Mercurial > hg > mpdl-xml-content
annotate texts/XML/diverse/de/Einst_Einig_de_1913.xml @ 10:d7b79f6537bb
Version vom 2009-02-14
| author | Klaus Thoden <kthoden@mpiwg-berlin.mpg.de> |
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| date | Thu, 02 May 2013 11:08:12 +0200 |
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1 <?xml version="1.0" encoding="utf-8" ?> |
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4 <!--http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd--> |
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5 <html> |
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7 <head><title></title> |
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8 <meta http-equiv="Content-Type" content="text/html; charset=iso-8859-1" /> |
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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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11 <!-- xhtml,html --> |
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12 <meta name="src" content="Einst_Einig_de_1913.tex" /> |
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13 <meta name="date" content="2005-03-10 17:29:00" /> |
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14 <link rel="stylesheet" type="text/css" href="Einst_Einig_de_1913.css" /> |
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15 </head><body > |
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17 <!--l. 17--><p class="noindent"><pb/></p> |
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18 <div class="center" > |
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20 <!--l. 18--><p class="noindent"> |
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21 </p><!--l. 19--><p class="noindent"><span |
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22 class="cmr-12x-x-120">8. </span><span |
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23 class="cmbxti-10x-x-144">Einige Argumente </span> <br/><span |
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24 class="cmbxti-10x-x-144">f</span><span |
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25 class="cmbxti-10x-x-144">ür die Annahme einer</span> |
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26 <span |
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27 class="cmbxti-10x-x-144">molekularen Agitation </span> <br/><span |
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28 class="cmbxti-10x-x-144">beim absoluten Nullpunkt; </span> <br/><span |
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29 class="cmbxti-10x-x-144">von</span> |
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30 <span |
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31 class="cmbxti-10x-x-144">A. Einstein und O. Stern.</span></p></div> |
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32 <div class="center" > |
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34 <!--l. 24--><p class="noindent"> |
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35 </p><!--l. 25--><p class="noindent">--------</p></div> |
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36 <!--l. 29--><p class="indent"> Der Ausdruck für die Energie eines Resonators lautet <br/>nach der ersten Planckschen Formel:</p> |
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37 <table width="100%" |
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38 class="equation"><tr><td><a |
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39 id="x1-2r1"></a> |
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40 <center class="math-display" > |
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41 <img |
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42 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Einig_de_1913/fulltext/img/Einst_Einig_de_19130x.png" alt=" h n E = -h-n----, ekT - 1 " class="math-display" /></center></td><td width="5%">(1)</td></tr></table> |
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43 <!--l. 35--><p class="nopar"> |
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44 </p><!--l. 39--><p class="noindent">nach der zweiten:</p> |
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45 <table width="100%" |
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46 class="equation"><tr><td><a |
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47 id="x1-3r2"></a> |
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48 <center class="math-display" > |
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49 <img |
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50 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Einig_de_1913/fulltext/img/Einst_Einig_de_19131x.png" alt="E = --h-n---+ h-n. ehkTn- 1 2 " class="math-display" /></center></td><td width="5%">(2)</td></tr></table> |
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52 <!--l. 43--><p class="nopar"> |
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53 </p><!--l. 47--><p class="indent"> Der Grenzwert für hohe Temperaturen wird, wenn wir <br/>die Entwickelung von |
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54 <span |
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55 class="cmmi-12">e</span><sup ><img |
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56 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Einig_de_1913/fulltext/img/Einst_Einig_de_19132x.png" alt="hknT-" class="frac" align="middle" /> |
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57 </sup> mit dem quadratischen Gliede ab-<br/>brechen, für (1):</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_Einig_de_1913/fulltext/img/Einst_Einig_de_19133x.png" alt=" lim E = k T - hn-, T=o o 2 " class="par-math-display" /></center> |
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61 <!--l. 55--><p class="nopar"> |
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62 </p><!--l. 59--><p class="noindent">für (2):</p> |
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63 <center class="par-math-display" > |
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64 <img |
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65 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Einig_de_1913/fulltext/img/Einst_Einig_de_19134x.png" alt=" lim E = kT . T=o o |
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66 " class="par-math-display" /></center> |
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67 <!--l. 65--><p class="nopar"> |
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68 Die Energie als Funktion der Temperatur, wie sie in <br/>Fig. 1 dargestellt ist, beginnt also nach Formel (1) für <span |
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69 class="cmmi-12">T </span>= 0 <br/>mit Null, dem von der klassischen Theorie <br/>geforderten Werte, bleibt aber bei hohen <br/>Temperaturen ständig um das Stück <span |
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70 class="cmmi-12">h<img |
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71 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Einig_de_1913/fulltext/img/cmmi12-17.png" alt="n" class="12x-x-17" /></span><img |
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72 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Einig_de_1913/fulltext/img/Einst_Einig_de_19136x.png" alt="/" class="left" align="middle" /> 2 |
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73 <br/>kleiner als dieser. Nach Formel (2) hat <br/>der Resonator beim absoluten Nullpunkt <br/>die Energie <span |
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74 class="cmmi-12">h<img |
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75 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Einig_de_1913/fulltext/img/cmmi12-17.png" alt="n" class="12x-x-17" /></span><img |
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76 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Einig_de_1913/fulltext/img/Einst_Einig_de_19137x.png" alt="/" class="left" align="middle" /> 2, im Widerspruch zur <br/>klassischen Theorie, erreicht aber bei hohen <br/>Temperaturen asymptotisch die von dieser <br/>geforderte Energie. Dagegen ist der Diffe-<br/>rentialquotient der Energie nach der Temperatur, d. h. die <br/>spezifische Wärme, in beiden Fällen gleich.<img |
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77 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Einig_de_1913/fulltext/img/Einst_Einig_de_19135x.png" alt="PIC" class="graphics" width="107.99805pt" height="128.0374pt" /><!--tex4ht:graphics |
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78 name="img/Einst_Einig_de_19135x.png" src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Einig_de_1913/fulltext/Einst_Einig_de_1913_001.eps" |
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79 --> |
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80 </p><!--l. 92--><p class="indent"> |
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81 <pb/> |
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82 </p><!--l. 93--><p class="indent"> Für Gebilde mit unveränderlichem <span |
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83 class="cmmi-12"><img |
