The Schrödinger equation takes the form,

Let us assume that
has the form,
Then
![{\displaystyle {\begin{aligned}-{\frac {\hbar ^{2}}{2m}}\left[{\frac {d^{2}\Phi (x)}{dx^{2}}}\Delta (y)\Omega (z)+\Phi (x){\frac {d^{2}\Delta (y)}{dy^{2}}}\Omega (z)+\Phi (x)\Delta (y){\frac {d^{2}\Omega (z)}{dz^{2}}}\right]\\+\left[X(x)+Y(y)+Z(z)\right]\Phi (x)\Delta (y)\Omega (z)&=E\Phi (x)\Delta (y)\Omega (z).\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a6ebb1eebcc6ff52d7a5c0c7d9ce717c22cecdaa)
Dividing by
we obtain

We may now separate the left-hand side into three parts, each depending on only one of the three coordinates
and
Each of these parts must be equal to a constant. Therefore,



where
,
and
are constants and
Hence, the three-dimensional problem has been divided into three one-dimensional problems where the total energy
is the sum of the energies
and
in each dimension.
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