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Supoose we have a repuslive potential well ($+V_0$) and, if a particle of energy $E(>V_0)$ is incident toward it, then it will feel repulsion, so intuitively the probability of finding particle inside the well should be less than finding it outside the well.

Now think same for attractive potential well ($-V_0$): in this case a particle of energy $E>(-V_0)$ should have more probability to find inside the well as compred to be found outside.

Now problem that I face is when I am trying to think of it using semiclassical techniques: we know that probability to find particle in a region is proportional to amount of time it spends there and that time is inversly proportional to momentum(or kinetic energy) of particle and that momentum is proportional to de Broglie wavelength. Now take the case of repulsive potential well ($+V_0$), here kinetic energy of particle is less when it is inside the well as compared to when it is outside the well so amount of time it spends inside the well should be more than it spends outside so probability of finding particle inside the well should be higher. This doesn't feel right intuitively. And problem persists for attractive potential well. This seems like a contradiction.

At which point I am making mistake in terms of my thinking?

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    $\begingroup$ Usually one calls well the case that you call attractive well, and one talks about barrier in the case that you call repulsive well. $\endgroup$
    – Roger V.
    Jan 28, 2021 at 8:24
  • $\begingroup$ Yeah, but question still stands. $\endgroup$
    – Barry
    Jan 28, 2021 at 8:29
  • $\begingroup$ It would be nice to make the calculations (both the quantum mechanical and the classical one) to see that the problem actually exists. $\endgroup$
    – Roger V.
    Jan 28, 2021 at 8:39

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The mistake in your thinking is assuming that lower kinetic energy corresponds to longer time spent somewhere, e.g. in the repulsive well. This probably stems from the "semiclassical" point of view you are talking about. The concept you are probably missing is the one of reflected matter waves at a step potential.

To not get into it too deeply and stay close to your picture: When speaking about potential wells, it is necessary to describe matter as waves to fully understand their dynamics. A matter wave with energy $E$, moving in free space (potential $V_{free space}=0$) "hits" the boundary of a repulsive potential well ($+V_0$). What happens at that boundary is that the matter wave gets partially reflected, so a part of this matter wave will move "back" in the direction where it came from and still will be be in free space. In the second last animation on this page you can see what this reflection of a wave looks like (only schematically, as matter waves will actually disperse).

As the wavefunction corresponds to the probability distribution of the particle, the particle will now be in the repulsive well with some probability, but it may also still be in free space due to the reflection! And when looking at the probabilities, we see that the slower the particle would be in the repulsive well ($E$ closer to $V_0$), the higher the probability of reflection and "staying in free space" is for that particle, which would fit to your more intuitive picture.

Here is a mathematically rigorous treatment of the topic, if you are interested.

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  • $\begingroup$ Gotcha. What about attractive potential well, will there be any scattering? $\endgroup$
    – Barry
    Jan 29, 2021 at 3:11
  • $\begingroup$ Yes, there will also be reflection! The same thing happens, you can see it actually in the last animation on the first link i provided. The exact transition probabilities will now be different though. I encourage you to sit down and calculate these, this will definitely deepen your knowledge about this topic (and this is actually a common exercise in quantum mechanics classes!) $\endgroup$
    – brewburger
    Jan 29, 2021 at 6:49

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