$$U(r)=-\frac{e^2}{4\pi\varepsilon _0}\frac{1}{r}$$

$$\text{d}V=r^2\sin(\vartheta)\text{d}r\text{d}\vartheta \text{d}\varphi\\ x=r\sin(\vartheta)\cos(\varphi)\\ y=r\sin(\vartheta)\sin(\varphi)\\ z=r\cos(\vartheta)\\ r=\sqrt{x^2+y^2+z^2}\\ \vartheta =\arccos\left(\frac{z}{r}\right)\\ \varphi=\arctan\left(\frac{y}{x}\right)$$

$$\nabla ^2 =\frac{1}{r^2}\frac{\partial}{\partial r}\left(r^2 \frac{\partial}{\partial r}\right)+\frac{1}{r^2\sin(\vartheta)}\frac{\partial}{\partial \vartheta}\left( \sin(\vartheta)\frac{\partial}{\partial \vartheta}\right)+\frac{1}{r^2\sin^2(\vartheta)}\left(\frac{\partial^2}{\partial \varphi^2} \right)$$

$$\boxed{-\frac{\hbar^2}{2m_e} \nabla ^2\psi(r,\vartheta,\varphi)+U(r,\vartheta,\varphi)\psi(r,\vartheta,\varphi)=E\psi(r,\vartheta,\varphi)}$$

$$\psi(r,\vartheta,\varphi)=R(r)Y(\vartheta, \varphi)$$

$$\ -\frac{\hbar^2}{2m}\frac{Y(\vartheta,\varphi)}{r^2}\frac{\partial }{\partial r}\left(r^2 \frac{\partial R(r)}{\partial r}\right)+\frac{R(r)}{r^2\sin(\vartheta)}\frac{\partial}{\partial \vartheta}\left( \sin(\vartheta)\frac{\partial Y(\vartheta,\varphi)}{\partial \vartheta}\right)+\frac{R(r)}{r^2\sin^2(\vartheta)}\left(\frac{\partial^2 Y(\vartheta,\varphi)}{\partial \varphi^2} \right)\\+V(r,\vartheta ,\varphi)R(r)Y(\vartheta,\varphi)=ER(r)Y(\vartheta,\varphi)$$

$$\underbrace{\frac{1}{R(r)}\frac{\partial }{\partial r}\left(r^2 \frac{\partial R(r)}{\partial r}\right)-\frac{2mr^2}{\hbar^2}\left(U(r)-E \right)}_\text{(i)}+\underbrace{\frac{1}{Y(\vartheta,\varphi)}\left\{ \frac{1}{\sin(\vartheta)}\frac{\partial}{\partial \vartheta}\left( \sin(\vartheta)\frac{\partial Y(\vartheta,\varphi)}{\partial \vartheta}\right)\\+\frac{1}{\sin^2(\vartheta)}\left(\frac{\partial^2 Y(\vartheta,\varphi)}{\partial \varphi^2} \right)\right\}}_\text{(ii)}=0\\ \text{(i)}=\ell(\ell+1),\ \text{(ii)}=-\ell(\ell+1)$$

$$\text{(ii)}\cdot Y(\vartheta, \varphi)=\frac{1}{\sin(\vartheta)}\frac{\partial}{\partial \vartheta}\left( \sin(\vartheta)\frac{\partial Y(\vartheta,\varphi)}{\partial \vartheta}\right)+\frac{1}{\sin^2(\vartheta)}\left(\frac{\partial^2 Y(\vartheta,\varphi)}{\partial \varphi^2} \right)=-\ell(\ell+1)\cdot Y(\vartheta, \varphi)$$

$$\hat{L}=\hat{r}\times \hat{p}\\ \hat{L}_x=\hat{y}\hat{p}_z-\hat{z}\hat{p}_y\\ \hat{L}_y=\hat{z}\hat{p}_x-\hat{x}\hat{p}_z\\ \hat{L}_z=\hat{x}\hat{p}_y-\hat{y}\hat{p}_x\\ \hat{L}_z=-i\hbar\frac{\partial}{\partial\varphi}\ \rightarrow \text{Übungsaufgabe}\\ \hat{L}_x=\hat{y}\left(-i\hbar\frac{\partial}{\partial z} \right)-\hat{z}\left(-i\hbar\frac{\partial}{\partial y} \right)\\ \hat{L}_y=\hat{z}\left(-i\hbar\frac{\partial}{\partial x} \right)-\hat{x}\left(-i\hbar\frac{\partial}{\partial z} \right)$$

