Should scattering states in 1D double delta function potential not lead to a steady-state problem?

Question

I have been trying for some time now to understand the double $\delta$-function potential, as solved here. One question in particular is why we don't consider scattering "back" from the second potential.

Thus: assume we have two $\delta$-function potentials, and a particle coming from the left, arriving at the first potential. Then the particle is either reflected or transmitted, with some probability. But surely the same holds true at the other $\delta$-potential? I.e. the particle could meet the second potential, and get reflected back towards the first potential? And at the first potential it might again get either reflected (going back towards the second potential) or transmitted (going out from the first potential, adding to the first transmission).

And surely this process could continue indefintely (albeit with smaller and smaller probabilities). Should this not lead to a steady state problem, where one needs to take into account the probability that the particle "bounces around" a certain number of times before being either transmitted or reflected? If so, then why do we define the transmission/reflection coefficients simply as the ratio of the amplitudes of the initial reflection/transmission? Or is this possibility somehow contained in this coefficients?

I have looked around and found e.g.this Stack Exchange question, which mentions the $S$-matrix for a single $\delta$-function potential. Could such an approach be used here?

Answer 1

This possibility is already contained in the coefficients.

When Griffiths is solving the coefficients, he is going for a general solution of the Schrodinger equation, which means that the solution will be an eigenstate of the equation and not evolve with time. He is already going for a steady-state solution. Working with the arbitrary coefficients takes into account the amplitude of the wave incident on the delta from a given side whatever it is, no matter what potential is on that side (wither a single delta, or a Gaussian mound, or whatever).

You could separately consider a spatially localized & time-dependent wave-packet incident on the two deltas, in which case if you mapped out the time-evolution carefully I'd indeed expect to see some sloshing back and forth inside the deltas as smaller and smaller parts of the wave-function bounce back and forward inside.

Roughly, your interpretation is correct, although I might gripe about your language which implies that when the wavefunction meets a delta it either reflects or transmits, as really the wavefunction coherently does both at the same time.

Also, yes S-matrices can be applied to one-dimensional scattering problems as well. Probably not what you're looking for at the moment though.