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84 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Einig_de_1913/fulltext/img/cmmi12-17.png" alt="n" class="12x-x-17" /> </span>sind diese Formeln <br/>also gleichwertig, während die Theorie solcher Gebilde, deren <span |
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85 class="cmmi-12"><img |
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86 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Einig_de_1913/fulltext/img/cmmi12-17.png" alt="n" class="12x-x-17" /> </span> <br/>für verschiedene Zustände verschiedene Werte hat, durch die <br/>Annahme einer Nullpunktsenergie wesentlich beeinflußt wird. <br/>Der ideale Fall wäre der eines aus monochromatischen Ge-<br/>bilden bestehenden Systems, dessen <span |
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87 class="cmmi-12"><img |
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88 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Einig_de_1913/fulltext/img/cmmi12-17.png" alt="n" class="12x-x-17" /></span>-Wert unabhängig von <br/>der Temperatur willkürlich geändert werden kann. Die Ab-<br/>hängigkeit der Energie von der Frequenz bei konstanter Tem-<br/>peratur würde wesentlich von der Existenz einer Nullpunkts-<br/>energie abhängen. Leider liegen Erfahrungen über ein der-<br/>artiges Gebilde nicht vor. Wohl aber kennen wir in den <br/>rotierenden Gasmolekülen Gebilde, deren thermische Bewegungen <br/>mit denen monochromatischer Gebilde eine weitgehende Ähn-<br/>lichkeit aufweisen<sup ><span |
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89 class="cmr-8">1</span></sup>), und bei welchen die mittlere Frequenz <br/>mit der Temperatur veränderlich ist. An diesen Gebilden ist <br/>also die Berechtigung der Annahme einer Nullpunktsenergie <br/>in erster Linie zu prüfen. Im folgenden soll zunächst unter-<br/>sucht werden, inwiefern wir aus der Planckschen Formel auf <br/>das theoretische Verhalten solcher Gebilde Rückschlüsse ziehen |
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90 <br/>können. |
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91 </p> |
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92 <div class="center" > |
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94 <!--l. 108--><p class="noindent"> |
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95 </p><!--l. 109--><p class="noindent">Die spezifische Wärme des Wasserstoffs bei tiefen <br/>Temperaturen.</p></div> |
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96 <!--l. 114--><p class="indent"> Es handelt sich um die Frage, wie die Energie der Rotation <br/>eines zweiatomigen Moleküls von der Temperatur abhängt. <br/>Analog wie bei der Theorie der spezifischen Wärme fester <br/>Stoffe sind wir zu der Annahme berechtigt, daß die mittlere <br/>kinetische Energie der Rotation davon unabhängig ist, ob das <br/>Molekül in Richtung seiner Symmetrieachse ein elektrisches Mo-<br/>ment besitzt oder nicht. Im Falle, daß das Molekül ein solches <br/>Moment besitzt, darf es das thermodynamische Gleichgewicht <br/>zwischen Gasmolekülen und Strahlung nicht stören. Hieraus <br/>kann man schließen, daß das Molekül unter der Einwirkung der <br/>Strahlung allein dieselbe kinetische Energie der Rotation an-<br/>nehmen muß, die es durch die Zusammenstöße mit anderen <br/>Molekülen erhalten würde. Die Frage ist also, bei welchem <br/>---------- |
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97 </p><!--l. 132--><p class="indent"> 1) Hierauf hat zuerst Nernst aufmerksam gemacht, vgl. Zeitschr. <br/>f. Elektroch. |
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98 <span |
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99 class="cmbx-12">17. </span>p. 270 u. 825. 1911. <pb/> |
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100 </p><!--l. 138--><p class="indent"> |
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102 </p><!--l. 139--><p class="noindent">Mittelwerte der Rotationsenergie sich eiu träger, starrer Dipol <br/>mit Strahlung von bestimmter Temperatur im Gleichgewicht <br/>befindet. Wie die Gesetze der Ausstrahlung auch sein mögen, <br/>so wird doch wohl daran festzuhalten sein, daß ein rotierender <br/>Dipol doppelt so viel Energie pro Zeiteinheit ausstrahlt als <br/>ein eindimensionaler Resonator, bei dem die Amplitude des |
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103 <br/>elektrischen und mechanischen Moments gleich dem elektri-<br/>schen und mechanischen Moment des Dipols ist. Analoges <br/>wird auch von dem Mittelwert der absorbierten Energie gelten. <br/>Machen wir nun noch die vereinfachende Näherungsannahme, <br/>daß bei gegebener Temperatur alle Dipole unseres Gases gleich <br/>rasch rotieren, so werden wir zu dem Schluß geführt, daß im |
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104 <br/>Gleichgewicht die kinetische Energie eines Dipols doppelt so <br/>groß sein muß, wie die eines eindimensionalen Resonators von <br/>gleicher Frequenz. Bei den gemachten Annahmen können wir <br/>die Ausdrücke (1) bzw. (2) direkt zur Berechnung der kine-<br/>tischen Energie eines mit zwei Freiheitsgraden rotierenden |
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105 <br/>Gasmoleküls anwenden, wobei bei jeder Temperatur zwischen <span |
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106 class="cmmi-12">E </span> <br/>und <span |
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107 class="cmmi-12"><img |
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108 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Einig_de_1913/fulltext/img/cmmi12-17.png" alt="n" class="12x-x-17" /> </span>die Gleichung |
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109 </p> |
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110 <center class="par-math-display" > |
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111 <img |
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112 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Einig_de_1913/fulltext/img/Einst_Einig_de_19138x.png" alt="E = J-(2p n)2 2 " class="par-math-display" /></center> |
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113 <!--l. 165--><p class="nopar"> |
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114 </p><!--l. 169--><p class="noindent">besteht (<span |
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115 class="cmmi-12">J </span>Trägheitsmoment des Moleküls). |
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116 </p><!--l. 172--><p class="indent"> So ergibt sich für die Energie der Rotation pro Mol:</p> |
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117 <table width="100%" |
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118 class="equation"><tr><td><a |
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119 id="x1-4r3"></a> |
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120 <center class="math-display" > |
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121 <img |
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122 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Einig_de_1913/fulltext/img/Einst_Einig_de_19139x.png" alt=" J h n E = N0 .--(2p n)2 = N0 -hn----- 2 ekT - 1 " class="math-display" /></center></td><td width="5%">(3)</td></tr></table> |
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123 <!--l. 178--><p class="nopar"> |
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124 </p><!--l. 182--><p class="noindent">bzw.</p> |
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125 <table width="100%" |
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126 class="equation"><tr><td><a |
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127 id="x1-5r4"></a> |
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128 |
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129 <center class="math-display" > |
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130 <img |
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131 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Einig_de_1913/fulltext/img/Einst_Einig_de_191310x.png" alt=" ( ) J- 2 --h-n--- hn- E = N0 . 2 (2p n) = N0 ehknT-- 1 + 2 . " class="math-display" /></center></td><td width="5%">(4)</td></tr></table> |
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132 <!--l. 190--><p class="nopar"> |
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133 </p><!--l. 193--><p class="noindent">Da nun <span |
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134 class="cmmi-12"><img |
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135 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Einig_de_1913/fulltext/img/cmmi12-17.png" alt="n" class="12x-x-17" /> </span>und <span |
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136 class="cmmi-12">T </span>durch eine transzendente Gleichung verknüpft <br/>sind, ist es nicht möglich, <span |
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137 class="cmmi-12">dE</span><img |
