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+%English
+## The Schrodinger Equation
+%German
+## Die Schr"odinger Gleichung
+
+----
+
+### Die harmonische Schwingung
+
+$$ a = \frac{d v}{dt} = \frac{F}{m} = -\frac{D*x}{m} = \frac{d^2 x}{d^2 t}$$
+;$$ x = A*\cos(\omega * t) \Leftrightarrow \frac{d^2 x}{d^2 t} = -\frac{D*x}{m} = -\omega^2 * x $$
+;$$\omega = \sqrt{\frac{D}{m}}$$
+
+---
+
+### Die harmonische Schwingung
+
+$$ x = A*\cos(\sqrt{\frac{D}{m}} * t) $$
+;$$ v = \frac{d x}{d t} = \omega * A * \sin(\sqrt{\frac{D}{m}} * t)$$
+;$$ E_{kin} = \frac{1}{2} * m * v^2 = \frac{1}{2} * D * A^2 * \sin^2(\omega * t)$$
+;$$ E_{Feder} = \frac{1}{2} * D * x^2 = \frac{1}{2} * D * A^2 * \cos^2(\omega * t)$$
+
+---
+
+### Die harmonische Schwingung
+
+$$ E_{kin} = \frac{1}{2} * m * v^2 = \frac{1}{2} * D * A^2 * \sin^2(\omega * t)$$
+
+$$ E_{Feder} = \frac{1}{2} * D * x^2 = \frac{1}{2} * D * A^2 * \cos^2(\omega * t)$$
+
+;$$ E_{ges} = E_{kin} + E_{Feder} = \frac{1}{2} * D * A^2 * $$
+$$(\sin^2(\omega * t) + \cos^2(\omega * t)) = \frac{1}{2} * D * A^2 $$
+
+---
+
+### Die harmonische Schwingung
+
+$$ E_{ges} = \frac{1}{2} * D * A^2 $$
+
+;$$ \Psi = \sqrt{\frac{D}{2}}*\begin{pmatrix} x \\ \frac{v}{\omega} \end{pmatrix} = \sqrt{\frac{D}{2}}*\begin{pmatrix}
+A*\cos(\omega * t) \\ \frac{\omega * A * \sin(\omega * t)}{\omega}\end{pmatrix}$$
+;$$ \Psi = \sqrt{\frac{D}{2}}*\begin{pmatrix} A \\ A \end{pmatrix} \odot \circlearrowleft(\omega * t)
+= \sqrt{\frac{D}{2}}*A *\circlearrowleft(\omega * t) $$
+
+---
+
+### Die harmonische Schwingung
+
+$$ \Psi = \sqrt{\frac{D}{2}}*\begin{pmatrix} A \\ A \end{pmatrix} \odot \circlearrowleft(\omega * t)
+= \sqrt{\frac{D}{2}}*A *\circlearrowleft(\omega * t) $$
+$$ \textcolor{red}{\Psi=\sqrt{\frac{D}{2}} * A * e^{i*\omega*t}}$$
+;$$|\Psi|^2 = E_{ges} = \frac{D*A^2}2 $$
+
+---
+
+### Die harmonische Schwingung
+<video autoplay loop>
+  <source src="video/oscillator.mp4" type="video/mp4">
+</video>
+
+----
+
+### Wanderwelle
+
+<video autoplay loop>
+  <source src="video/travelling.mp4" type="video/mp4">
+</video>
+
+---
+
+### Wanderwelle
+
+<video autoplay loop width="50%" height="50%">
+  <source src="video/travelling.mp4" type="video/mp4">
+</video>
+
+$$ \Psi = A*\circlearrowleft(k*x - \omega * t) \textcolor{red}{= A*e^{i(k*x-\omega*t)}}$$
+
+---
+
+### Wanderwelle
+
