https://wikiskripta.fjfi.cvut.cz/wiki/index.php?title=02KVANCV:Kapitola3&feed=atom&action=history 02KVANCV:Kapitola3 - Historie editací 2024-03-29T01:27:40Z Historie editací této stránky MediaWiki 1.25.2 https://wikiskripta.fjfi.cvut.cz/wiki/index.php?title=02KVANCV:Kapitola3&diff=7893&oldid=prev Steffy v 13. 9. 2017, 13:07 2017-09-13T13:07:57Z <p></p> <table class='diff diff-contentalign-left'> <col class='diff-marker' /> <col class='diff-content' /> <col class='diff-marker' /> <col class='diff-content' /> <tr style='vertical-align: top;'> <td colspan='2' style="background-color: white; color:black; text-align: center;">← Starší verze</td> <td colspan='2' style="background-color: white; color:black; text-align: center;">Verze z 13. 9. 2017, 13:07</td> </tr><tr><td colspan="2" class="diff-lineno" id="L27" >Řádka 27:</td> <td colspan="2" class="diff-lineno">Řádka 27:</td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>kde $\tilde{\psi}(\vec{p},0)$ je FT počáteční podmínky $\psi(\vec x,0)$, tj.</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>kde $\tilde{\psi}(\vec{p},0)$ je FT počáteční podmínky $\psi(\vec x,0)$, tj.</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>$$</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>$$</div></td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>\tilde{\psi}(\vec{p},<del class="diffchange diffchange-inline">t_{</del>0<del class="diffchange diffchange-inline">}</del>) = \frac{1}{(\sqrt{2\pi\hbar})^3}\int\limits_{\mathds{R}^3} e^{-\frac{i}{\hbar} \vec{p}\cdot\vec{x}} \psi(\vec{x},<del class="diffchange diffchange-inline">t_0</del>) d^3 x = \frac{C}{(\sqrt{2 A \hbar})^3} e^\frac{(\vec{B}-\frac{i}{\hbar}\vec{p})^2}{4 A}</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>\tilde{\psi}(\vec{p},0) = \frac{1}{(\sqrt{2\pi\hbar})^3}\int\limits_{\mathds{R}^3} e^{-\frac{i}{\hbar} \vec{p}\cdot\vec{x}} \psi(\vec{x},<ins class="diffchange diffchange-inline">0</ins>) d^3 x = \frac{C}{(\sqrt{2 A \hbar})^3} e^\frac{(\vec{B}-\frac{i}{\hbar}\vec{p})^2}{4 A}</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>$$</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>$$</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>Řešení v proměnné $\vec{x}$ získáme inverzní FT</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>Řešení v proměnné $\vec{x}$ získáme inverzní FT</div></td></tr> <tr><td colspan="2" class="diff-lineno" id="L70" >Řádka 70:</td> <td colspan="2" class="diff-lineno">Řádka 70:</td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>Doba letu částice od štěrbin na stínítko je $t = \frac{d M}{p}$. Hustota pravděpodobnosti nalezení částice v místě $x$ na stínítku je tedy rovna</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>Doba letu částice od štěrbin na stínítko je $t = \frac{d M}{p}$. Hustota pravděpodobnosti nalezení částice v místě $x$ na stínítku je tedy rovna</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>$$</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>$$</div></td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>|\psi(x,t = \frac{d M}{p})|^2 = |\psi_1(x) + \psi_2(x)|^2 = |\psi_1(x)|^2 + |\psi_2(x)|^2 + \psi_1(x)\overline{\psi_2}(x) + \overline{\psi_1}(x)\psi_2(x).</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">\left</ins>|\psi(x,t = \frac{d M}{p})<ins class="diffchange diffchange-inline">\right</ins>|^2 = |\psi_1(x) + \psi_2(x)|^2 = |\psi_1(x)|^2 + |\psi_2(x)|^2 + \psi_1(x)\overline{\psi_2}(x) + \overline{\psi_1}(x)\psi_2(x).</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>$$</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>$$</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>První dva členy odpovídají situaci jen s horní (resp. spodní) štěrbinou</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>První dva členy odpovídají situaci jen s horní (resp. spodní) štěrbinou</div></td></tr> </table> Steffy https://wikiskripta.fjfi.cvut.cz/wiki/index.php?title=02KVANCV:Kapitola3&diff=7890&oldid=prev Steffy v 12. 9. 2017, 08:40 2017-09-12T08:40:25Z <p></p> <table class='diff diff-contentalign-left'> <col class='diff-marker' /> <col class='diff-content' /> <col class='diff-marker' /> <col class='diff-content' /> <tr style='vertical-align: top;'> <td colspan='2' style="background-color: white; color:black; text-align: center;">← Starší verze</td> <td colspan='2' style="background-color: white; color:black; text-align: center;">Verze z 12. 9. 2017, 08:40</td> </tr><tr><td colspan="2" class="diff-lineno" id="L36" >Řádka 36:</td> <td colspan="2" class="diff-lineno">Řádka 36:</td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>&#160; e^{-A\frac{[\vec x-\vec B/(2A)]^2}{\chi(t)}},</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>&#160; e^{-A\frac{[\vec x-\vec B/(2A)]^2}{\chi(t)}},</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\end{equation}</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\end{equation}</div></td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>kde $\chi(t)=1+\frac{2iA\hbar}{M}<del class="diffchange diffchange-inline">(</del>t<del class="diffchange diffchange-inline">-t_0)</del>$.</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>kde $\chi(t)=1+\frac{2iA\hbar}{M} t$.