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138 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Einig_de_1913/fulltext/img/Einst_Einig_de_191311x.png" alt="/" class="left" align="middle" /> <span |
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139 class="cmmi-12">dT</span> als explizite Funktion von <span |
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140 class="cmmi-12">T </span> <br/>auszudrücken, sondern man erhält, falls man zur Abkürzung <br/>2 <span |
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141 class="cmmi-12"><img |
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142 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Einig_de_1913/fulltext/img/cmmi12-19.png" alt="p" class="12x-x-19" /></span><sup ><span |
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143 class="cmr-8">2</span></sup><span |
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144 class="cmmi-12">J </span>= <span |
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145 class="cmmi-12">p </span>setzt, als Formel für die spezifische Wärme der |
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146 <br/>Rotation:</p> |
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147 <table width="100%" |
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148 class="equation"><tr><td><a |
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149 id="x1-6r5"></a> |
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150 <center class="math-display" > |
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151 <img |
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152 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Einig_de_1913/fulltext/img/Einst_Einig_de_191312x.png" alt="c = dE--= d-E-.d-n--= N 2 pn --(----n------)- t dT dn dT 0 T 1 + --k-T-- pn2+ hn " class="math-display" /></center></td><td width="5%">(5)</td></tr></table> |
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153 <!--l. 207--><p class="nopar"> |
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154 <pb/> |
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155 </p><!--l. 214--><p class="indent"> |
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156 |
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157 </p><!--l. 215--><p class="noindent">bzw.</p> |
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158 <table width="100%" |
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159 class="equation"><tr><td><a |
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160 id="x1-7r6"></a> |
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161 <center class="math-display" > |
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162 <img |
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163 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Einig_de_1913/fulltext/img/Einst_Einig_de_191313x.png" alt="c = dE--= d-E- .d-n-= N 2p n --(----n------)-, t dT dn d T 0 kT T 1 + p-n2--h2 4p " class="math-display" /></center></td><td width="5%">(6)</td></tr></table> |
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164 <!--l. 224--><p class="nopar"> |
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165 </p><!--l. 228--><p class="noindent">wobei <span |
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166 class="cmmi-12"><img |
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167 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Einig_de_1913/fulltext/img/cmmi12-17.png" alt="n" class="12x-x-17" /> </span>und <span |
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168 class="cmmi-12">T </span>durch die Gleichung:</p> |
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169 <table width="100%" |
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170 class="equation"><tr><td><a |
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171 id="x1-8r7"></a> |
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172 <center class="math-display" > |
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173 <img |
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174 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Einig_de_1913/fulltext/img/Einst_Einig_de_191314x.png" alt=" h n T = -----(-------) k ln hpn + 1 " class="math-display" /></center></td><td width="5%">(5a)</td></tr></table> |
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175 <!--l. 234--><p class="nopar"> |
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176 </p><!--l. 238--><p class="noindent">bzw.</p> |
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177 <table width="100%" |
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178 class="equation"><tr><td><a |
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179 id="x1-9r7"></a> |
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180 <center class="math-display" > |
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181 <img |
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182 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Einig_de_1913/fulltext/img/Einst_Einig_de_191315x.png" alt="T = h----(---n-----)- k ln --h-- + 1 p n- h2 " class="math-display" /></center></td><td width="5%">(6a)</td></tr></table> |
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183 <!--l. 244--><p class="nopar"> |
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184 </p><!--l. 247--><p class="noindent">verbunden sind. In Fig. 2 stellt die Kurve I die auf Grund <br/>von (6) und (6a) berechnete spezifische Wärme dar, wobei <span |
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185 class="cmmi-12">p</span> |
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186 <img |
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187 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Einig_de_1913/fulltext/img/Einst_Einig_de_191316x.png" alt="PIC" class="graphics" width="284.52756pt" height="212.46811pt" /><!--tex4ht:graphics |
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188 name="img/Einst_Einig_de_191316x.png" src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Einig_de_1913/fulltext/Einst_Einig_de_1913_002.eps" |
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189 --> |
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190 </p><!--l. 254--><p class="noindent">den Wert 2<span |
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191 class="cmmi-12">, </span>90 <span |
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192 class="cmmi-12">. </span>10<sup ><span |
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193 class="cmsy-8">-</span><span |
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194 class="cmr-8">40</span></sup> hat;<sup ><span |
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195 class="cmr-8">1</span></sup>) Kurve II ist aus (5) und (5a) mit <br/>Hilfe von |
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196 <span |
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197 class="cmmi-12">p </span>= 2 <span |
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198 class="cmsy-10x-x-120"><sup class="htf"><strong>.</strong></sup> </span>10<sup ><span |
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199 class="cmsy-8">-</span><span |
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200 class="cmr-8">40</span></sup> berechnet. Die Kreuzchen bezeichnen <br/>die von Eucken<sup ><span |
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201 class="cmr-8">2</span></sup>) gemessenen Werte. Wie man sieht, zeigt <br/>die Kurve II einen Verlauf, der mit den Versuchen in völligem <br/>---------- |
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202 </p><!--l. 262--><p class="indent"> 1) Berechnet man den zu diesem Trägheitsmoment gehörigen Molekül-<br/>durchmesser, so ergibt er sich zu 9 <span |
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203 class="cmsy-10x-x-120"><sup class="htf"><strong>.</strong></sup> </span>10<sup ><span |
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204 class="cmsy-8">-</span><span |
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205 class="cmr-8">9</span></sup>, etwa halb so groß, als der <br/>gastheoretisch ermittelte Wert. |
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206 </p><!--l. 267--><p class="indent"> 2) Eucken, Sitzungsber. d. preuß. Akad. p. 141. 1912. <pb/> |
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207 </p><!--l. 272--><p class="indent"> |
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208 |
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209 </p><!--l. 273--><p class="noindent">Widerspruch steht, während Kurve I, die auf der Annahme <br/>einer Nullpunktsenergie basiert, die Resultate der Messungen <br/>in vorzüglicher Weise widergibt. Um festzustellen, welchen <br/>Wert nach Formel (4) <span |
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210 class="cmmi-12"><img |
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211 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Einig_de_1913/fulltext/img/cmmi12-17.png" alt="n" class="12x-x-17" /> </span>für die Grenze <span |
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212 class="cmmi-12">T </span>= 0 annimmt, |
