This means that it is infinitely unfavorable for the particle-wave to exist outside the box, and so it never does. 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. Despite being unrealistic, this simplification is quite useful for gaining an understanding of the Schrödinger equation.

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This is a particle that has properties of a wave…so it is unlike the macroscopic particle that you’re probably imagining. 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”. 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 \(x>a\)). Although it does not represent a real situation, we can limit our model to just one dimension (the x-dimension, for instance) such that the Schrödinger equation becomes significantly simplified. Because the \(y\) and \(z\) values are zero, we can drop \(y\) and \(z\) out of our Hamiltonian equation.

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. org%2FBookshelves%2FInorganic_Chemistry%2FInorganic_Chemistry_(LibreTexts)%2F02%253A_Atomic_Structure%2F2.

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Both of these equations are described in the previous section and are written below for convenience. The particle-wave can only exist inside the walls (where \(0