</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\begin{cvi}</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\begin{cvi}</div></td></tr> <tr><td colspan="2" class="diff-lineno" id="L50" >Řádka 50:</td> <td colspan="2" class="diff-lineno">Řádka 50:</td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\navod Je zapotřebí spočítat $| \psi(x,t) |^2 \sim \left|e^{-A\frac{[\vec x-\vec B/(2A)]^2}{\chi(t)}}\right|^2$ (nezajímá nás časový vývoj normalizace, i v dalším počítání je vhodné vynechávat celkové faktory nezávisející na $x$). Odvoďte si a využijte $|e^{z}|^2 = e^{2 {\rm Re} z}$. Pro určení střední kvadratické odchylky atd. porovnejte výsledek s tvarem Gaussovy rozdělovací funkce a najdete</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\navod Je zapotřebí spočítat $| \psi(x,t) |^2 \sim \left|e^{-A\frac{[\vec x-\vec B/(2A)]^2}{\chi(t)}}\right|^2$ (nezajímá nás časový vývoj normalizace, i v dalším počítání je vhodné vynechávat celkové faktory nezávisející na $x$). Odvoďte si a využijte $|e^{z}|^2 = e^{2 {\rm Re} z}$. Pro určení střední kvadratické odchylky atd. porovnejte výsledek s tvarem Gaussovy rozdělovací funkce a najdete</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\begin{eqnarray}</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\begin{eqnarray}</div></td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>\nonumber&#160; \vec{x}_0(t) &amp; = &amp; \frac{{\rm Re} \vec{ B}}{2 {\rm Re}A} + \frac{\hbar}{M}{\rm Im} \vec{B} t - \frac{\hbar}{M}\frac{{\rm Im}A}{{\rm Re}A}{\rm Re}\vec{B}t ,\\</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>\nonumber&#160; \vec{x}_0(t) &amp; = &amp; \frac{{\rm Re} \vec{ B}}{2 {\rm Re}A} + \frac{\hbar}{M}{\rm Im} \vec{B}<ins class="diffchange diffchange-inline">\ </ins>t - \frac{\hbar}{M}\frac{{\rm Im}A}{{\rm Re}A}{\rm Re}\vec{B}<ins class="diffchange diffchange-inline">\ </ins>t ,\\</div></td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>\nonumber \sigma^2(t) &amp; = &amp; \frac{1}{4{\rm Re} A} + \frac{\hbar^2}{M^2}{\rm Re}A t^2 + \frac{\hbar^2}{M^2}\frac{({\rm Im}A)^2}{{\rm Re}A} t^2 - \frac{\hbar}{M}\frac{{\rm Im}A}{{\rm Re}A} t.</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>\nonumber \sigma^2(t) &amp; = &amp; \frac{1}{4{\rm Re} A} + \frac{\hbar^2}{M^2}{\rm Re}A<ins class="diffchange diffchange-inline">\ </ins>t^2 + \frac{\hbar^2}{M^2}\frac{({\rm Im}A)^2}{{\rm Re}A}<ins class="diffchange diffchange-inline">\ </ins>t^2 - \frac{\hbar}{M}\frac{{\rm Im}A}{{\rm Re}A}<ins class="diffchange diffchange-inline">\ </ins>t.</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\end{eqnarray}</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\end{eqnarray}</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>Neurčitost polohy je stejná ve všech směrech, tj. $(\Delta x_j) = \sigma(t)$. Vlnový balík se může po konečnou dobu zužovat, pokud je ${\rm Im}A\neq 0$. Pro $A&gt;0$ se pouze rozšiřuje, vztahy se zjednoduší na</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>Neurčitost polohy je stejná ve všech směrech, tj. $(\Delta x_j) = \sigma(t)$. Vlnový balík se může po konečnou dobu zužovat, pokud je ${\rm Im}A\neq 0$. Pro $A&gt;0$ se pouze rozšiřuje, vztahy se zjednoduší na</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\begin{eqnarray}</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\begin{eqnarray}</div></td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>\nonumber&#160; \vec{x}_0(t) &amp; = &amp; \frac{{\rm Re} \vec{ B}}{2 A} + \frac{\hbar}{M}{\rm Im} \vec{B} t,\\</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>\nonumber&#160; \vec{x}_0(t) &amp; = &amp; \frac{{\rm Re} \vec{ B}}{2 A} + \frac{\hbar}{M}{\rm Im} \vec{B}<ins class="diffchange diffchange-inline">\ </ins>t,\\</div></td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>\nonumber \sigma^2(t) &amp; = &amp; \frac{1}{4A} + \frac{\hbar^2}{M^2}A t^2.</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>\nonumber \sigma^2(t) &amp; = &amp; \frac{1}{4A} + \frac{\hbar^2}{M^2}A<ins class="diffchange diffchange-inline">\ </ins>t^2.</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\end{eqnarray}</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\end{eqnarray}</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>Zdvojnásobení: pro elektron cca $ 3 \, {\rm s}$, pro částici cca $10^{12} \, {\rm let}$.</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>Zdvojnásobení: pro elektron cca $ 3 \, {\rm s}$, pro částici cca $10^{12} \, {\rm let}$.</div></td></tr> </table> Steffy https://wikiskripta.fjfi.cvut.cz/wiki/index.php?title=02KVANCV:Kapitola3&diff=7882&oldid=prev Steffy v 11. 9. 2017, 10:21 2017-09-11T10:21:42Z <p></p> <table class='diff diff-contentalign-left'> <col class='diff-marker' /> <col class='diff-content' /> <col class='diff-marker' /> <col class='diff-content' /> <tr style='vertical-align: top;'> <td colspan='2' style="background-color: white; color:black; text-align: center;">← Starší verze</td> <td colspan='2' style="background-color: white; color:black; text-align: center;">Verze z 11. 9. 