By introducing the separation of variables

(11) |

we obtain

(12) |

If we introduce one of the standard wave equation solutions for

Now we have an ordinary differential equation describing the spatial amplitude of the matter wave as a function of position. The energy of a particle is the sum of kinetic and potential parts

(14) |

which can be solved for the momentum,

(15) |

Now we can use the de Broglie formula (4) to get an expression for the wavelength

(16) |

The term in equation (13) can be rewritten in terms of if we recall that and .

(17) |

When this result is substituted into equation (13) we obtain the famous

(18) |

which is almost always written in the form

(19) |

This single-particle one-dimensional equation can easily be extended to the case of three dimensions, where it becomes

(20) |

A two-body problem can also be treated by this equation if the mass

It is important to point out that this analogy with the classical wave
equation only goes so far. We cannot, for instance, derive the
time-*dependent* Schrödinger equation in an analogous fashion
(for instance, that equation involves the partial first derivative
with respect to time instead of the partial second derivative). In
fact, Schrödinger presented his time-independent equation first, and
then went back and postulated the more general time-dependent
equation.