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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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Despite being unrealistic, this simplification is quite useful for gaining an understanding of the Schrödinger equation. The particle-wave can only exist inside the walls (where \(0

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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”. 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 \(x>a\)). 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 (\(V=\infty\)).

Before we simplify, let's take another look at the full Hamiltonian for a particle-wave in three dimensions (see equation 2. 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.

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And, since \(V=0\) inside the box, we can drop the whole part of the Hamiltonian equation that describes the potential energy (\(\frac{-Ze