2017, 10:21</td> </tr><tr><td colspan="2" class="diff-lineno" id="L5" >Řádka 5:</td> <td colspan="2" class="diff-lineno">Řádka 5:</td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\begin{cvi}</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\begin{cvi}</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>Pomocí Fourierovy transformace určete řešení \sv y</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>Pomocí Fourierovy transformace určete řešení \sv y</div></td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>\rc e pro volnou částici, které v čase $<del class="diffchange diffchange-inline">t_0</del>$ má tvar</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>\rc e pro volnou částici, které v čase $<ins class="diffchange diffchange-inline">t=0</ins>$ má tvar</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\be</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\be</div></td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>\psi(\vec x,<del class="diffchange diffchange-inline">t_0</del>)=g(\vec x)=C\exp[-Ax^2+\vec B\vec x]</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>\psi(\vec x,<ins class="diffchange diffchange-inline">0</ins>)=g(\vec x)=C\exp[-Ax^2+\vec B\vec x]</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\ll{mvb}</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\ll{mvb}</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\ee</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\ee</div></td></tr> <tr><td colspan="2" class="diff-lineno" id="L18" >Řádka 18:</td> <td colspan="2" class="diff-lineno">Řádka 18:</td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>která převede \sv u \rc i na obyčejnou diferenciální rovnici 1. řádu v čase</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>která převede \sv u \rc i na obyčejnou diferenciální rovnici 1. řádu v čase</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>$$</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>$$</div></td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>i\hbar\frac{\partial \tilde{\psi}}{\partial t} = \frac{p^2}{<del class="diffchange diffchange-inline">2m</del>}\tilde{\psi}.</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>i\hbar\frac{\partial \tilde{\psi}}{\partial t} = \frac{p^2}{<ins class="diffchange diffchange-inline">2M</ins>}\tilde{\psi}.</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>$$</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>$$</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>Řešení této rovnice je</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>Řešení této rovnice je</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\begin{equation}</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\begin{equation}</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\label{free:p}</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\label{free:p}</div></td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>\tilde{\psi}(\vec{p},t) = e^{-\frac{i}{\hbar}\frac{p^2}{<del class="diffchange diffchange-inline">2m</del>}<del class="diffchange diffchange-inline">(</del>t<del class="diffchange diffchange-inline">-t_0)</del>}\tilde{\psi}(\vec{p},<del class="diffchange diffchange-inline">t_0</del>),</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>\tilde{\psi}(\vec{p},t) = e^{-\frac{i}{\hbar}\frac{p^2}{<ins class="diffchange diffchange-inline">2M</ins>}t}\tilde{\psi}(\vec{p},<ins class="diffchange diffchange-inline">0</ins>),</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\end{equation}</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\end{equation}</div></td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>kde $\tilde{\psi}(\vec{p},<del class="diffchange diffchange-inline">t_0</del>)$ je FT počáteční podmínky $\psi(\vec x,<del class="diffchange diffchange-inline">t_0</del>)$, tj.</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>kde $\tilde{\psi}(\vec{p},<ins class="diffchange diffchange-inline">0</ins>)$ je FT počáteční podmínky $\psi(\vec x,<ins class="diffchange diffchange-inline">0</ins>)$, tj.</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>$$</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>$$</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\tilde{\psi}(\vec{p},t_{0}) = \frac{1}{(\sqrt{2\pi\hbar})^3}\int\limits_{\mathds{R}^3} e^{-\frac{i}{\hbar} \vec{p}\cdot\vec{x}} \psi(\vec{x},t_0) d^3 x = \frac{C}{(\sqrt{2 A \hbar})^3} e^\frac{(\vec{B}-\frac{i}{\hbar}\vec{p})^2}{4 A}</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\tilde{\psi}(\vec{p},t_{0}) = \frac{1}{(\sqrt{2\pi\hbar})^3}\int\limits_{\mathds{R}^3} e^{-\frac{i}{\hbar} \vec{p}\cdot\vec{x}} \psi(\vec{x},t_0) d^3 x = \frac{C}{(\sqrt{2 A \hbar})^3} e^\frac{(\vec{B}-\frac{i}{\hbar}\vec{p})^2}{4 A}</div></td></tr> <tr><td colspan="2" class="diff-lineno" id="L36" >Řádka 36:</td> <td colspan="2" class="diff-lineno">Řádka 36:</td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>&#160; e^{-A\frac{[\vec x-\vec B/(2A)]^2}{\chi(t)}},</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>&#160; e^{-A\frac{[\vec x-\vec B/(2A)]^2}{\chi(t)}},</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\end{equation}</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\end{equation}</div></td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>kde $\chi(t)=1+\frac{2iA\hbar}{<del class="diffchange diffchange-inline">m</del>}(t-t_0)$.</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>kde $\chi(t)=1+\frac{2iA\hbar}{<ins class="diffchange diffchange-inline">M</ins>}(t-t_0)$.