Intuition | Implementation |

Young's double slit experiment for particles and for waves yield similar interference patterns. | Postulate a wave function
associated with quantum particles . |

The interference pattern is observed via intensity variations on a screen. The intensity of a wave is its amplitude squared. | Postulate that the probability
of finding a quantum particle at a given
position and time is related to
the mod squared of its wave function. . |

We need to respect energy quantisation as prompted by the study of black body radiation. | For quantum particles . |

The wave nature of particles is evidenced by the de Broglie relation. | For quantum particles . |

The simplest wave function is the wave function of a free-particle (absence of forces
acting on the particle). The particle can be considered to be moving in the -direction.
A logical way to postulate wave function for a free particle is then :

(22) |

(23) |

(24) |

We need to develop a mathematical
procedure that allows us to calculate observables from the wave function.
The first step towards this is to note that

(25) |

(26) |

Verify these equations.

Because the momentum and energy can be ``extracted'' from the wave function by
``operating'' on it appropriately, we define the energy and momentum ``operators''
as

(27) |

(28) |

Note that the operator for momentum squared is

(29) |