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213 <br/>schreiben wir (4) in folgender Form: |
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214 </p> |
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215 <center class="par-math-display" > |
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216 <img |
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217 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Einig_de_1913/fulltext/img/Einst_Einig_de_191317x.png" alt=" h e hknT-= ---h--- - 1 = pn-+-2-. p n- h2 pn - h2 " class="par-math-display" /></center> |
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218 <!--l. 284--><p class="nopar"> |
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219 </p><!--l. 286--><p class="noindent">Dann sieht man, daß für <span |
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220 class="cmmi-12">T </span>= 0 <span |
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221 class="cmmi-12"><img |
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222 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Einig_de_1913/fulltext/img/cmmi12-17.png" alt="n" class="12x-x-17" /> </span>nicht gleich Null werden <br/>kann, da die rechte Seite dann gegen <span |
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223 class="cmsy-10x-x-120">-</span>1 konvergieren würde, <br/>während auf der linken eine Potenz von |
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224 <span |
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225 class="cmmi-12">e </span>steht. Es muß also <br/>für lim <span |
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226 class="cmmi-12">T </span>= 0 <span |
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227 class="cmmi-12"><img |
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228 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Einig_de_1913/fulltext/img/cmmi12-17.png" alt="n" class="12x-x-17" /> </span>endlich bleiben, und zwar muß die rechte |
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229 <br/>Seite ebenso wie die linke gegen <span |
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230 class="cmsy-10x-x-120"><img |
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231 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Einig_de_1913/fulltext/img/cmsy10-c-31.png" alt=" oo " class="10-120x-x-31" /> </span>konvergieren, es muß <br/>daher <span |
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232 class="cmmi-12">p<img |
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233 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Einig_de_1913/fulltext/img/cmmi12-17.png" alt="n" class="12x-x-17" /></span><sub ><span |
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234 class="cmr-8">0</span></sub> <span |
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235 class="cmsy-10x-x-120">- </span><span |
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236 class="cmmi-12">h</span><img |
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237 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Einig_de_1913/fulltext/img/Einst_Einig_de_191318x.png" alt="/" class="left" align="middle" /> 2 = 0 sein, falls wir mit <span |
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238 class="cmmi-12"><img |
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239 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Einig_de_1913/fulltext/img/cmmi12-17.png" alt="n" class="12x-x-17" /></span><sub ><span |
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240 class="cmr-8">0</span></sub> den Grenzwert <br/>von <span |
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241 class="cmmi-12"><img |
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242 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Einig_de_1913/fulltext/img/cmmi12-17.png" alt="n" class="12x-x-17" /> </span>für <span |
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243 class="cmmi-12">T </span>= 0 bezeichnen. Es ist also |
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244 <span |
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245 class="cmmi-12"><img |
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246 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Einig_de_1913/fulltext/img/cmmi12-17.png" alt="n" class="12x-x-17" /></span><sub ><span |
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247 class="cmr-8">0</span></sub> = <span |
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248 class="cmmi-12">h</span><img |
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249 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Einig_de_1913/fulltext/img/Einst_Einig_de_191319x.png" alt="/" class="left" align="middle" /> 2 <span |
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250 class="cmmi-12">p</span>. Im vor-<br/>liegenden Falle ergibt sich <span |
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251 class="cmmi-12"><img |
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252 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Einig_de_1913/fulltext/img/cmmi12-17.png" alt="n" class="12x-x-17" /></span><sub ><span |
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253 class="cmr-8">0</span></sub> zu 11<span |
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254 class="cmmi-12">, </span>3<span |
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255 class="cmsy-10x-x-120"><sup class="htf"><strong>.</strong></sup></span>10<sup ><span |
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256 class="cmr-8">12</span></sup>. Der Wert von <span |
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257 class="cmmi-12"><img |
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258 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Einig_de_1913/fulltext/img/cmmi12-17.png" alt="n" class="12x-x-17" /></span> |
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259 <br/>ändert sich zunächst auch sehr wenig mit steigender Tempe-<br/>ratur; so ist bei 102<sup ><span |
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260 class="cmr-8">0</span></sup> |
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261 abs. <span |
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262 class="cmmi-12"><img |
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263 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Einig_de_1913/fulltext/img/cmmi12-17.png" alt="n" class="12x-x-17" /> </span>= 11<span |
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264 class="cmmi-12">, </span>4<span |
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265 class="cmsy-10x-x-120"><sup class="htf"><strong>.</strong></sup></span>10<sup ><span |
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266 class="cmr-8">12</span></sup>, bei 189<sup ><span |
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267 class="cmr-8">0</span></sup> <span |
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268 class="cmmi-12"><img |
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269 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Einig_de_1913/fulltext/img/cmmi12-17.png" alt="n" class="12x-x-17" /> </span>= 12<span |
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270 class="cmmi-12">, </span>3<span |
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271 class="cmsy-10x-x-120"><sup class="htf"><strong>.</strong></sup></span>10<sup ><span |
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272 class="cmr-8">12</span></sup> , <br/>bei 323<sup ><span |
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273 class="cmr-8">0</span></sup> <span |
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274 class="cmmi-12"><img |
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275 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Einig_de_1913/fulltext/img/cmmi12-17.png" alt="n" class="12x-x-17" /> </span>= 14<span |
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276 class="cmmi-12">, </span>3<span |
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277 class="cmsy-10x-x-120"><sup class="htf"><strong>.</strong></sup></span>10<sup ><span |
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278 class="cmr-8">12</span></sup>. Dies erklärt nun, weshalb Eucken <br/>seine Messungen verhältnismäßig noch am besten durch die <br/>einfache Einsteinsche Formel mit von der Temperatur un-<br/>abhängigem <span |
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279 class="cmmi-12"><img |
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280 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Einig_de_1913/fulltext/img/cmmi12-17.png" alt="n" class="12x-x-17" /> </span>(Kurve III, Fig. 2) darstellen konnte. Jedoch <br/>sieht man, daß auch diese Formel, namentlich bei höheren <br/>Temperaturen, versagt, abgesehen davon, daß ohne die An-<br/>nahme der Nullpunktsenergie die Konstanz von <span |
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281 class="cmmi-12"><img |