</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\begin{cvi}</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\begin{cvi}</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\label{vlnbal:pr}</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\label{vlnbal:pr}</div></td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Čemu je úměrná hustota pravděpodobnosti pro řešení</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Čemu je úměrná hustota pravděpodobnosti pro řešení (\<ins class="diffchange diffchange-inline">ref</ins>{<ins class="diffchange diffchange-inline">free:</ins>x})</div></td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline">$$</del></div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div></div></td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline">{\LARGE \psi</del>(\<del class="diffchange diffchange-inline">vec x,t)=C\chi(t)^</del>{<del class="diffchange diffchange-inline">-3/2}e^{\frac{\vec</del></div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div></div></td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline">B^2}{4A}}</del></div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div></div></td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline"> e^{-A\frac{[\vec </del>x<del class="diffchange diffchange-inline">-\vec B/(2A)]^2</del>}<del class="diffchange diffchange-inline">{\chi(t)}}</del></div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div></div></td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline">},\quad \chi(t)=1+\frac{2iA\hbar}{m}(t-t_0</del>)</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div></div></td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline">$$</del></div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div></div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>z příkladu \ref{ex:vlnbal}? Jak se mění poloha jejího maxima s časem? Čemu je</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>z příkladu \ref{ex:vlnbal}? Jak se mění poloha jejího maxima s časem? Čemu je</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>rovna její střední kvadratická odchylka? Jak se mění s časem?</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>rovna její střední kvadratická odchylka? Jak se mění s časem?</div></td></tr> <tr><td colspan="2" class="diff-lineno" id="L56" >Řádka 56:</td> <td colspan="2" class="diff-lineno">Řádka 50:</td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\navod Je zapotřebí spočítat $| \psi(x,t) |^2 \sim \left|e^{-A\frac{[\vec x-\vec B/(2A)]^2}{\chi(t)}}\right|^2$ (nezajímá nás časový vývoj normalizace, i v dalším počítání je vhodné vynechávat celkové faktory nezávisející na $x$). Odvoďte si a využijte $|e^{z}|^2 = e^{2 {\rm Re} z}$. Pro určení střední kvadratické odchylky atd. porovnejte výsledek s tvarem Gaussovy rozdělovací funkce a najdete</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\navod Je zapotřebí spočítat $| \psi(x,t) |^2 \sim \left|e^{-A\frac{[\vec x-\vec B/(2A)]^2}{\chi(t)}}\right|^2$ (nezajímá nás časový vývoj normalizace, i v dalším počítání je vhodné vynechávat celkové faktory nezávisející na $x$). Odvoďte si a využijte $|e^{z}|^2 = e^{2 {\rm Re} z}$. Pro určení střední kvadratické odchylky atd. porovnejte výsledek s tvarem Gaussovy rozdělovací funkce a najdete</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\begin{eqnarray}</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\begin{eqnarray}</div></td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>\nonumber&#160; \vec{x}_0(t) &amp; = &amp; \frac{{\rm Re} \vec{ B}}{2 {\rm Re}A} + \frac{\hbar}{<del class="diffchange diffchange-inline">m</del>}{\rm Im} \vec{B} <del class="diffchange diffchange-inline">(</del>t<del class="diffchange diffchange-inline">-t_0) </del>- \frac{\hbar}{<del class="diffchange diffchange-inline">m</del>}\frac{{\rm Im}A}{{\rm Re}A}{\rm Re}\vec{B}<del class="diffchange diffchange-inline">(</del>t<del class="diffchange diffchange-inline">-t_0) </del>,\\</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>\nonumber&#160; \vec{x}_0(t) &amp; = &amp; \frac{{\rm Re} \vec{ B}}{2 {\rm Re}A} + \frac{\hbar}{<ins class="diffchange diffchange-inline">M</ins>}{\rm Im} \vec{B} t - \frac{\hbar}{<ins class="diffchange diffchange-inline">M</ins>}\frac{{\rm Im}A}{{\rm Re}A}{\rm Re}\vec{B}t ,\\</div></td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>\nonumber \sigma^2(t) &amp; = &amp; \frac{1}{4{\rm Re} A} + \frac{\hbar^2}{<del class="diffchange diffchange-inline">m</del>^2}{\rm Re}A <del class="diffchange diffchange-inline">(</del>t<del class="diffchange diffchange-inline">-t_0)</del>^2 + \frac{\hbar^2}{<del class="diffchange diffchange-inline">m</del>^2}\frac{({\rm Im}A)^2}{{\rm Re}A} <del class="diffchange diffchange-inline">(</del>t<del class="diffchange diffchange-inline">-t_0)</del>^2 - \frac{\hbar}{<del class="diffchange diffchange-inline">m</del>}\frac{{\rm Im}A}{{\rm Re}A} <del class="diffchange diffchange-inline">(</del>t<del class="diffchange diffchange-inline">-t_0)</del>.