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In quantum mechanics we have the system of an infinite potential well and then we find out the energy of the particle inside the well using Schrödinger's equation which gives,

$$E=\frac{n^2π^2\hbar^2}{2ma}.$$

I was wondering where does the particle inside the well come from? Was it always there? Can there be particles in the region of infinite potential?

Edit: From the answer, suppose we have a particle outside the well which has energy greater than the outside potential. Now, according to the inside of the well both the outside potential and the outside particle have an infinite parameter. So can this particle enter the well?

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Studying the particle in an infinite potential well is a pedagogical tool, allowing derivation of quantized energy levels and wave functions and demonstrating a lot of the properties of problems in QM: the solutions form an orthonormal basis, taking matrix elements, calculating "simple" problems such as time evolution etc. In that sense, we do not have to concern ourselves with the question of how did the particle came to be in the well. It is there because we wrote the problem that way, and we can imagine whatever we like.

However, this is not completely made-up problem. One can think of it as an approximation to a real setup where there is a "deep" potential well (deep in the sense that the potential energy there is much lower than the surroundings, when compared to characteristic energy scale such as temperature), and we trap there a particle, let's say by depositing it with STM or by manipulations of the potential. Experimentalists can actually manufacture such devices and examine them. In that case, while we discuss the low-energy properties of the particle, the solutions of the infinite well approximation are very good and can be used to many applications.

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  • $\begingroup$ Okay, but what about the outside. As you said that, if we have a deep well we can use infinite potential well for a good approximation. Meaning that outside the well there can be particles. Now suppose one of them have an energy more than the potential then according to the inside of the well both the outside potential and the particle in that outside potential is infinite. I guess I will put this in the question. $\endgroup$ – Korra Feb 12 at 16:45
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    $\begingroup$ for a "true" infinite potential you cannot have particles outside. For the realizations there is not problem with particles outside - they have high energy, and that's that, and we focus on the particle inside which has very low energy and is trapped. $\endgroup$ – user245141 Feb 12 at 16:51
  • $\begingroup$ So basically if we are considering those outside particles, we have to use the finite potential well no matter how large the potential is? $\endgroup$ – Korra Feb 12 at 16:53
  • $\begingroup$ yes. a particle cannot have infinite energy. if we treat the particles outside the well we will treat the potential well as finite. $\endgroup$ – user245141 Feb 12 at 16:55

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