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282 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Einig_de_1913/fulltext/img/cmmi12-17.png" alt="n" class="12x-x-17" /> </span>völlig un-<br/>verständlich bleibt. Man sieht also, daß die spezifische Wärme <br/>des Wasserstoffs die Existenz einer Nullpunktsenergie wahr-<br/>scheinlich macht, und es handelt sich nur noch darum, zu <br/>prüfen, wie weit der spezielle Wert von <span |
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283 class="cmmi-12">h<img |
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284 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Einig_de_1913/fulltext/img/cmmi12-17.png" alt="n" class="12x-x-17" /></span><img |
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285 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Einig_de_1913/fulltext/img/Einst_Einig_de_191320x.png" alt="/" class="left" align="middle" /> 2 als gesichert <br/>anzusehen ist. Da nun in der folgenden Untersuchung über <br/>das Strahlungsgesetz der Betrag der Nullpunktsenergie zu <span |
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286 class="cmmi-12">h<img |
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287 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Einig_de_1913/fulltext/img/cmmi12-17.png" alt="n" class="12x-x-17" /> </span> <br/>angenommen werden muß, haben wir die spezifische Wärme des <br/>Wasserstoffs auch für diese Annahme berechnet (<span |
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288 class="cmmi-12">p </span>= 5<span |
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289 class="cmmi-12">, </span>60<span |
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290 class="cmsy-10x-x-120"><sup class="htf"><strong>.</strong></sup></span>10<sup ><span |
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291 class="cmsy-8">-</span><span |
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292 class="cmr-8">40</span></sup>, <br/>Kurve IV, Fig. 2). Es ist ersichtlich, daß die Kurve bei <br/>höheren Temperaturen zu steil und zu hoch ist. Andererseits <br/>ist zu bemerken, daß bei Berücksichtigung der Geschwindig-<br/>keitsverteilung unter den Molekülen die Kurve jedenfalls etwas <br/>flacher ausfallen dürfte. Es ist demnach zwar unwahrschein-<br/><pb/> |
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293 </p><!--l. 333--><p class="indent"> |
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294 |
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295 </p><!--l. 334--><p class="noindent">lich, aber nicht mit Sicherheit auszuschließen, daß die Null-<br/>punktsenergie den Wert <span |
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296 class="cmmi-12">h<img |
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297 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Einig_de_1913/fulltext/img/cmmi12-17.png" alt="n" class="12x-x-17" /> </span>besitzt.<sup ><span |
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298 class="cmr-8">1</span></sup>) </p> |
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299 <div class="center" > |
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300 |
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301 <!--l. 337--><p class="noindent"> |
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302 </p><!--l. 338--><p class="noindent">Die Ableitung des Strahlungsgesetzes.</p></div> |
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303 <!--l. 341--><p class="indent"> Im folgenden soll gezeigt werden, wie sich auf Grund <br/>der Annahme einer Nullpunktsenergie die Plancksche Strah-<br/>lungsformel in ungezwungener, wenn auch nicht ganz strenger <br/>Weise ableiten läßt, und zwar ohne jede Annahme über irgend-<br/>welche Diskontinuitäten. Der Weg, den wir hierzu einschlagen, <br/>ist im wesentlichen derselbe, den Einstein und Hopf<sup ><span |
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304 class="cmr-8">2</span></sup>) in <br/>einer vor 2 Jahren erschienenen Abhandlung benutzten. Wir <br/>betrachten die fortschreitende Bewegung eines freibeweglichen <br/>Resonators, der etwa an einem Gasmolekül festsitzt, unter <br/>dem Einflusse eines ungeordneten Strahlungsfeldes. Im ther-<br/>mischen Gleichgewicht muß dann die mittlere kinetische Energie, <br/>die das Gasmolekül durch die Strahlung erhält, gleich der-<br/>jenigen sein, die es durch Zusammenstöße mit anderen Mole-<br/>külen bekommen würde. Man erhält so den Zusammenhang <br/>zwischen der Dichte der schwarzen Strahlung und der mitt-<br/>leren kinetischen Energie einer Gasmolekel, d. h. der Tem-<br/>peratur. Einstein und Hopf finden auf diese Weise das <br/>Rayleigh-Jeanssche Gesetz. Wir wollen nun dieselbe Be-<br/>---------- |
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305 </p><!--l. 365--><p class="indent"> 1) Nimmt man die Entropie rotierender Gebilde gleich der fester <br/>Stoffe nach dem Nernstschen Theorem für <span |
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306 class="cmmi-12">T </span>= 0 zu Null an, so ergibt <br/>sich der gesamte von der Rotation der zweiatomigen Moleküle her-<br/>rübrende Anteil der Entropie eines Mols zu</p> |
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307 <center class="par-math-display" > |
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308 <img |
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309 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Einig_de_1913/fulltext/img/Einst_Einig_de_191321x.png" alt=" integral T integral n 2 [( ) ] S = ctd T = ln n +-n0d n = 2-pn--+ k ln p-n 2- 1 . t T n- n0 T h 0 n0 " class="par-math-display" /></center> |
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310 <!--l. 377--><p class="nopar"> |
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311 </p><!--l. 380--><p class="noindent">Für hohe Temperaturen wird: |
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312 </p> |
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313 <center class="par-math-display" > |
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314 <img |
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315 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Einig_de_1913/fulltext/img/Einst_Einig_de_191322x.png" alt=" 2 St = R ln T + 2R + R ln2-p--J k-. h2 " class="par-math-display" /></center> |
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316 <!--l. 386--><p class="nopar"> |
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317 </p><!--l. 389--><p class="noindent">Nach Sackur (Nernst-Festschrift p. 414. 1912) ist die Entropiekonstante <br/>der Rotation: |
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318 </p> |
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319 <center class="par-math-display" > |
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320 <img |
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321 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Einig_de_1913/fulltext/img/Einst_Einig_de_191323x.png" alt=" 16 p3 J k R + R ln-----2--- , h " class="par-math-display" /></center> |
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322 <!--l. 395--><p class="nopar"> |
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323 </p><!--l. 399--><p class="noindent">in der Hauptsache, nämlich dem Ausdruck <span |
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324 class="cmmi-12">J k</span><img |
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325 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Einig_de_1913/fulltext/img/Einst_Einig_de_191324x.png" alt="/" class="left" align="middle" /> <span |
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326 class="cmmi-12">h</span><sup ><span |
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327 class="cmr-8">2</span></sup>, mit dem obigen Aus-<br/>druck übereinstimmend. Dasselbe Resultat erhält man übrigens, wenn <br/>man für <span |
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328 class="cmmi-12">c</span><sub ><span |
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329 class="cmmi-8">r</span></sub> nicht Formel (5), sondern Formel (6) einsetzt. |
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330 </p><!--l. 406--><p class="indent"> 2) A. Einstein u. L. Hopf, Ann. d. Phys. <span |
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331 class="cmbx-12">33. </span>p. 1105--1115. 1910. |
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332 <pb/> |
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333 </p><!--l. 410--><p class="indent"> |
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334 |
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335 </p><!--l. 411--><p class="noindent">trachtung unter der Annahme einer Nullpunktsenergie durch-<br/>führen. Der Einfluß, den die Strahlung ausübt, läßt sich nach <br/>Einstein und Hopf in zwei verschiedene Wirkungen zer-<br/>legen. Erstens einmal erleidet die geradlinig fortschreitende |
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336 <br/>Bewegung des Resonatormoleküls eine Art Reibung, veran-<br/>laßt durch den Strahlungsdruck auf den bewegten Oszillator. <br/>Diese Kraft <span |