$$\hat{L}_x=-i\hbar\left[\hat{y}\frac{\partial}{\partial z}-\hat{z}\frac{\partial}{\partial y} \right]=-i\hbar\left[\hat{y}\left(\frac{\partial r}{\partial z}\frac{\partial }{\partial r} + \frac{\partial \vartheta}{\partial z}\frac{\partial }{\partial \vartheta} +\frac{\partial \varphi}{\partial z}\frac{\partial }{\partial \varphi}\right)-\hat{z}\left(\frac{\partial r}{\partial y}\frac{\partial }{\partial r} + \frac{\partial \vartheta}{\partial y}\frac{\partial }{\partial \vartheta} +\frac{\partial \varphi}{\partial y}\frac{\partial }{\partial \varphi}\right) \right]\\ \hat{L}_y=-i\hbar\left[z \left(\frac{\partial r}{\partial x}\frac{\partial }{\partial r} + \frac{\partial \vartheta}{\partial x}\frac{\partial }{\partial \vartheta} +\frac{\partial \varphi}{\partial x}\frac{\partial }{\partial \varphi}\right)-x\left(\frac{\partial r}{\partial z}\frac{\partial }{\partial r} + \frac{\partial \vartheta}{\partial z}\frac{\partial }{\partial \vartheta} +\frac{\partial \varphi}{\partial z}\frac{\partial }{\partial \varphi}\right)\right]\\ x=r\sin(\vartheta)\cos(\varphi),\ y=r\sin(\vartheta)\sin(\varphi),\ z=r\cos(\vartheta)\\ r=\sqrt{x^2+y^2+z^2},\ \vartheta=\arccos\left(\frac{z}{\sqrt{x^2+y^2+z^2}}\right),\ \varphi=\arctan\left(\frac{y}{x}\right)\\ \frac{\partial r}{\partial x}= \sin(\vartheta)\cos(\varphi),\ \frac{\partial r}{\partial y}= \sin(\vartheta)\sin(\varphi),\ \frac{\partial r}{\partial z}= \cos(\vartheta)\\ \frac{\partial \vartheta }{\partial x}=\frac{\cos(\vartheta)\cos(\varphi)}{r},\ \frac{\partial \vartheta }{\partial y}=\frac{\cos(\vartheta)\sin(\varphi)}{r},\ \frac{\partial \vartheta }{\partial z}=-\frac{\sin(\vartheta)}{r}\\ \frac{\partial \varphi }{\partial x}=\frac{\sin(\varphi)}{r\sin(\vartheta)},\ \frac{\partial \varphi }{\partial y}=\frac{\cos(\varphi)}{r\sin(\vartheta)},\ \frac{\partial \varphi }{\partial z}=0\\ \Rightarrow\hat{L}_x=-i\hbar\left[-\sin(\varphi)\frac{\partial}{\partial \vartheta}-\underbrace{\cot(\vartheta)}_\text{cos/sin}\cos(\varphi) \frac{\partial }{\partial \varphi}\right]\\ \hat{L}_y=-i\hbar\left[\cos(\varphi)\frac{\partial}{\partial \vartheta}-\cot(\vartheta)\sin(\varphi) \frac{\partial }{\partial \varphi}\right]\\ \hat{L}^2=\hat{L}^2_x+\hat{L}^2_y+\hat{L}^2_z=-\hbar^2\left[\frac{1}{\sin(\vartheta)}\frac{\partial}{\partial \vartheta}\left( \sin(\vartheta)\frac{\partial Y(\vartheta,\varphi)}{\partial \vartheta}\right)+\frac{1}{\sin^2(\vartheta)}\left(\frac{\partial^2 Y(\vartheta,\varphi)}{\partial \varphi^2} \right) \right]\\ \Rightarrow\text{(ii)}\cdot Y(\vartheta ,\varphi):\ \hat{L}^2Y(\vartheta ,\varphi)=\underbrace{\hbar^2\ell(\ell +1)}_\text{Quadrat des Gesamtdrehimpuls}Y(\vartheta ,\varphi)$$

$$\sin(\vartheta)\frac{\partial}{\partial \vartheta}\left( \sin(\vartheta)\frac{\partial Y(\vartheta,\varphi)}{\partial \vartheta}\right)+\left(\frac{\partial^2 Y(\vartheta,\varphi)}{\partial \varphi^2} \right)=-\ell(\ell+1)\sin^2(\vartheta) Y(\vartheta ,\varphi)$$

$$Y(\vartheta ,\varphi)=\Theta(\vartheta)\Phi(\varphi)$$

$$\Rightarrow \underbrace{\frac{1}{\Theta(\vartheta)}\left[\sin(\vartheta) \frac{\partial}{\partial\vartheta}\left( \sin(\vartheta) \frac{\partial \Theta(\vartheta )}{\partial \vartheta}\right) \right]+\ell(\ell +1 )\sin^2(\vartheta)}_{\text{nur abhängig von } \vartheta}+\underbrace{\frac{1}{\Phi(\varphi)}\frac{\partial^2\Phi(\varphi)}{\partial \varphi^2}}_{\text{nur abhängig von } \varphi}=0$$

$$\frac{1}{\Theta(\vartheta)}\left[\sin(\vartheta) \frac{\partial}{\partial\vartheta}\left( \sin(\vartheta) \frac{\partial \Theta(\vartheta )}{\partial \vartheta}\right) \right]+\ell(\ell +1 )\sin^2(\vartheta)=m^2\\ \frac{1}{\Phi(\varphi)}\frac{\partial^2\Phi(\varphi)}{\partial \varphi^2}=-m^2\Rightarrow \frac{\partial^2\Phi(\varphi)}{\partial \varphi^2}=-m^2\Phi(\varphi)\Rightarrow \Phi(\varphi)=e^{+im\varphi}$$

$$\varphi'=\varphi +2\cdot n\cdot \pi\\ \Phi(\varphi +2\cdot n\cdot \pi)=\Phi(\varphi)\Rightarrow e ^{+im(\varphi+2\pi)}=e^{+im\varphi}\Rightarrow e^{+2\pi im}=1\Rightarrow m\in\Z\ (m=0,\pm1,\pm2,...)$$

$$\lambda=\frac{h}{p_\text{tan}}\Rightarrow \lambda = \frac{h}{\underbrace{M}_\text{Masse} v_\text{tan}}\\ 2\pi r=\frac{mh}{Mv_\text{tan}}\Rightarrow m\hbar=\underbrace{Mv_\text{tan} r}_{\vec{r} \times \vec{p_t}=\vec{L_z}}\Rightarrow L_z=m\hbar \, \,\,\,\,m=0,\pm1,\pm2,...,?$$