$$-\frac{\hbar^2}{2m}\frac{\text{d}^2\psi(x)}{\text{d}x^2}=E\psi(x)\\ \frac{\text{d}^2\psi(x)}{\text{d}x^2}=-\underbrace{\frac{2mE}{\hbar^2}}_{k^2}\psi(x)\\ \Rightarrow k=\sqrt{\frac{2mE}{k^2}}$$

$$\psi(x) = 0=\psi(L)\Rightarrow A\sin(k\cdot L)= 0\Rightarrow kL=n\pi$$

$$\sqrt{\frac{2mE}{\hbar^2}}L=n\pi\\ \Rightarrow E_n\frac{n^2\pi\hbar}{smL^2}=E_\text{kin}\\ \psi(x)=\begin{cases} A\cdot\sin(\frac{n\pi x}{L})&\text{, für } 0<x<L\\0&\text{, sonst}\end{cases}$$

$$\int_{-\infty}^\infty\left| \psi(x,t)\right|^2\text{d}x= \int_0^L A^2\cdot \sin^2(\frac{n\pi x}{L})\text{d} x =A^2\frac{L}{2}=1\\ \Rightarrow A =\sqrt{\frac{2}{L}}\\ \psi(x)=\begin{cases}\sqrt{\frac{2}{L}} \cdot\sin(\frac{n\pi x}{L})&\text{, für } 0<x<L\\0&\text{, sonst}\end{cases}$$