[Physics] Quantum Mechanics – Rectangular Potential Barrier – Normalisation

hilbert-spacenormalizationquantum mechanicsscatteringwavefunction

I have a quick question regarding the normalisation of the wave function of a particle incident on a potential barrier specifically regarding the normalisation of the wave functions.

The problem is set up as on this webpage:

https://www.ntmdt.com/spm-basics/view/tunneling-effect

And through consideration of the boundary conditions (continuity of wave function and derivative of wave function) the final wave function looks like this:

Diagram 1

My question is: are the wave functions either side of the barrier normalisable? And if not does this mean the situation is not physical?

Best Answer

They are not normalisable because they either come from, or extend to infinity. This essentially means that the probability density blows off and gives non-physical results.