+$$ \Psi = A*\circlearrowleft(k*x - \omega * t) \textcolor{red}{= A*e^{i(k*x-\omega*t)}}$$
+;$$ k = \frac{2*\pi}{\lambda}, \omega = 2*\pi*f $$
+;$$ \lambda = \frac{h}{p} \Rightarrow k = \frac{2*\pi*p}{h} $$
+;$$ \omega = 2 * \pi * \frac{v}{\lambda} = \frac{2 * \pi * v * p}{h} $$
+
+---
+
+### Wanderwelle
+
+$$ \Psi = A*\circlearrowleft(k*x - \omega * t) \textcolor{red}{= A*e^{i(k*x-\omega*t)}}$$
+
+$$ k = \frac{2*\pi*p}{h}, \omega = \frac{2 * \pi * v * p}{h} $$
+;$$ \Psi = A*\circlearrowleft(\frac{2 * \pi * p}{h}*x - \frac{2 * \pi * v * p}{h} * t) $$
+;$$ \Psi = A*\circlearrowleft(\frac{2*\pi*p}{h}*(x - v*t)) \textcolor{red}{= A*e^{\frac{i*p}{\hbar}*(x-v*t)}} $$
+
+----
+
+### Energieerhaltungssatz
+
+$$ \Psi = A*\circlearrowleft(\frac{2*\pi*p}{h}*(x - v*t)) \textcolor{red}{= A*e^{\frac{i*p}{\hbar}*(x-v*t)}} $$
+;$$ E_{ges} = E_{kin} + E_{pot} = \frac{p^2}{2*m} + V $$
+;$$ E_{ges} *\Psi = \frac{p^2}{2*m} * \Psi + V * \Psi $$
+
+---
+
+### Energieerhaltungssatz
+
+$$ \Psi = A*\circlearrowleft(\frac{2*\pi*p}{h}*(x - v*t)) \textcolor{red}{= A*e^{\frac{i*p}{\hbar}*(x-v*t)}} $$
+
+$$ E_{ges} *\Psi = \frac{p^2}{2*m} * \Psi + V * \Psi $$
+;$$ \frac{\partial \Psi}{\partial x} = \circlearrowleft(90\degree) * \frac{2*\pi*p}{h} * \Psi \textcolor{red}{=
+\frac{i*p}{\hbar} * \Psi} $$
+;$$ p = \circlearrowleft(-90\degree) * \frac{h}{2*\pi} * \frac{\partial \Psi}{\partial x} \textcolor{red}{=
+-i * \hbar * \frac{\partial \Psi}{\partial x}} $$
+
+---
+
+### Energieerhaltungssatz
+
+$$ \Psi = A*\circlearrowleft(\frac{2*\pi*p}{h}*(x - v*t)) \textcolor{red}{= A*e^{\frac{i*p}{\hbar}*(x-v*t)}} $$
+
+$$ E_{ges} *\Psi = \circlearrowleft(180\degree) * \frac{h^2}{8*m*\pi^2} * \frac{\partial^2 \Psi}{\partial^2 x} + V *
+\Psi $$
+;
+<div class=" small">
+  $$ E_{ges} *\Psi =- \frac{h^2}{8*m*\pi^2} * \frac{\partial^2 \Psi}{\partial^2 x} + V *\Psi
+  \textcolor{red}{=-\frac{\hbar^2}{2*m} * \frac{\partial^2 \Psi}{\partial^2 x} + V * \Psi } $$
+</div>
+
+---
+
+### Energieerhaltungssatz
+
+<div class=" small">
+  $$ E_{ges} *\Psi =- \frac{h^2}{8*m*\pi^2} * \frac{\partial^2 \Psi}{\partial^2 x} + V *\Psi
+  \textcolor{red}{=-\frac{\hbar^2}{2*m} * \frac{\partial^2 \Psi}{\partial^2 x} + V * \Psi } $$
+</div>
+
+$$ \Psi = A*\circlearrowleft(\frac{2*\pi*p}{h}*(x - v*t)) \textcolor{red}{= A*e^{\frac{i*p}{\hbar}*(x-v*t)}} $$
+
