3. The attempt at a solution
The energy ##100eV## must be the kinetic energy of the electron. So i said ok this kinetic energy is very small compared to the rest energy and i can say that ##pc \ll E_0## which means i have a classical limit where:

So if i want to get the speciffic solution i need to calculate the constant ##L## and then normalise the ##\psi##. Because ##E=E_0## i calculated the constant ##L## like this:

$$L=\sqrt{\frac{2mE_0}{\hbar^2}}$$

while my professor states that i should do it like this:

$$L=\sqrt{\frac{2mE_k}{\hbar^2}}$$

where ##E_k## is the kinetic energy of the electron. Who is wrong? I mean whaaaaat? The constant ##L## is afterall defined using the full energy and not kinetic energy...

You're wrong. Both your expression for L and the Schrodinger equation are non-relativistic. It's not appropriate to use the relativistic expression for the energy of the electron here.

But can you explain to me why don't we generaly use the rest energy when dealing with a classical approximations. In books everyone explains the rest energy but what it realy is? I mean i know how to calculate it and whatsoever but when does it appear and why do we have to take it into the calculation?

I am confused only because of that Lorentz invariance which says that if a particle is mooving slowly most of its energy is the rest energy. On the other hand we just neglect it like it isn't there... It seems to me like a contradiction...

It's just the way you're doing the classical approximation isn't correct. A plane wave is given by $$\psi = e^{i(kx-\omega t)}$$ where ##p= \hbar k## and ##E = \hbar \omega##. For the spatial dependence, you need to find the classical approximation for the momentum p:
\begin{align*}
pc &= \sqrt{E^2 - (mc^2)^2} \\
&= \sqrt{(mc^2+K)^2 - (mc^2)^2} \\
&= \sqrt{2mc^2 K + K^2} \\
&\cong \sqrt{2mc^2 K}
\end{align*} where K is the kinetic energy. The rest energy term cancels out.

The other way to look at it is that ##k = \sqrt{\frac{2mE}{\hbar^2}}## is equivalent to the relationship ##E = \frac{(\hbar k)^2}{2m} = \frac{p^2}{2m}##. This latter expression is clearly the classical quantity for the kinetic energy of a particle of mass m, not the total energy.