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337 class="cmmi-12">K </span>ist proportional der Geschwindigkeit <span |
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338 class="cmmi-12">v</span>, also <br/><span |
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339 class="cmmi-12">K </span>= <span |
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340 class="cmsy-10x-x-120">-</span><span |
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341 class="cmmi-12">P v</span>, wenigstens falls <span |
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342 class="cmmi-12">v </span>klein gegen die Lichtgeschwin-<br/>digkeit ist. Der Impuls, den das Resonatormolekül in der |
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343 <br/>kleinen Zeit <span |
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344 class="cmmi-12"><img |
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345 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Einig_de_1913/fulltext/img/cmmi12-1c.png" alt="t" class="12x-x-1c" /></span>, während deren sich <span |
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346 class="cmmi-12">v </span>nicht merklich ändern <br/>soll, erhält, ist also <span |
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347 class="cmsy-10x-x-120">-</span><span |
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348 class="cmmi-12">P v <img |
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349 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Einig_de_1913/fulltext/img/cmmi12-1c.png" alt="t" class="12x-x-1c" />. </span>Zweitens erteilt die Strahlung <br/>dem Resonatormolekül Impulsschwankungen <img |
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350 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Einig_de_1913/fulltext/img/cmr12-1.png" alt="D" class="12x-x-1" />, die von der <br/>Bewegung des Moleküls in erster Annäherung unabhängig und <br/>für alle Richtungen gleich sind, so daß nur ihr quadratischer <br/>Mittelwert <span class="overline"><img |
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351 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Einig_de_1913/fulltext/img/cmr12-1.png" alt="D" class="12x-x-1" /><sup ><span |
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352 class="cmr-8">2</span></sup></span> während der Zeit <span |
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353 class="cmmi-12"><img |
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354 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Einig_de_1913/fulltext/img/cmmi12-1c.png" alt="t" class="12x-x-1c" /> </span>für die kinetische Energie <br/>maßgebend ist. Soll nun diese den von der statistischen <br/>Mechanik geforderten Wert <span |
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355 class="cmmi-12">k </span>(<span |
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356 class="cmmi-12">T/</span>2) besitzen (der Oszillator soll <br/>der Einfachheit halber nur in der <span |
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357 class="cmmi-12">x</span>-Richtung beweglich sein <br/>und nur in der <span |
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358 class="cmmi-12">z</span>-Richtung schwingen), so muß nach Einstein <br/>und Hopf (l. c. p. 1107) folgende Gleichung gelten: |
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359 </p> |
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360 <center class="par-math-display" > |
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361 <img |
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362 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Einig_de_1913/fulltext/img/Einst_Einig_de_191325x.png" alt="--- D2 = 2 kT P t . " class="par-math-display" /></center> |
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363 <!--l. 441--><p class="nopar"> |
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364 </p><!--l. 444--><p class="noindent">Was nun die Berechnung von <span |
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365 class="cmmi-12">P </span>anlangt, so können wir an-<br/>nehmen, daß hierfür nur die von der Strahlung selbst an-<br/>geregten Schwingungen in Betracht kommen, und daß man <br/>diese so berechnen kann, als ob die Nullpunktsenergie nicht |
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366 <br/>vorhanden wäre. Wir können also den von Einstein und <br/>Hopf berechneten Wert (l. c. p. 1111): |
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367 </p> |
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368 <center class="par-math-display" > |
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369 <img |
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370 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Einig_de_1913/fulltext/img/Einst_Einig_de_191326x.png" alt=" ( ) P = -3c-s- r - n-d-r 10 p n 3 d n " class="par-math-display" /></center> |
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371 <!--l. 454--><p class="nopar"> |
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372 </p><!--l. 458--><p class="noindent">benutzen. |
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373 </p><!--l. 461--><p class="indent"> Um nun <span class="overline"><img |
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374 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Einig_de_1913/fulltext/img/cmr12-1.png" alt="D" class="12x-x-1" /><sup ><span |
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375 class="cmr-8">2</span></sup></span> zu berechnen, setzen wir (l. c. p. 1111) den <br/>Impuls, welchen der Oszillator während der Zeit <span |
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376 class="cmmi-12"><img |
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377 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Einig_de_1913/fulltext/img/cmmi12-1c.png" alt="t" class="12x-x-1c" /> </span>in der <br/><span |
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378 class="cmmi-12">x</span>-Richtung erfährt: |
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379 </p> |
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380 <center class="par-math-display" > |
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381 <img |
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382 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Einig_de_1913/fulltext/img/Einst_Einig_de_191327x.png" alt=" integral t integral t @-Ez- J = kx dt = @ x f d t, 0 0 " class="par-math-display" /></center> |
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383 <!--l. 470--><p class="nopar"> |
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384 </p><!--l. 474--><p class="noindent">wobei <span |
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385 class="cmmi-12">f </span>das Moment des Oszillators ist. Wir wollen zunächst <br/>nur den Fall betrachten, daß die Energie der durch die Strah-<br/><pb/> |
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386 </p><!--l. 479--><p class="indent"> |
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387 |
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388 </p><!--l. 480--><p class="noindent">lung angeregten Schwingung zu vernachlässigen ist gegen die <br/>Nullpunktsenergie des Resonators, was bei genügend tiefen <br/>Temperaturen sicher erlaubt ist. Bezeichnen wir mit <span |
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389 class="cmmi-12">f</span><sub ><span |
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390 class="cmr-8">0</span></sub> das <br/>maximale Moment des Resonators, so ist: |
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391 </p> |
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392 <center class="par-math-display" > |
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393 <img |
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394 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Einig_de_1913/fulltext/img/Einst_Einig_de_191328x.png" alt=" 2-pn0-t f = f0 cos T , " class="par-math-display" /></center> |
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395 <!--l. 489--><p class="nopar"> |
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396 </p><!--l. 493--><p class="noindent">wobei <span |
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397 class="cmmi-12">T </span>eine große Zeit und <span |
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398 class="cmmi-12">n</span><sub ><span |
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399 class="cmr-8">0</span></sub><img |
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400 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Einig_de_1913/fulltext/img/Einst_Einig_de_191329x.png" alt="/" class="left" align="middle" /> <span |
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401 class="cmmi-12">T </span>= <span |
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402 class="cmmi-12"><img |
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403 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Einig_de_1913/fulltext/img/cmmi12-17.png" alt="n" class="12x-x-17" /></span><sub ><span |