</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>\nonumber \sigma^2(t) &amp; = &amp; \frac{1}{4{\rm Re} A} + \frac{\hbar^2}{<ins class="diffchange diffchange-inline">M</ins>^2}{\rm Re}A t^2 + \frac{\hbar^2}{<ins class="diffchange diffchange-inline">M</ins>^2}\frac{({\rm Im}A)^2}{{\rm Re}A} t^2 - \frac{\hbar}{<ins class="diffchange diffchange-inline">M</ins>}\frac{{\rm Im}A}{{\rm Re}A} t.</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\end{eqnarray}</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\end{eqnarray}</div></td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Vlnový balík se může po konečnou dobu zužovat, pokud je ${\rm Im}A\neq 0$. Pro $A&gt;0$ se pouze rozšiřuje, vztahy se zjednoduší na</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">Neurčitost polohy je stejná ve všech směrech, tj. $(\Delta x_j) = \sigma(t)$. </ins>Vlnový balík se může po konečnou dobu zužovat, pokud je ${\rm Im}A\neq 0$. Pro $A&gt;0$ se pouze rozšiřuje, vztahy se zjednoduší na</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\begin{eqnarray}</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\begin{eqnarray}</div></td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>\nonumber&#160; \vec{x}_0(t) &amp; = &amp; \frac{{\rm Re} \vec{ B}}{2 A} + \frac{\hbar}{<del class="diffchange diffchange-inline">m</del>}{\rm Im} \vec{B} <del class="diffchange diffchange-inline">(</del>t<del class="diffchange diffchange-inline">-t_0)</del>,\\</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>\nonumber&#160; \vec{x}_0(t) &amp; = &amp; \frac{{\rm Re} \vec{ B}}{2 A} + \frac{\hbar}{<ins class="diffchange diffchange-inline">M</ins>}{\rm Im} \vec{B} t,\\</div></td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>\nonumber \sigma^2(t) &amp; = &amp; \frac{1}{4A} + \frac{\hbar^2}{<del class="diffchange diffchange-inline">m</del>^2}A <del class="diffchange diffchange-inline">(</del>t<del class="diffchange diffchange-inline">-t_0)</del>^2.</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>\nonumber \sigma^2(t) &amp; = &amp; \frac{1}{4A} + \frac{\hbar^2}{<ins class="diffchange diffchange-inline">M</ins>^2}A t^2.</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\end{eqnarray}</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\end{eqnarray}</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>Zdvojnásobení: pro elektron cca $ 3 \, {\rm s}$, pro částici cca $10^{12} \, {\rm let}$.</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>Zdvojnásobení: pro elektron cca $ 3 \, {\rm s}$, pro částici cca $10^{12} \, {\rm let}$.</div></td></tr> <tr><td colspan="2" class="diff-lineno" id="L72" >Řádka 72:</td> <td colspan="2" class="diff-lineno">Řádka 66:</td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\navod Vlnová funkce popisující stav částice po průchodu štěrbinami je superpozicí vlnových balíků</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\navod Vlnová funkce popisující stav částice po průchodu štěrbinami je superpozicí vlnových balíků</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>$$</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>$$</div></td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>\psi(x,t) = \psi_1(x,t) + \psi_2(x,t),\quad \psi_{1,2}(x,t) = e^{-\frac{(x\mp x_0)^2}{4 \sigma_0^2\chi(t)}} = e^{\frac{(x\mp x_0)^2(2\sigma_0^2-i\frac{\hbar}{<del class="diffchange diffchange-inline">m</del>}t)}{2(4\sigma_0^4 + \frac{\hbar^2 t^2}{<del class="diffchange diffchange-inline">m</del>^2})}}.</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>\psi(x,t) = \psi_1(x,t) + \psi_2(x,t),\quad \psi_{1,2}(x,t) = e^{-\frac{(x\mp x_0)^2}{4 \sigma_0^2\chi(t)}} = e^{\frac{(x\mp x_0)^2(2\sigma_0^2-i\frac{\hbar}{<ins class="diffchange diffchange-inline">M</ins>}t)}{2(4\sigma_0^4 + \frac{\hbar^2 t^2}{<ins class="diffchange diffchange-inline">M</ins>^2})}}.</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>$$</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>$$</div></td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Doba letu částice od štěrbin na stínítko je $t = \frac{d <del class="diffchange diffchange-inline">m</del>}{p}$. Hustota pravděpodobnosti nalezení částice v místě $x$ na stínítku je tedy rovna</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Doba letu částice od štěrbin na stínítko je $t = \frac{d <ins class="diffchange diffchange-inline">M</ins>}{p}$. Hustota pravděpodobnosti nalezení částice v místě $x$ na stínítku je tedy rovna</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>$$</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>$$</div></td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>|\psi(x,t = \frac{d <del class="diffchange diffchange-inline">m</del>}{p})|^2 = |\psi_1(x) + \psi_2(x)|^2 = |\psi_1(x)|^2 + |\psi_2(x)|^2 + \psi_1(x)\overline{\psi_2}(x) + \overline{\psi_1}(x)\psi_2(x).</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>|\psi(x,t = \frac{d <ins class="diffchange diffchange-inline">M</ins>}{p})|^2 = |\psi_1(x) + \psi_2(x)|^2 = |\psi_1(x)|^2 + |\psi_2(x)|^2 + \psi_1(x)\overline{\psi_2}(x) + \overline{\psi_1}(x)\psi_2(x).</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>$$</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>$$</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>První dva členy odpovídají situaci jen s horní (resp. spodní) štěrbinou</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>První dva členy odpovídají situaci jen s horní (resp. spodní) štěrbinou</div></td></tr> </table> Steffy https://wikiskripta.fjfi.cvut.cz/wiki/index.php?title=02KVANCV:Kapitola3&diff=7874&oldid=prev Steffy v 11. 