Three-Dimensional Wave Mechanics

(C.80) |

where is a complex constant, and . Here, the wavevector, , and the angular frequency, , are related to the particle momentum, , and energy, , according to [cf., Equations (C.3) and (C.27)]

and [cf., Equation (C.1)]

(C.82) |

respectively. Generalizing the analysis of Section C.5, the three-dimensional version of Schrödinger's equation is [cf., Equations (C.24) and (C.29)]

where

(C.84) |

is the Hamiltonian, and the differential operator

(C.85) |

is known as the

(C.86) |

Moreover, the normalization condition for the wavefunction becomes [cf., Equation (C.32)]

It can be demonstrated that Schrödinger's equation, (C.83), preserves the normalization condition, (C.87), of a localized wavefunction. Heisenberg's uncertainty principle generalizes to [cf., Equation (C.60)]

(C.88) | ||

(C.89) | ||

(C.90) |

Finally, a stationary state of energy is written [cf., Equation (C.64)]

(C.91) |

where the stationary wavefunction, , satisfies [cf., Equation (C.66)]

(C.92) |

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

(C.93) |

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 (C.95). The remaining boundary conditions, (C.96), are satisfied provided

where the quantum numbers , , and are (independent) positive integers. Note that a stationary state generally possesses as many quantum numbers as there are degrees of freedom of the system (in this case, three). Substitution of the wavefunction (C.97) into Equation (C.94) yields

(C.101) |

Thus, it follows from Equations (C.98)-(C.100) that the particle energy is quantized, and that the allowed

The properly normalized [see Equation (C.87)] stationary wavefunctions corresponding to these energy levels are

(C.103) |

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

(C.104) |

Here,

(C.105) |

is the