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\begin{document}
Transformation der Schrodinger-Gleichung in Schwerpunkt- und Relativkoordinaten:
\begin{align*}
\frac{\partial \psi}{\partial x_{1}} &= \frac{\partial \psi}{\partial X} \cdot \frac{\partial X}{\partial x_{1}} + \frac{\partial \psi}{\partial x} \cdot \frac{\partial x}{\partial x_{1}} &\frac{\partial \psi}{\partial x_{2}} &= \frac{\partial \psi}{\partial X} \cdot \frac{\partial X}{\partial x_{2}} + \frac{\partial \psi}{\partial x} \cdot \frac{\partial x}{\partial x_{2}}  \\
\frac{\partial \psi}{\partial y_{1}} &= \frac{\partial \psi}{\partial Y} \cdot \frac{\partial Y}{\partial y_{1}} + \frac{\partial \psi}{\partial y} \cdot \frac{\partial y}{\partial y_{1}} &\frac{\partial \psi}{\partial y_{2}} &= \frac{\partial \psi}{\partial Y} \cdot \frac{\partial Y}{\partial y_{2}} + \frac{\partial \psi}{\partial y} \cdot \frac{\partial y}{\partial y_{2}}  \\
\frac{\partial \psi}{\partial z_{1}} &= \frac{\partial \psi}{\partial Z} \cdot \frac{\partial Z}{\partial z_{1}} + \frac{\partial \psi}{\partial z} \cdot \frac{\partial z}{\partial z_{1}} &\frac{\partial \psi}{\partial z_{2}} &= \frac{\partial \psi}{\partial Z} \cdot \frac{\partial Z}{\partial z_{2}} + \frac{\partial \psi}{\partial z} \cdot \frac{\partial z}{\partial z_{2}}  \\
\end{align*}
Schwerpunkt- und Relativkoordinaten:
\begin{align*}
X &= \frac{m_{1} \cdot x_{1} + m_{2} \cdot x_{2}}{M} & x = x_{1} - x_{2} \\
Y &= \frac{m_{1} \cdot y_{1} + m_{2} \cdot y_{2}}{M} & y = y_{1} - y_{2} \\
Z &= \frac{m_{1} \cdot z_{1} + m_{2} \cdot z_{2}}{M} & z = z_{1} - z_{2} \\
\end{align*}
Erste Ableitungen von Schwerpunkt- und Relativkoordinaten:
\begin{flalign*}
\frac{\partial X}{\partial x_{1}} &= \frac{m_{1}}{M} &  \frac{\partial X}{\partial x_{2}} &= \frac{m_{2}}{M} & \frac{\partial x}{\partial x_{1}} &= + 1 &  \frac{\partial x}{\partial x_{2}} &= -1 \\
\frac{\partial Y}{\partial y_{1}} &= \frac{m_{1}}{M} &  \frac{\partial Y}{\partial y_{2}} &= \frac{m_{2}}{M} & \frac{\partial y}{\partial y_{1}} &= + 1 &  \frac{\partial y}{\partial y_{2}} &= -1 \\
\frac{\partial Z}{\partial z_{1}} &= \frac{m_{1}}{M} &  \frac{\partial Z}{\partial z_{2}} &= \frac{m_{2}}{M} & \frac{\partial z}{\partial z_{1}} &= + 1 &  \frac{\partial z}{\partial z_{2}} &= -1 \\
\end{flalign*}
Erste (partielle) Ableitungen der Wellenfunktion:
\begin{align*}
\frac{\partial \psi}{\partial x_{1}} &= \frac{m_{1}}{M} \cdot \frac{\partial \psi}{\partial X} + \frac{\partial \psi}{\partial x} & \frac{\partial \psi}{\partial x_{2}} &= \frac{m_{2}}{M} \cdot \frac{\partial \psi}{\partial X} - \frac{\partial \psi}{\partial x}\\
\frac{\partial \psi}{\partial y_{1}} &= \frac{m_{1}}{M} \cdot \frac{\partial \psi}{\partial Y} + \frac{\partial \psi}{\partial y} & \frac{\partial \psi}{\partial y_{2}} &= \frac{m_{2}}{M} \cdot \frac{\partial \psi}{\partial Y} - \frac{\partial \psi}{\partial y}\\
\frac{\partial \psi}{\partial z_{1}} &= \frac{m_{1}}{M} \cdot \frac{\partial \psi}{\partial Z} + \frac{\partial \psi}{\partial z} & \frac{\partial \psi}{\partial z_{2}} &= \frac{m_{2}}{M} \cdot \frac{\partial \psi}{\partial Z} - \frac{\partial \psi}{\partial z}\\
\end{align*}
\end{document}