9. 2017, 08:30 2017-09-11T08:30:02Z <p></p> <table class='diff diff-contentalign-left'> <col class='diff-marker' /> <col class='diff-content' /> <col class='diff-marker' /> <col class='diff-content' /> <tr style='vertical-align: top;'> <td colspan='2' style="background-color: white; color:black; text-align: center;">← Starší verze</td> <td colspan='2' style="background-color: white; color:black; text-align: center;">Verze z 11. 9. 2017, 08:30</td> </tr><tr><td colspan="2" class="diff-lineno" id="L21" >Řádka 21:</td> <td colspan="2" class="diff-lineno">Řádka 21:</td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>$$</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>$$</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>Řešení této rovnice je</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>Řešení této rovnice je</div></td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline">$$</del></div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">\begin{equation}</ins></div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">\label{free:p}</ins></div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\tilde{\psi}(\vec{p},t) = e^{-\frac{i}{\hbar}\frac{p^2}{2m}(t-t_0)}\tilde{\psi}(\vec{p},t_0),</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\tilde{\psi}(\vec{p},t) = e^{-\frac{i}{\hbar}\frac{p^2}{2m}(t-t_0)}\tilde{\psi}(\vec{p},t_0),</div></td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline">$$</del></div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">\end{equation}</ins></div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>kde $\tilde{\psi}(\vec{p},t_0)$ je FT počáteční podmínky $\psi(\vec x,t_0)$, tj.</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>kde $\tilde{\psi}(\vec{p},t_0)$ je FT počáteční podmínky $\psi(\vec x,t_0)$, tj.</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>$$</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>$$</div></td></tr> <tr><td colspan="2" class="diff-lineno" id="L29" >Řádka 29:</td> <td colspan="2" class="diff-lineno">Řádka 30:</td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>$$</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>$$</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>Řešení v proměnné $\vec{x}$ získáme inverzní FT</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>Řešení v proměnné $\vec{x}$ získáme inverzní FT</div></td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline">$$</del></div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">\begin{equation}</ins></div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">\label{free:x}</ins></div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\psi(\vec{x},t) = \frac{1}{(\sqrt{2\pi\hbar})^3}\int\limits_{\mathds{R}^3} e^{\frac{i}{\hbar} \vec{p}\cdot\vec{x}} \tilde{\psi}(\vec{p},t) d^3 p =</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\psi(\vec{x},t) = \frac{1}{(\sqrt{2\pi\hbar})^3}\int\limits_{\mathds{R}^3} e^{\frac{i}{\hbar} \vec{p}\cdot\vec{x}} \tilde{\psi}(\vec{p},t) d^3 p =</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>C\chi(t)^{-3/2}e^{\frac{\vec B^2}{4A}}</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>C\chi(t)^{-3/2}e^{\frac{\vec B^2}{4A}}</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>&#160; e^{-A\frac{[\vec x-\vec B/(2A)]^2}{\chi(t)}},</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>&#160; e^{-A\frac{[\vec x-\vec B/(2A)]^2}{\chi(t)}},</div></td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline">$$</del></div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">\end{equation}</ins></div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>kde $\chi(t)=1+\frac{2iA\hbar}{m}(t-t_0)$.</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>kde $\chi(t)=1+\frac{2iA\hbar}{m}(t-t_0)$.</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr> </table> Steffy https://wikiskripta.fjfi.cvut.cz/wiki/index.php?title=02KVANCV:Kapitola3&diff=5553&oldid=prev Steffy v 8. 9. 2015, 13:55 2015-09-08T13:55:06Z <p></p> <table class='diff diff-contentalign-left'> <col class='diff-marker' /> <col class='diff-content' /> <col class='diff-marker' /> <col class='diff-content' /> <tr style='vertical-align: top;'> <td colspan='2' style="background-color: white; color:black; text-align: center;">← Starší verze</td> <td colspan='2' style="background-color: white; color:black; text-align: center;">Verze z 8. 9. 2015, 13:55</td> </tr><tr><td colspan="2" class="diff-lineno" id="L18" >Řádka 18:</td> <td colspan="2" class="diff-lineno">Řádka 18:</td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>která převede \sv u \rc i na obyčejnou diferenciální rovnici 1. řádu v čase</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>která převede \sv u \rc i na obyčejnou diferenciální rovnici 1. řádu v čase</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>$$</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>$$</div></td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>\frac{\partial \tilde{\psi}}{\partial t} = <del class="diffchange diffchange-inline">-\frac{i}{\hbar} </del>\frac{p^2}{2m}\tilde{\psi}.