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The particle-wave can only exist inside the walls (where \(0

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On the other hand, outside the box, the particle cannot exist and the potential energy is infinitely large (\(V=\infty\)) outside the walls (where \(x<0\) or \(x>a\)). The particle in the box is a hypothetical situation with a particle trapped in a one-dimensional “box”.Before we simplify, let's take another look at the full Hamiltonian for a particle-wave in three dimensions (see equation 2. Despite being unrealistic, this simplification is quite useful for gaining an understanding of the Schrödinger equation. This is a particle that has properties of a wave…so it is unlike the macroscopic particle that you’re probably imagining.

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org%2FBookshelves%2FInorganic_Chemistry%2FInorganic_Chemistry_(LibreTexts)%2F02%253A_Atomic_Structure%2F2. Both of these equations are described in the previous section and are written below for convenience.

The particle-wave is trapped between the walls, along the 1-dimensional \(x\) axis, and there are no forces acting on the particle-wave inside this “box”.

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Because the \(y\) and \(z\) values are zero, we can drop \(y\) and \(z\) out of our Hamiltonian equation. This “box” is more like a line, or an x-axis; it is just a one-dimensional space in which a particle-wave is trapped. This means that it is infinitely unfavorable for the particle-wave to exist outside the box, and so it never does. Since no forces act on the particle inside the box, the particle's potential energy inside the box is zero (\(V=0\)) and its potential energy outside the box is infinite (\(V=\infty\)).