(1154) |

where is a complex constant, and . Here, the wavevector, , and the angular frequency, , are related to the particle momentum, , and energy, , according to [cf., Equation (1082)]

(1155) |

and [cf., Equation (1080)]

(1156) |

respectively. Generalizing the analysis of Section 12.5, the three-dimensional version of Schrödinger's equation is [cf., Equation (1102)]

where the differential operator

(1158) |

is known as the

(1159) |

Moreover, the normalization condition for the wavefunction becomes [cf., Equation (1107)]

It can be demonstrated that Schrödinger's equation, (1157), preserves the normalization condition, (1160), of a localized wavefunction (Gasiorowicz 1996). Heisenberg's uncertainty principle generalizes to [cf., Equation (1135)]

(1161) | ||

(1162) | ||

(1163) |

Finally, a stationary state of energy is written [cf., Equation (1139)]

(1164) |

where the stationary wavefunction, , satisfies [cf., Equation (1141)]

(1165) |

As an example of a three-dimensional problem in wave mechanics, consider a particle trapped in a square potential well of infinite depth, such that

(1166) |

Within the well, the stationary wavefunction, , satisfies

subject to the boundary conditions

and

because outside the well. Let us try a separable wavefunction of the form

This expression automatically satisfies the boundary conditions (1168). The remaining boundary conditions, (1169), are satisfied provided

where , , and are (independent) positive integers. Substitution of the wavefunction (1170) into Equation (1167) yields

(1174) |

Thus, it follows from Equations (1171)-(1173) that the particle energy is quantized, and that the allowed

The properly normalized [see Equation (1160)] stationary wavefunctions corresponding to these energy levels are

(1176) |

As is the case for a particle trapped in a one-dimensional potential well, the lowest energy level for a particle trapped in a three-dimensional well is not zero, but rather

(1177) |

Here,

(1178) |

is the