</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">i\hbar</ins>\frac{\partial \tilde{\psi}}{\partial t} = \frac{p^2}{2m}\tilde{\psi}.</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>$$</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>$$</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>Řešení této rovnice je</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>Řešení této rovnice je</div></td></tr> <tr><td colspan="2" class="diff-lineno" id="L65" >Řádka 65:</td> <td colspan="2" class="diff-lineno">Řádka 65:</td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\begin{cvi}</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\begin{cvi}</div></td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Částice s hmotností $m$ a hybností $p$ letí kolmo proti stěně se dvěma štěrbinami v bodech $\pm x_0$. Šířka štěrbin je $\sigma_0$. Ve vzdálenosti $d$ od štěrbin je stínítko. Určete hustotu pravděpodobnosti nalezení částice na stínítku. Předpokládejte, že po průchodu horní, resp. spodní štěrbinou, je stav částice možné popsat vlnovým balíkem se střední hodnotou polohy $\pm x_0$ a <del class="diffchange diffchange-inline">její </del>střední kvadratickou odchylkou rovnou $\sigma_0$.</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Částice s hmotností $m$ a hybností $p$ letí kolmo proti stěně se dvěma štěrbinami v bodech $\pm x_0$. Šířka štěrbin je $\sigma_0$. Ve vzdálenosti $d$ od štěrbin je stínítko. Určete hustotu pravděpodobnosti nalezení částice na stínítku. Předpokládejte, že po průchodu horní, resp. spodní štěrbinou, je stav částice možné popsat vlnovým balíkem se střední hodnotou polohy $\pm x_0$ a střední kvadratickou odchylkou rovnou $\sigma_0$.</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\end{cvi}</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\end{cvi}</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\navod Vlnová funkce popisující stav částice po průchodu štěrbinami je superpozicí vlnových balíků</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\navod Vlnová funkce popisující stav částice po průchodu štěrbinami je superpozicí vlnových balíků</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>$$</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>$$</div></td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>\psi(x,t) = \psi_1(x,t) + \psi_2(x,t),\quad \psi_{1,2}(x,t) = e^{-\frac{(x\mp x_0)^2}{4 \sigma_0^2\chi(t)}} = e^{\frac{(x\mp x_0)^2(2\sigma_0^2-i\frac{\hbar}{m}t)}{2(4\sigma_0^<del class="diffchange diffchange-inline">2 </del>+ \frac{\hbar^2 t^2}{m^2})}}.</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>\psi(x,t) = \psi_1(x,t) + \psi_2(x,t),\quad \psi_{1,2}(x,t) = e^{-\frac{(x\mp x_0)^2}{4 \sigma_0^2\chi(t)}} = e^{\frac{(x\mp x_0)^2(2\sigma_0^2-i\frac{\hbar}{m}t)}{2(4\sigma_0^<ins class="diffchange diffchange-inline">4 </ins>+ \frac{\hbar^2 t^2}{m^2})}}.</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>$$</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>$$</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>Doba letu částice od štěrbin na stínítko je $t = \frac{d m}{p}$. Hustota pravděpodobnosti nalezení částice v místě $x$ na stínítku je tedy rovna</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>Doba letu částice od štěrbin na stínítko je $t = \frac{d m}{p}$. Hustota pravděpodobnosti nalezení částice v místě $x$ na stínítku je tedy rovna</div></td></tr> <tr><td colspan="2" class="diff-lineno" id="L82" >Řádka 82:</td> <td colspan="2" class="diff-lineno">Řádka 82:</td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>Zbylé dva členy jsou zodpovědné za interferenci</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>Zbylé dva členy jsou zodpovědné za interferenci</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>$$</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>$$</div></td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>\psi_1(x)\overline{\psi_2}(x) + \overline{\psi_1}(x)\psi_2(x) = 2 e^{-\frac{x^2 +&#160; x_0^2}{2\sigma^2}} \cos\left(\frac{<del class="diffchange diffchange-inline">2</del>\hbar d p x x_0}{4 p^2\sigma_0^4 + \hbar^2 d^2}\right).</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>\psi_1(x)\overline{\psi_2}(x) + \overline{\psi_1}(x)\psi_2(x) = 2 e^{-\frac{x^2 +&#160; x_0^2}{2\sigma^2}} \cos\left(\frac{<ins class="diffchange diffchange-inline">4</ins>\hbar d p x x_0}{4 p^2\sigma_0^4 + \hbar^2 d^2}\right).</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>$$</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>$$</div></td></tr> </table> Steffy https://wikiskripta.fjfi.cvut.cz/wiki/index.php?title=02KVANCV:Kapitola3&diff=5008&oldid=prev Steffy: Založena nová stránka: %\wikiskriptum{02KVANCV} \chapter{Volná částice} \begin{cvi} Pomocí Fourierovy transformace určete řešení \sv y \rc e pro volnou částici, které v čase $t_0$ m... 2013-08-29T13:09:29Z <p>Založena nová stránka: %\wikiskriptum{02KVANCV} \chapter{Volná částice} \begin{cvi} Pomocí Fourierovy transformace určete řešení \sv y \rc e pro volnou částici, které v čase $t_0$ m...