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The Schrodinger Equation

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Die Schr"odinger Gleichung

Die harmonische Schwingung

$$ a = \frac{d v}{dt} = \frac{F}{m} = -\frac{Dx}{m} = \frac{d^2 x}{d^2 t}$$ ;$$ x = A\cos(\omega t) \Leftrightarrow \frac{d^2 x}{d^2 t} = -\frac{Dx}{m} = -\omega^2 * x $$ ;$$\omega = \sqrt{\frac{D}{m}}$$

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

$$ \Psi = \sqrt{\frac{D}{2}}\begin{pmatrix} A \ A \end{pmatrix} \odot \circlearrowleft(\omega t) = \sqrt{\frac{D}{2}}A \circlearrowleft(\omega t) $$ $$ \textcolor{red}{\Psi=\sqrt{\frac{D}{2}} A e^{i\omegat}}$$ ;$$|\Psi|^2 = E_{ges} = \frac{DA^2}2 $$

Die harmonische Schwingung

Wanderwelle

Wanderwelle

$$ \Psi = A\circlearrowleft(kx - \omega t) \textcolor{red}{= Ae^{i(kx-\omegat)}}$$

Wanderwelle

$$ \Psi = A\circlearrowleft(kx - \omega t) \textcolor{red}{= Ae^{i(kx-\omegat)}}$$ ;$$ k = \frac{2\pi}{\lambda}, \omega = 2\pif $$ ;$$ \lambda = \frac{h}{p} \Rightarrow k = \frac{2\pip}{h} $$ ;$$ \omega = 2 \pi \frac{v}{\lambda} = \frac{2 \pi v p}{h} $$

Wanderwelle

$$ \Psi = A\circlearrowleft(kx - \omega t) \textcolor{red}{= Ae^{i(kx-\omegat)}}$$

$$ k = \frac{2\pip}{h}, \omega = \frac{2 \pi v p}{h} $$ ;$$ \Psi = A\circlearrowleft(\frac{2 \pi p}{h}x - \frac{2 \pi v p}{h} t) $$ ;$$ \Psi = A\circlearrowleft(\frac{2\pip}{h}(x - vt)) \textcolor{red}{= Ae^{\frac{ip}{\hbar}(x-vt)}} $$