi7-1d.png" alt="v" class="7x-x-1d" /></span></sub><sup ><span
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410 class="cmr-7">2</span></sup> über viele aufe nander <br/>folgende Glieder ohne merkbaren Fehler gemittelt werden, <br/>und es wird jener Mittelwert <span class="overline"><span
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411 class="cmmi-10"><img
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412 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Antwo_de_1915/fulltext/img/cmmi10-b.png" alt="a" class="10x-x-b" /></span><sub ><span
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413 class="cmmi-7"><img
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414 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Antwo_de_1915/fulltext/img/cmmi7-17.png" alt="n" class="7x-x-17" /></span></sub><sup ><span
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415 class="cmr-7">2</span></sup></span> als Konstante aus der Summe <br/>herausgesetzt werden können, da die Summation überhaupt <br/>nur über einen engen Spektralbereich zu erstrecken ist. Die
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416 <br/>über den Bruch erstreckte Summe kann dann noch in ein <br/>Integral verwandelt werden, so daß man erhält:
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417 </p>
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418 <table width="100%"
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419 class="equation"><tr><td><a
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420 id="x1-15r12"></a>
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421 <center class="math-display" >
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422 <img
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423 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Antwo_de_1915/fulltext/img/Einst_Antwo_de_191522x.png" alt="------ 1 --2-h- integral -----sin2-x------ Am An = 2 an pT (x - m p)(x - np)d x. " class="math-display" /></center></td><td width="5%">(12)</td></tr></table>
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424 <!--l. 466--><p class="nopar">
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425 </p><!--l. 470--><p class="indent"> Das Integral kann ohne merklichen Fehler zwischen <span
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426 class="cmsy-10">-<img
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427 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Antwo_de_1915/fulltext/img/cmsy10-31.png" alt=" oo " class="10x-x-31" /></span> <br/>und + <span
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428 class="cmsy-10"><img
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429 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Antwo_de_1915/fulltext/img/cmsy10-31.png" alt=" oo " class="10x-x-31" /> </span>genommen werden, statt zwischen der durch den <br/>vorerwähnten Spektralbereich bestimmten Grenzen.
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430 </p><!--l. 475--><p class="indent"> Dieses Integral hat für <span
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431 class="cmmi-10">m </span>= <span
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432 class="cmmi-10">n </span>den Wert <span
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433 class="cmmi-10"><img
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434 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Antwo_de_1915/fulltext/img/cmmi10-19.png" alt="p" class="10x-x-19" /></span>, verschwindet <br/>aber stets<sup ><span
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435 class="cmr-7">1</span></sup>), wenn <span
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436 class="cmmi-10">m</span><img
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437 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Antwo_de_1915/fulltext/img/Einst_Antwo_de_191523x.png" alt="/=" class="neq" align="middle" /><span
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438 class="cmmi-10">n</span>
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439 (<span
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440 class="cmmi-10">m </span>und <span
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441 class="cmmi-10">n </span>sind ganze Zahlen). Damit <br/>ist zunächst das Verschwinden von
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442 <span class="overline"><span
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443 class="cmmi-10">A</span><sub ><span
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444 class="cmmi-7">m</span></sub> <span
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445 class="cmmi-10">A</span><sub ><span
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446 class="cmmi-7">n</span></sub></span> (für <span
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447 class="cmmi-10">m</span><img
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448 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Antwo_de_1915/fulltext/img/Einst_Antwo_de_191524x.png" alt="/=" class="neq" align="middle" /><span
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449 class="cmmi-10">n</span>) bewiesen; <br/>der Beweis für das Verschwinden von <span class="overline"><span
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450 class="cmmi-10">B</span><sub ><span
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451 class="cmmi-7">m</span></sub> <span
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452 class="cmmi-10">B</span><sub ><span
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453 class="cmmi-7">n</span></sub></span>
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454 (für <span
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455 class="cmmi-10">m</span><img
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456 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Antwo_de_1915/fulltext/img/Einst_Antwo_de_191525x.png" alt="/=" class="neq" align="middle" /><span
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457 class="cmmi-10">n</span>) und <br/><span class="overline"><span
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458 class="cmmi-10">A</span><sub ><span
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459 class="cmmi-7">m</span></sub> <span
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460 class="cmmi-10">B</span><sub ><span
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461 class="cmmi-7">n</span></sub></span> ist analog zu führen. Aus dem Verschwinden dieser
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462 <br/>Mittelwerte folgt nach <span
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463 class="cmsy-10">§</span>1 die behauptete statistische Un-<br/>abhängigkeit der Fourierkoeffizienten.
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464 </p><!--l. 486--><p class="noindent">----------
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465 </p><!--l. 489--><p class="indent"> 1) Das Integral ist nämlich gleich
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466 </p>
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467 <center class="par-math-display" >
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468 <img
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469 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Antwo_de_1915/fulltext/img/Einst_Antwo_de_191526x.png" alt=" ( ) 1 + integral oo sin2x + integral o o sin2x (m---n)p- x--mp--dx- x---npdx . - oo - oo
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470 " class="par-math-display" /></center>
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471 <!--l. 496--><p class="nopar">
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472 </p><!--l. 499--><p class="noindent">Jedes der letzteren Integrale ist gleich
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473 </p>
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474 <center class="par-math-display" >
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475 <img
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476 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Antwo_de_1915/fulltext/img/Einst_Antwo_de_191527x.png" alt=" + integral oo 2 sin-ydy = 0. - oo y " class="par-math-display" /></center>
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477 <!--l. 505--><p class="nopar">
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478 </p><!--l. 509--><p class="indent"> <span
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479 class="cmti-10">Bemerkung zur Korrektur</span>: Statt bei der Auswertung von (11) über <br/>viele aufeinanderfolgende Summenglieder zu mitteln, kann man auch un-<br/>endlich viele, voneinander unabhängige Entwicklungen (8) zugrunde legen <br/>und über diese mitteln. Nimmt man an (11) jene Mittelwertbildung vor, <br/>so tritt der dementsprechend verstandene Mittelwert <span class="overline"><span
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480 class="cmmi-10"><img
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481 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Antwo_de_1915/fulltext/img/cmmi10-b.png" alt="a" class="10x-x-b" /></span><sub ><span
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482 class="cmmi-7"><img
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483 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Antwo_de_1915/fulltext/img/cmmi7-17.png" alt="n" class="7x-x-17" /></span></sub><sup ><span
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484 class="cmr-7">2</span></sup></span> vor das Summen-<br/>zeichen. Das Endresultat bleibt natürlich dasselbe.
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485 </p>
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486 <div class="center" >
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487
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488 <!--l. 519--><p class="noindent">
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489 </p><!--l. 520--><p class="noindent">(Eingegangen 24. Juni 1915.)</p></div>
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490 <div class="center" >
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491
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492 <!--l. 523--><p class="noindent">
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493 </p><!--l. 524--><p class="noindent">----------</p></div>
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494
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495 </body></html>
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496
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497
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498