</p> <p><b>Nová stránka</b></p><div>%\wikiskriptum{02KVANCV}<br /> <br /> \chapter{Volná částice}<br /> <br /> \begin{cvi}<br /> Pomocí Fourierovy transformace určete řešení \sv y<br /> \rc e pro volnou částici, které v čase $t_0$ má tvar<br /> \be<br /> \psi(\vec x,t_0)=g(\vec x)=C\exp[-Ax^2+\vec B\vec x]<br /> \ll{mvb}<br /> \ee<br /> kde $Re\ A&gt;0,\ \vec B\in\complex^3,\ C\in\complex$.<br /> \ll{ex:vlnbal}<br /> \end{cvi}<br /> \navod<br /> Při řešení používáme Fourierovu transformaci (FT) ve tvaru<br /> $$\tilde\psi(\vec{p},t)=\frac{1}{(\sqrt{2\pi\hbar})^3}\int\limits_{\mathds{R}^3} e^{-\frac{i}{\hbar} \vec{p}\cdot\vec{x}} \psi(\vec{x},t) d^3 x,$$<br /> která převede \sv u \rc i na obyčejnou diferenciální rovnici 1. řádu v čase<br /> $$<br /> \frac{\partial \tilde{\psi}}{\partial t} = -\frac{i}{\hbar} \frac{p^2}{2m}\tilde{\psi}.<br /> $$<br /> Řešení této rovnice je<br /> $$<br /> \tilde{\psi}(\vec{p},t) = e^{-\frac{i}{\hbar}\frac{p^2}{2m}(t-t_0)}\tilde{\psi}(\vec{p},t_0),<br /> $$<br /> kde $\tilde{\psi}(\vec{p},t_0)$ je FT počáteční podmínky $\psi(\vec x,t_0)$, tj.<br /> $$<br /> \tilde{\psi}(\vec{p},t_{0}) = \frac{1}{(\sqrt{2\pi\hbar})^3}\int\limits_{\mathds{R}^3} e^{-\frac{i}{\hbar} \vec{p}\cdot\vec{x}} \psi(\vec{x},t_0) d^3 x = \frac{C}{(\sqrt{2 A \hbar})^3} e^\frac{(\vec{B}-\frac{i}{\hbar}\vec{p})^2}{4 A}<br /> $$<br /> Řešení v proměnné $\vec{x}$ získáme inverzní FT<br /> $$<br /> \psi(\vec{x},t) = \frac{1}{(\sqrt{2\pi\hbar})^3}\int\limits_{\mathds{R}^3} e^{\frac{i}{\hbar} \vec{p}\cdot\vec{x}} \tilde{\psi}(\vec{p},t) d^3 p =<br /> C\chi(t)^{-3/2}e^{\frac{\vec B^2}{4A}}<br /> e^{-A\frac{[\vec x-\vec B/(2A)]^2}{\chi(t)}},<br /> $$<br /> kde $\chi(t)=1+\frac{2iA\hbar}{m}(t-t_0)$.<br /> <br /> \begin{cvi}<br /> \label{vlnbal:pr}<br /> Čemu je úměrná hustota pravděpodobnosti pro řešení<br /> $$<br /> {\LARGE \psi(\vec x,t)=C\chi(t)^{-3/2}e^{\frac{\vec<br /> B^2}{4A}}<br /> e^{-A\frac{[\vec x-\vec B/(2A)]^2}{\chi(t)}}<br /> },\quad \chi(t)=1+\frac{2iA\hbar}{m}(t-t_0)<br /> $$<br /> z příkladu \ref{ex:vlnbal}? Jak se mění poloha jejího maxima s časem? Čemu je<br /> rovna její střední kvadratická odchylka? Jak se mění s časem?<br /> Za jak dlouho se zdvojnásobí &quot;šířka&quot; vlnového balíku<br /> pro elektron lokalisovaný s přesností 1 cm a pro částici s hmotností 1 gram,<br /> jejíž těžiště je lokalizováno s přesností $10^{-6}$m?<br /> \ll{ex:pstvb}<br /> \end{cvi}<br /> \navod Je zapotřebí spočítat $| \psi(x,t) |^2 \sim \left|e^{-A\frac{[\vec x-\vec B/(2A)]^2}{\chi(t)}}\right|^2$ (nezajímá nás časový vývoj normalizace, i v dalším počítání je vhodné vynechávat celkové faktory nezávisející na $x$). Odvoďte si a využijte $|e^{z}|^2 = e^{2 {\rm Re} z}$. Pro určení střední kvadratické odchylky atd. porovnejte výsledek s tvarem Gaussovy rozdělovací funkce a najdete<br /> \begin{eqnarray}<br /> \nonumber \vec{x}_0(t) &amp; = &amp; \frac{{\rm Re} \vec{ B}}{2 {\rm Re}A} + \frac{\hbar}{m}{\rm Im} \vec{B} (t-t_0) - \frac{\hbar}{m}\frac{{\rm Im}A}{{\rm Re}A}{\rm Re}\vec{B}(t-t_0) ,\\<br /> \nonumber \sigma^2(t) &amp; = &amp; \frac{1}{4{\rm Re} A} + \frac{\hbar^2}{m^2}{\rm Re}A (t-t_0)^2 + \frac{\hbar^2}{m^2}\frac{({\rm Im}A)^2}{{\rm Re}A} (t-t_0)^2 - \frac{\hbar}{m}\frac{{\rm Im}A}{{\rm Re}A} (t-t_0).<br /> \end{eqnarray}<br /> Vlnový balík se může po konečnou dobu zužovat, pokud je ${\rm Im}A\neq 0$. Pro $A&gt;0$ se pouze rozšiřuje, vztahy se zjednoduší na<br /> \begin{eqnarray}<br /> \nonumber \vec{x}_0(t) &amp; = &amp; \frac{{\rm Re} \vec{ B}}{2 A} + \frac{\hbar}{m}{\rm Im} \vec{B} (t-t_0),\\<br /> \nonumber \sigma^2(t) &amp; = &amp; \frac{1}{4A} + \frac{\hbar^2}{m^2}A (t-t_0)^2.<br /> \end{eqnarray}<br /> Zdvojnásobení: pro elektron cca $ 3 \, {\rm s}$, pro částici cca $10^{12} \, {\rm let}$.<br /> <br /> \begin{cvi}<br /> Částice s hmotností $m$ a hybností $p$ letí kolmo proti stěně se dvěma štěrbinami v bodech $\pm x_0$. Šířka štěrbin je $\sigma_0$. Ve vzdálenosti $d$ od štěrbin je stínítko. Určete hustotu pravděpodobnosti nalezení částice na stínítku. Předpokládejte, že po průchodu horní, resp. spodní štěrbinou, je stav částice možné popsat vlnovým balíkem se střední hodnotou polohy $\pm x_0$ a její střední kvadratickou odchylkou rovnou $\sigma_0$.<br /> \end{cvi}<br /> <br /> \navod Vlnová funkce popisující stav částice po průchodu štěrbinami je superpozicí vlnových balíků<br /> $$<br /> \psi(x,t) = \psi_1(x,t) + \psi_2(x,t),\quad \psi_{1,2}(x,t) = e^{-\frac{(x\mp x_0)^2}{4 \sigma_0^2\chi(t)}} = e^{\frac{(x\mp x_0)^2(2\sigma_0^2-i\frac{\hbar}{m}t)}{2(4\sigma_0^2 + \frac{\hbar^2 t^2}{m^2})}}.<br /> $$<br /> Doba letu částice od štěrbin na stínítko je $t = \frac{d m}{p}$. Hustota pravděpodobnosti nalezení částice v místě $x$ na stínítku je tedy rovna<br /> $$<br /> |\psi(x,t = \frac{d m}{p})|^2 = |\psi_1(x) + \psi_2(x)|^2 = |\psi_1(x)|^2 + |\psi_2(x)|^2 + \psi_1(x)\overline{\psi_2}(x) + \overline{\psi_1}(x)\psi_2(x).<br /> $$<br /> První dva členy odpovídají situaci jen s horní (resp. spodní) štěrbinou<br /> $$<br /> |\psi_{1,2}(x)|^2 = e^{-\frac{(x\mp x_0)^2}{2\sigma^2}},\quad \sigma^2 = \sigma_0^2 + \left(\frac{\hbar d}{4p\sigma_0}\right)^2.<br /> $$<br /> Zbylé dva členy jsou zodpovědné za interferenci<br /> $$<br /> \psi_1(x)\overline{\psi_2}(x) + \overline{\psi_1}(x)\psi_2(x) = 2 e^{-\frac{x^2 + x_0^2}{2\sigma^2}} \cos\left(\frac{2\hbar d p x x_0}{4 p^2\sigma_0^4 + \hbar^2 d^2}\right).<br /> $$</div> Steffy