Quantenmechanik 1
Abstract
cheatsheet
based on Quantenmechanik 1 HS25
Time Dependant Schrödinger Equation (TDSE):
Wave:
$$ \begin{split} \Psi(x, t) = \Psi(0, 0)e^{\frac{2\pi i x}{\lambda} - i\omega t}, \quad \lambda =\frac{2\pi\hbar}{p}, \omega = \frac{E}{\hbar} \\ \leadsto \begin{cases} \partial_x\Psi(x, t) = \frac{2\pi i}{\lambda}\Psi(x, t) = i\frac{p}{\hbar}\Psi(x, t) \\ \partial_t\Psi(x, t) = -i\omega\Psi(x, t) = -i\frac{E}{\hbar}\Psi(x, t) \end{cases} \end{split} $$ $$ \begin{split} i\hbar \partial_t\Psi (x, t) &= \left(-\frac{\hbar^2}{2m}\partial_x^2 + V(x, t)\right)\Psi (x, t) \\ &= H\Psi (x, t) \end{split} $$
$$ \begin{split} \Psi(x, t) = \Psi(0, 0)e^{\frac{2\pi i x}{\lambda} - i\omega t}, \quad \lambda =\frac{2\pi\hbar}{p}, \omega = \frac{E}{\hbar} \\ \leadsto \begin{cases} \partial_x\Psi(x, t) = \frac{2\pi i}{\lambda}\Psi(x, t) = i\frac{p}{\hbar}\Psi(x, t) \\ \partial_t\Psi(x, t) = -i\omega\Psi(x, t) = -i\frac{E}{\hbar}\Psi(x, t) \end{cases} \end{split} $$ $$ \begin{split} i\hbar \partial_t\Psi (x, t) &= \left(-\frac{\hbar^2}{2m}\partial_x^2 + V(x, t)\right)\Psi (x, t) \\ &= H\Psi (x, t) \end{split} $$
Normalization Condition (on $\Psi$):
$$\int_\mathbb{R}dx |\Psi(x, t)|^2 = 1 \quad \forall t$$