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404 class="cmr-8">0</span></sub> die Frequenz des <br/>Resonators ist. <span |
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405 class="cmmi-12"><img |
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406 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Einig_de_1913/fulltext/img/cmmi12-40.png" alt="@" class="12x-x-40" /></span> <span |
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407 class="cmmi-12">G</span><sub ><span |
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408 class="cmmi-8">z</span></sub><img |
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409 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Einig_de_1913/fulltext/img/Einst_Einig_de_191330x.png" alt="/" class="left" align="middle" /> <span |
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410 class="cmmi-12"><img |
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411 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Einig_de_1913/fulltext/img/cmmi12-40.png" alt="@" class="12x-x-40" /> x</span> |
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412 setzen wir als Fouriersche Reihe an: |
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413 </p> |
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414 <center class="par-math-display" > |
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415 <img |
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416 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Einig_de_1913/fulltext/img/Einst_Einig_de_191331x.png" alt=" sum ( ) @-Gz-= Cn cos 2p n t-- hn . @ x T " class="par-math-display" /></center> |
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417 <!--l. 505--><p class="nopar"> |
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418 </p><!--l. 508--><p class="noindent">Dann wird:</p> |
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419 <center class="par-math-display" > |
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420 <img |
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421 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Einig_de_1913/fulltext/img/Einst_Einig_de_191332x.png" alt=" t integral sum ( t ) ( t) J = Cn cos 2 p n-- - hn f0 cos 2p n0 -- d t 0 T T sum ( ) ( ) = f0 Cn -----T----- sin p n0---n-t .cos p n0---n-t- hn , 2 p(n0- n) T T " class="par-math-display" /></center> |
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422 <!--l. 522--><p class="nopar"> |
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423 </p><!--l. 526--><p class="noindent">da das mit 1<img |
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424 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Einig_de_1913/fulltext/img/Einst_Einig_de_191333x.png" alt="/" class="left" align="middle" /> <span |
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425 class="cmmi-12">n</span><sub ><span |
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426 class="cmr-8">0</span></sub> + <span |
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427 class="cmmi-12">n</span> behaftete Glied wegfällt, weil <span |
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428 class="cmmi-12">n</span><sub ><span |
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429 class="cmr-8">0</span></sub> + <span |
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430 class="cmmi-12">n </span>eine <br/>sehr große Zahl ist. Setzt man nun <span |
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431 class="cmmi-12">n</span><img |
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432 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Einig_de_1913/fulltext/img/Einst_Einig_de_191334x.png" alt="/" class="left" align="middle" /><span |
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433 class="cmmi-12">T </span>= <span |
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434 class="cmmi-12"><img |
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435 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Einig_de_1913/fulltext/img/cmmi12-17.png" alt="n" class="12x-x-17" /> </span>und quadriert, <br/>so wird: |
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436 </p> |
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437 <center class="par-math-display" > |
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438 <img |
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439 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Einig_de_1913/fulltext/img/Einst_Einig_de_191335x.png" alt=" integral + oo |
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440 --2 -2- 2 --2-T- sin2-p(n0---n)t- J = D = f0 Cn 8 2 d n , - oo [p(n0 - n)] " class="par-math-display" /></center> |
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441 <!--l. 539--><p class="nopar"> |
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442 </p><!--l. 543--><p class="noindent">oder:</p> |
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443 <center class="par-math-display" > |
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444 <img |
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445 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Einig_de_1913/fulltext/img/Einst_Einig_de_191336x.png" alt="--- 1 ---- D2 = --f02 .Cn2 T .t. 8 " class="par-math-display" /></center> |
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447 </p><!--l. 553--><p class="noindent">Nun ist (l. c. p. 1114):</p> |
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448 <center class="par-math-display" > |
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449 <img |
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450 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Einig_de_1913/fulltext/img/Einst_Einig_de_191337x.png" alt="---- 64 p3 n2 Cn2 T = ------2- r. 15 c " class="par-math-display" /></center> |
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451 <!--l. 559--><p class="nopar"> |
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452 </p><!--l. 562--><p class="noindent">Also ist:</p> |
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453 <center class="par-math-display" > |
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454 <img |
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455 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Einig_de_1913/fulltext/img/Einst_Einig_de_191338x.png" alt="--- 8 p3 n2 D2 = -----2--r t .f02. 15 c " class="par-math-display" /></center> |
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456 <!--l. 568--><p class="nopar"> |
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457 </p><!--l. 571--><p class="noindent">Besitzt nun der Resonator die Nullpunktsenergie <span |
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458 class="cmmi-12">h<img |
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459 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Einig_de_1913/fulltext/img/cmmi12-17.png" alt="n" class="12x-x-17" /></span><sup ><span |
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460 class="cmr-8">1</span></sup>), so ist: |
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461 </p> |
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462 <center class="par-math-display" > |
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463 <img |
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464 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Einig_de_1913/fulltext/img/Einst_Einig_de_191339x.png" alt=" 3 1-K f 2 = h n2) oder f 2 = 2-h-n = 3-h-sc-.2) 2 0 0 K 8 n4 n2 " class="par-math-display" /></center> |
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465 <!--l. 579--><p class="nopar"> |
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466 </p><!--l. 582--><p class="noindent">---------- |
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467 </p><!--l. 585--><p class="indent"> 1) Es hat sich gezeigt, daß bei der hier skizzierten Rechnungs-<br/>weise die Nullpunktsenergie gleich <span |
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468 class="cmmi-12">h<img |
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469 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Einig_de_1913/fulltext/img/cmmi12-17.png" alt="n" class="12x-x-17" /> </span>gesetzt werden muß, um zur <br/>Planckschen Strahlungsformel zu gelangen. Spätere Untersuchungen <br/>müssen zeigen, ob die Diskrepanz zwischen dieser Annahme und der <br/>bei der Untersuchung über den Wasserstoff zugrunde gelegten Annahme <br/>bei strengerer Rechnung verschwindet. |
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470 </p><!--l. 594--><p class="indent"> 2) M. Planck, Wärmestrahlung 6. Aufl. p. 112 (Gleichung (168)). <pb/> |
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471 </p><!--l. 599--><p class="indent"> |
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472 |
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473 </p><!--l. 600--><p class="noindent">Mithin ist: |
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474 </p> |
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475 <center class="par-math-display" > |
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476 <img |
