Lets say we have a potential step as in the picture:(adsbygoogle = window.adsbygoogle || []).push({});

In the regionIthere is a free particle with a wavefunction ##\psi_I## while in the regionIIthe wave function will be ##\psi_{II}##. Let me now take the Schrödinger equation and try to derive ##\psi_I## which bugs me:

\begin{align}

&~~W \psi = -\frac{\hbar^2}{2m}\, \frac{d^2 \Psi}{d\, x^2} + W_p \psi ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\nonumber \\

&~~W \psi = -\frac{\hbar^2}{2m}\, \frac{d^2 \Psi}{d\, x^2}\nonumber \\

&\frac{d^2 \Psi}{d\, x^2} = -\frac{2m W}{\hbar^2}\,\psi \nonumber\\

{\scriptsize \text{DE: }} &\boxed{\frac{d^2 \Psi}{d\, x^2} = -\mathcal L\,\psi}~\boxed{\mathcal{L} \equiv \sqrt{\tfrac{2mW}{\hbar^2}}} \nonumber\\

&~\phantom{\line(1,0){18.3}}\Downarrow \nonumber\\

{\scriptsize \text{general solution of DE: }} &\boxed{\psi_{I} = C \sin\left(\mathcal{L}\, x \right) + D \cos \left(\mathcal{L}\, x \right)}\nonumber

\end{align}

I got the general solution for the intervalI, but this is nothing like the solution they use in all the books: ##\psi_{I} = C e^{i\mathcal L x} + D e^{-i \mathcal L x}## where ##\mathcal L \equiv \sqrt{{\scriptsize 2mW/\hbar^2}}##. I have a personal issue with this because if ##x= -\infty## part ##De^{-i \mathcal L x}## would become infinite and this is impossible for a wavefunction! I know that i would get exponential form if i defined constant ##\mathcal L## a bit differently as i did above:

\begin{align}

{\scriptsize \text{DE: }} &\boxed{\frac{d^2 \Psi}{d\, x^2} = \mathcal L\,\psi}~\boxed{\mathcal{L} \equiv -\sqrt{\tfrac{2mW}{\hbar^2}}} \nonumber\\

&~\phantom{\line(1,0){18.3}}\Downarrow \nonumber\\

{\scriptsize \text{general solution of DE: }} &\boxed{ \psi_{I} = C e^{\mathcal L x } + D^{-\mathcal L x} }\nonumber

\end{align}

This general solution looks more like the one they use in the books but it lacks an imaginary ##i## and ##\mathcal L## is defined with a - while in all the books it is positive. Could anyone tell me what am i missing here so i could connect all this into a solid one piece of knowledge?

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# Solutions of the schrödinger eq. for a potential step

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