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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. 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 \(x>a\)).

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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 (\(V=\infty\)). 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”. org%2FBookshelves%2FInorganic_Chemistry%2FInorganic_Chemistry_(LibreTexts)%2F02%253A_Atomic_Structure%2F2.

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

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

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Before we simplify, let's take another look at the full Hamiltonian for a particle-wave in three dimensions (see equation 2. And, since \(V=0\) inside the box, we can drop the whole part of the Hamiltonian equation that describes the potential energy (\(\frac{-Ze