+;$$ E_{ph} = h*f = \frac{h*v}{\lambda} = \frac{h*v*p}{h} = v*p $$
+;$$ \frac{\partial \Psi}{\partial t} = -\circlearrowleft(90\degree) * \frac{2*\pi*p * v}{h} * \Psi$$
+
+---
+
+### Energieerhaltungssatz
+
+<div class=" small">
+  $$ E_{ges} *\Psi =- \frac{h^2}{8*m*\pi^2} * \frac{\partial^2 \Psi}{\partial^2 x} + V *\Psi
+  \textcolor{red}{=-\frac{\hbar^2}{2*m} * \frac{\partial^2 \Psi}{\partial^2 x} + V * \Psi } $$
+</div>
+
+$$ \Psi = A*\circlearrowleft(\frac{2*\pi*p}{h}*(x - v*t)) \textcolor{red}{= A*e^{\frac{i*p}{\hbar}*(x-v*t)}} $$
+
+$$ \frac{\partial \Psi}{\partial t} = -\circlearrowleft(90\degree) * \frac{2*\pi*p * v}{h} * \Psi$$
+
+;$$ p*v*\Psi = -\frac{\partial \Psi}{\partial t} * \frac{h}{2*\pi*\circlearrowleft(90\degree)}$$
+
+---
+
+### Energieerhaltungssatz
+
+<div class=" small">
+  $$ E_{ges} *\Psi =- \frac{h^2}{8*m*\pi^2} * \frac{\partial^2 \Psi}{\partial^2 x} + V *\Psi
+  \textcolor{red}{=-\frac{\hbar^2}{2*m} * \frac{\partial^2 \Psi}{\partial^2 x} + V * \Psi } $$
+</div>
+
+$$ \Psi = A*\circlearrowleft(\frac{2*\pi*p}{h}*(x - v*t)) \textcolor{red}{= A*e^{\frac{i*p}{\hbar}*(x-v*t)}} $$
+
+$$ p*v*\Psi = -\frac{\partial \Psi}{\partial t} * \frac{h}{2*\pi*\circlearrowleft(90\degree)}$$
+;$$ E_{ges} *\Psi = \circlearrowleft(90\degree)* \frac{h}{2*\pi} * \frac{\partial \Psi}{\partial t} \textcolor{red}{ =
+i*\hbar*\frac{\partial \Psi}{\partial t}}$$
+
+----
+
+### Die Schr"odinger Gleichung
+
+<div class=" small">
+  $$ \circlearrowleft(90\degree)* \frac{h}{2*\pi} * \frac{\partial \Psi}{\partial t} =- \frac{h^2}{8*m*\pi^2} *
+  \frac{\partial^2 \Psi}{\partial^2 x} + V *\Psi $$
+</div>
+<div class=" small">
+  $$\textcolor{red}{i*\hbar*\frac{\partial \Psi}{\partial t}=-\frac{\hbar^2}{2*m} * \frac{\partial^2 \Psi}{\partial^2
+  x} + V * \Psi } $$
+</div>
+
+----
+
+### Teilchen im Potentialtopf
+
+
+$$ V = 0 \forall x \in [0; l], V = \infty \forall x \notin [0; l] $$
+
+;$$ \Psi = 0 \forall x \notin [0, l], \Psi \in \mathbb{C} \forall x \in [0, l] $$
+;$$ \Psi = \Psi_x * \Psi_t $$
+
+---
+
+### Teilchen im Potentialtopf
+
+$$ \Psi = \Psi_x * \Psi_t $$
+
+;$$E*\Psi = -\frac{h^2}{8*m*\pi^2} * \frac{\partial^2 \Psi}{\partial^2 x}$$
+;$$E*\Psi_t * \Psi_x = -\frac{h^2}{8*m*\pi^2} *\Psi_t* \frac{\partial^2 \Psi_x}{\partial^2 x}$$
+;$$E* \Psi_x = -\frac{h^2}{8*m*\pi^2} * \frac{\partial^2 \Psi_x}{\partial^2 x}$$
+
