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Gert
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The Schrödinger equation for the 1D potential well is given by:

$$-\frac{\hbar}{2m}\frac{\text{d}^2 \psi}{\text{d}x^2}=E\psi$$

Slightly re-written:

$$\psi''+k^2\psi=0\tag{1}$$

where:

$$k^2=\frac{2mE}{\hbar}$$

$(1)$ is a second order, linear, homogeneous differential equation and any decent math textbook will tell you it has the following solution, thus not experimentally obtained:

$$\psi=A\sin kx + B\cos kx$$

where $A$ and $B$ are integration constants.

Their value is determined by means of the boundary conditions $\psi(0)=\psi(L)=0$ (assuming the wave function acts on the domain$[0,L]$.

Gert
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