Partikel i en boks: Forskelle mellem versioner

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Tag: 2017-kilderedigering
Tag: 2017-kilderedigering
Det ses, at det tredje led er tidsafhængigt, og denne blandede tilstand er derfor ikke stationær. I animationen vises, hvordan sandsynlighedsfordelingen ændrer sig over tid.
 
==== Forventningsværdien ====
Udtrykket for sandsynlighedsfordelingen kan dog godt skrives lidt mere kompaktud:
:<math>\begin{align}\rho(x,t)&=\frac{1}{L}\sin^2\left(\frac{\pi}{L}x\right)+\frac{1}{L}\sin^2\left(\frac{2\pi}{L}x\right)+\frac{2}{L}\sin\left(\frac{\pi}{L}x\right)\sin\left(\frac{2\pi}{L}x\right)\cos\left(\frac{E_2-E_1}{\hbar}t\right)\end{align}</math>
:<math>\begin{align}\rho(x,t)&=\frac{1}{L}\left[\left(\frac{1}{2i}\left(e^{i\frac{\pi}{L}x}-e^{-i\frac{\pi}{L}x}\right)\right)^2+\left(\frac{1}{2i}\left(e^{i\frac{2\pi}{L}x}-e^{-i\frac{2\pi}{L}x}\right)\right)^2+2\left(\frac{1}{2i}\left(e^{i\frac{\pi}{L}x}-e^{-i\frac{\pi}{L}x}\right)\right)\left(\frac{1}{2i}\left(e^{i\frac{2\pi}{L}x}-e^{-i\frac{2\pi}{L}x}\right)\right)\cos\left(\frac{\frac{\pi^2 \hbar^2 2^2}{2mL^2}-\frac{\pi^2 \hbar^2 1^2}{2mL^2}}{\hbar}t\right)\right]\\
\rho(x,t)&=-\frac{1}{4L}\left[\left(e^{i\frac{\pi}{L}x}-e^{-i\frac{\pi}{L}x}\right)^2+\left(e^{i\frac{2\pi}{L}x}-e^{-i\frac{2\pi}{L}x}\right)^2+2\left(e^{i\frac{\pi}{L}x}-e^{-i\frac{\pi}{L}x}\right)\left(e^{i\frac{2\pi}{L}x}-e^{-i\frac{2\pi}{L}x}\right)\cos\left(\frac{3\pi^2 \hbar}{2mL^2}t\right)\right]\\
\rho(x,t)&=-\frac{1}{4L}\left[e^{i\frac{2\pi}{L}x}+e^{-i\frac{2\pi}{L}x}-2+e^{i\frac{4\pi}{L}x}+e^{-i\frac{4\pi}{L}x}-2+2\left(e^{i\frac{3\pi}{L}x}+e^{-i\frac{3\pi}{L}x}-e^{i\frac{\pi}{L}x}-e^{-i\frac{\pi}{L}x}\right)\cos\left(\frac{3\pi^2 \hbar}{2mL^2}t\right)\right]\\
\rho(x,t)&=-\frac{1}{4L}\left[2\cos\left(\frac{2\pi}{L}x\right)-4+2\cos\left(\frac{4\pi}{L}x\right)+2\left(2\cos\left(\frac{3\pi}{L}x\right)-2\cos\left(\frac{\pi}{L}x\right)\right)\cos\left(\frac{3\pi^2 \hbar}{2mL^2}t\right)\right]\\
:<math>\begin{align}\rho(x,t)&=-\frac{1}{L2L}\sin^2left[\cos\left(\frac{2\pi}{L}x\right)-2+\frac{1}{L}\sin^2cos\left(\frac{24\pi}{L}x\right)+\frac{2}{L}\sinleft(\cos\left(\frac{3\pi}{L}x\right)-\sincos\left(\frac{2\pi}{L}x\right)\right)\cos\left(\frac{E_2-E_1}{3\pi^2 \hbar}{2mL^2}t\right)\right]\end{align}</math>
 
== Kildehenvisninger ==
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