QM-106 Transmission Symmetry in Barrier and Double Potential Well

Is transmission probability same for particles approaching a one-dimensional barrier from either side ? Imagine a potential barrier of an arbitrary shape with a maximum height V0. Assume the barrier is in a finite region. If particles of energy E less than V0 are incident on the barrier, then some will tunnel through the barrier. The fraction that tunnel through is transmission coefficient. What happens if the particles are incident from the other direction? Intuitively, I think the transmission coefficient should be the same. Let us prove it, using an interesting mathematical tool Wronskian. Let us also expand out knowledge about finite potential well. Consider two of such wells, separated faaaaaaar away from each other. What will happen? Nothing ! The particles in the well remain unaffected by a faraway similar well. Why? Because the wave function of such particle vanishes much before reaching the other well. What happens as the second well comes close to the first well….meaning, what if the two wells are now separated by a barrier of width comparable to the size of the well, and not tooooo much. Lots of fun happens. Let us watch that.