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477 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Einig_de_1913/fulltext/img/Einst_Einig_de_191340x.png" alt="--- 1 D2 = ---h cs r t . 5p " class="par-math-display" /></center> |
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478 <!--l. 606--><p class="nopar"> |
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479 </p><!--l. 609--><p class="noindent">Setzt man dies in die Gleichung |
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480 </p> |
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482 <img |
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483 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Einig_de_1913/fulltext/img/Einst_Einig_de_191341x.png" alt="--2 D = 2k T P t " class="par-math-display" /></center> |
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484 <!--l. 614--><p class="nopar"> |
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485 </p><!--l. 618--><p class="noindent">ein, so gelangt man zum Wienschen Strahlungsgesetz. Wir <br/>wollen hier jedoch gleich die Voraussetzung, daß die durch <br/>die Strahlung angeregte Schwingung des Resonators zu ver-<br/>nachlässigen sei, aufgeben. Nehmen wir nun an, daß die |
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486 <br/>Energie der dem Resonator von der Strahlung erteilten Schwin-<br/>gungen Impulsschwankungen liefert, die von den der Null-<br/>punktsenergie entsprechenden Schwankungen unabhängig sind, <br/>so können wir den quadratischen Mittelwert beider Impuls-<br/>schwankungen addieren.<sup ><span |
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487 class="cmr-8">1</span></sup>) Wir haben also zu dem oben be-<br/>rechneten Wert für <span class="overline"><img |
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488 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Einig_de_1913/fulltext/img/cmr12-1.png" alt="D" class="12x-x-1" /><sup ><span |
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489 class="cmr-8">2</span></sup></span> noch den von Einstein und Hopf <br/>(l. c. p. 1114, Gleichung (15)) hinzuzufügen und erhalten: |
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490 </p> |
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491 <center class="par-math-display" > |
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492 <img |
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493 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Einig_de_1913/fulltext/img/Einst_Einig_de_191342x.png" alt="--- 1 c4s t 2 D2 = 5p-h c s r t + 40-p2n3-r . " class="par-math-display" /></center> |
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494 <!--l. 637--><p class="nopar"> |
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495 </p><!--l. 640--><p class="noindent">Andererseits ist: |
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496 </p> |
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497 <center class="par-math-display" > |
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498 <img |
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499 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Einig_de_1913/fulltext/img/Einst_Einig_de_191343x.png" alt="--- ( ) D2 = 2 kT P t = 2k T t .3-cs-- r- n-d-r . 10p n 3 d n " class="par-math-display" /></center> |
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500 <!--l. 647--><p class="nopar"> |
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501 </p><!--l. 650--><p class="noindent">Es ergibt sich demnach als Differentialgleichung für <span |
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502 class="cmmi-12"><img |
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503 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Einig_de_1913/fulltext/img/cmmi12-25.png" alt="r" class="cmmi-12x-x-25" align="middle" /> </span>: |
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504 </p> |
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507 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Einig_de_1913/fulltext/img/Einst_Einig_de_191344x.png" alt=" 3 ( ) hr + -c----r2 = 3k T r - n-d-r . 8p n3 3 d n " class="par-math-display" /></center> |
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508 <!--l. 658--><p class="nopar"> |
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509 </p><!--l. 661--><p class="noindent">Die Auflösung dieser Gleichung liefert: |
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510 </p> |
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513 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Einig_de_1913/fulltext/img/Einst_Einig_de_191345x.png" alt=" 8p-n2---h-n--- r = c3 hknT- , e - 1 " class="par-math-display" /></center> |
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514 <!--l. 667--><p class="nopar"> |
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515 </p><!--l. 671--><p class="noindent">das Plancksche Strahlungsgesetz, und die Energie des Reso-<br/>nators ergibt sich zu:</p> |
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518 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Einig_de_1913/fulltext/img/Einst_Einig_de_191346x.png" alt=" hn E = --hn---- + h n . e kT- 1 " class="par-math-display" /></center> |
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519 <!--l. 677--><p class="nopar"> |
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520 </p><!--l. 680--><p class="noindent">---------- |
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521 </p><!--l. 683--><p class="indent"> 1) Es braucht kaum betont zu werden, daß diese Art des Vor-<br/>gehens sich nur durch unsere Unkenntnis der tatsächlichen Resonator-<br/>gesetze rechtfertigen läßt. |
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522 <pb/> |
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528 <!--l. 691--><p class="noindent"> |
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529 </p><!--l. 692--><p class="noindent">Zusammenfassung.</p></div> |
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530 <!--l. 695--><p class="indent"> 1. Die Euckensche Resultate über die spezifische Wärme <br/>des Wasserstoffs machen die Existenz einer Nullpunktsenergie <br/>vom Betrage <span |
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532 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Einig_de_1913/fulltext/img/cmmi12-17.png" alt="n" class="12x-x-17" /></span><img |
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533 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Einig_de_1913/fulltext/img/Einst_Einig_de_191347x.png" alt="/" class="left" align="middle" /> 2 wahrscheinlich. |
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534 </p><!--l. 701--><p class="indent"> 2. Die Annahme der Nullpunktsenergie eröffnet einen <br/>Weg, die Plancksche Strahlungsformel ohne Zuhilfenahme <br/>irgendwelcher Diskontinuitäten abzuleiten. Es erscheint je-<br/>doch zweifelhaft, ob auch die anderen Schwierigkeiten sich <br/>ohne die Annahme von Quanten werden bewältigen lassen. |
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535 </p><!--l. 708--><p class="indent"> Zürich, Dezember 1912. |
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536 </p> |
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540 </p><!--l. 711--><p class="noindent">(Eingegangen 5. Januar 1913.)</p></div> |
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544 </p><!--l. 715--><p class="noindent">----------</p></div> |
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548 </p><!--l. 719--><p class="noindent">Anmerkung bei der Korrektur:</p></div> |
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549 <!--l. 722--><p class="indent"> Hr. Prof. Weiß machte uns darauf aufmerksam, daß auch <br/>die Curieschen Messungen über den Paramagnetismus des <br/>gasförmigen Sauerstoffs darauf hinweisen, daß dessen Rotations-<br/>energie bei hohen Temperaturen den von der klassischen <br/>Theorie geforderten Wert und nicht einen um <span |
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551 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Einig_de_1913/fulltext/img/cmmi12-17.png" alt="n" class="12x-x-17" /></span><img |
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552 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Einig_de_1913/fulltext/img/Einst_Einig_de_191348x.png" alt="/" class="left" align="middle" /> 2 kleineren <br/>besitzt, wie dies ohne die Annahme einer Nullpunktsenergie <br/>zu erwarten sein würde. Es läßt sich leicht zeigen, daß in <br/>letzterem Falle bei der Genauigkeit der Curieschen Mes-<br/>sungen sich Abweichungen vom Curieschen Gesetz hätten <br/>zeigen müssen. |
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553 </p> |
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557 </p><!--l. 738--><p class="noindent">----------</p></div> |
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558 <!--l. 740--><p class="noindent"><pb/> |
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563 </body></html> |
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