I've been thinking about quantum particle waves interacting with potential barriers, specifically the case of a particle interacting with a step potential, reading from a website which deals with the time-independent solutions for individual eigenfunctions of the momentum operator.(adsbygoogle = window.adsbygoogle || []).push({});

We can solve Schrodinger's equation for both regions: in region I (before the barrier) we have the solution for a free particle, before it hits the barrier, and can model a 'reflected' wave by one with momentum away from the barrier and a lower amplitude than the 'incident' wave. With the solution beyond the potenial step, we can model a wave that has made it through, and which will have a lower amplitude than the incident wave, and a longer wavelength. The ratio of reflection to transmission, as the site puts it, is calculated using the ratio of the reflected and transmitted particles' amplitudes. But what exactly does this mean?

I've heard quantum tunnelling, whereby there is a finite chance that a particle with energy less than that of a thin potential barrier will make it through, and I suppose this is in some ways analogous. But what I wonder is this: after the collision with the barrier does the wavefuntion evolve so that there are two wave packets, each going away from the barrier, but in opposite directions? This would imply not only that there is a probability that the particle will go through or not, but also that each time we measure it, that probability will be there, and we may detect the particle on one side of the barrier the first time round, on the other the next. But even as I write the idea comes to me: do we disturb the wavefunction the first time we measure it, thereby destroying the wavepacket on the side of the barrier where we haven't detected the particle? Is this how the probability of the twoeventsis described in the theory?

One last thing - how do we specifiy the amplitude of the reflected and transmitted waves? Is this purely experimental? It seems that we cannot do it using Schrodinger's equation. If not, how can the equation claim to be able to describe completely the evolution of a wavefunction? Thanks for taking the time to read all this - I would just like some clarification on the issues raised here. Also, recommendations for any good websites/books on this subject would be appreciated. Thanks again.

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# Potential barriers, wavepackets, probabilities etc.

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