+---
+
+### Teilchen im Potentialtopf
+
+$$ \Psi = A*\sin(k*x + \alpha_0) * \Psi_t $$
+
+$$E * \Psi_x = -\frac{h^2}{8*m*\pi^2} * \frac{\partial^2 \Psi_x}{\partial^2 x}$$
+;$$E * \Psi_x = \frac{h^2}{8*m*\pi^2} * k^2 \Psi_x$$
+;$$E = \frac{h^2*k^2}{8*m*\pi^2}$$
+
+---
+
+### Teilchen im Potentialtopf
+
+$$ \Psi = A*\sin(k*x + \alpha_0) * \Psi_t $$
+
+;$$ \circlearrowleft(90\degree)* \frac{h}{2*\pi} * \frac{\partial \Psi}{\partial t} = E*\Psi =
+\frac{h^2*k^2*\Psi_t}{8*m*\pi^2}
+\textcolor{red}{ = i*\hbar*\frac{\partial \Psi}{\partial t}} $$
+;$$ \circlearrowleft(90\degree)* \frac{h}{2*\pi} * \frac{\partial \Psi_t}{\partial t} = E*\Psi_t =
+\frac{h^2*k^2*\Psi_t}{8*m*\pi^2}\textcolor{red}{ = i*\hbar*\frac{\partial \Psi_t}{\partial t}} $$
+
+;$$ \Psi_t = \circlearrowleft(\omega * t) \textcolor{red}{=e^{i*\omega*t}} \Rightarrow \omega = -\frac{h*k^2}{4*\pi*m}
+$$
+
+---
+
+### Teilchen im Potentialtopf
+
+$$ \Psi = A*\sin(k*x + \alpha_0) * \circlearrowleft(-\frac{h*k^2}{4*\pi*m} * t)$$
+;$$ \Psi(0) = 0 \Rightarrow \alpha_0 = 0 $$
+;$$ \Psi(l) = 0 = A*\sin(k*x*l) \Rightarrow k = \frac{n*\pi}{l} \forall n\in \mathbb{N} $$
+;$$ E_n = \frac{h^2*n^2}{8*m*l^2}, \Psi = A*\sin(\frac{n*\pi*x}{l}) * \circlearrowleft(-\frac{h*k^2}{4*\pi*m} * t)$$
+$$\textcolor{red}{\Psi=A*\sin(\frac{n*\pi*x}{l}) * e^{-\frac{i*\hbar*k^2*t}{2*m}}} $$
+
+----
+
+### "Uberlagerung von Zust"anden
+
+$$ \Psi = \Psi_1 + \Psi_2, \rho = |\Psi|^2 = |\Psi_1 + \Psi_2|^2 $$
+;$ |\Psi|^2 = (\Psi_{1x}+\Psi_{2x})^2 + (\Psi_{1y} + \Psi_{2y})^2 $
+;$ |\Psi|^2 = A^2((\sin(k_1*x)*\cos(\omega_1*t) +
+\sin(k_2*x)*\cos(\omega_2*t))^2 +
+((\sin(k_1*x)*\sin(\omega_1*t) +
+\sin(k_2*x)*\sin(\omega_2*t))^2$
+
+---
+
+### "Uberlagerung von Zust"anden
+
+$$ |\Psi|^2 = A^2*(\sin^2(k_1*x) + \sin^2(k_2*x) + (2*\sin(k_1*x) * sin(k_2*x) * \cos((\omega_1-\omega_2)*t)$$
+;
+<video autoplay loop width="50%" height="50%">
+  <source src="video/photon.mp4" type="video/mp4">
+</video>
+
+---
+
+### "Uberlagerung von Zust"anden
+
+$$ |\Psi|^2 = A^2*(\sin^2(k_1*x) + \sin^2(k_2*x) + (2*\sin(k_1*x) * sin(k_2*x) * \cos((\omega_1-\omega_2)*t)$$
+
+;$$\omega = \omega_1-\omega_2, f = \frac{\omega}{2*\pi} = \frac{\omega_1 - \omega_2}{2*\pi}$$
+;$$ f = \frac{h}{2*\pi*4*\pi*m} *(k_1^2 - k_2^2) $$
+;$$ E_{ph} = h*f = \frac{h^2}{8*\pi^2*m} *(k_1^2 - k_2^2) $$
+;$$ E_{ph} = E_1 - E_2 $$