Particle in an infinite potential well

Question

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?

Answer 1

I don't really like to call this problem the Particle in an Infinite Potential Well for precisely this reason. It invites questions like, "how does a particle behave in a region of infinite potential?" to which the answer is invariably that by infinite potential, we simply mean that the particle cannot access that region of space. My response would then be, "well why didn't you just say that in the first place?"

I prefer to call this system the Free Particle on an Interval. It's a system which consists of a particle which is not under the influence of any potential at all, but whose wavefunction lives in $L^2\big([0,a]\big)$ rather than $L^2(\mathbb R)$. This makes it clear that it doesn't make sense to talk about the particle being outside the well, and sidesteps any (reasonable) questions about what it means for the potential to be infinite everywhere except a small interval.

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Now, there is a sense in which the name Particle in an Infinite Potential Well is a very good name. If you treat the perfectly reasonable Particle in a Finite Potential Well, you can find its bound-state energy eigenfunctions (of which there is always at least one). If you take the limit as the potential $V_0$ outside the box goes to infinity, the aforementioned eigenfunctions converge to the energy eigenfunctions of a particle resticted to an interval$^\dagger$. In this sense, "infinite potential well" can be interpreted as the system being a limiting case of a finite potential well as $V_0\rightarrow \infty$.

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$^\dagger$As an obligatory side note, the Free Particle on an Interval is defined not only by its Hilbert space $L^2([0,a])$ and the form of its Hamiltonian $-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}$, but also by the domain of its Hamiltonian, which is essentially the twice-differentiable functions $\psi$ such that $\psi(0)=\psi(a)=0$. The boundary conditions are extremely important, but are inserted by hand because we simply choose them to be so.

We could in principle choose the periodic boundary conditions $\psi(0)=\psi(a)$ without setting these values to zero. This would define the Free Particle on a Ring.

However, if we view the Free Particle on an Interval as a limiting case of the finite potential well, then we find that the eigenfunctions of the Hamiltonian are exponentially suppressed outside of the interval $[0,a]$, tending to zero as $V_0\rightarrow \infty$. Therefore, the boundary conditions $\psi(0)=\psi(a)=0$ are a natural choice from this point of view.

Answer 2

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.