Non-Interacting Particles

Here, the first term on the right-hand side represents the total kinetic energy of the system, whereas the potential specifies the nature of the interaction between the various particles making up the system, as well as the interaction of the particles with any external forces.

Suppose that the particles do not interact with one another. This
implies that each particle moves in a common potential: *i.e.*,

(427) |

where

In other words, for the case of non-interacting particles, the multi-particle Hamiltonian of the system can be written as the sum of independent single-particle Hamiltonians. Here, represents the energy of the th particle, and is completely unaffected by the energies of the other particles. Furthermore, given that the various particles which make up the system are non-interacting, we expect their instantaneous positions to be completely

Here, is the probability of finding the th particle between and at time . This probability is completely unaffected by the positions of the other particles. It is evident that must satisfy the normalization constraint

(431) |

According to Eqs. (428) and (430), the time-dependent
Schrödinger equation (423) for a system of non-interacting
particles factorizes into independent equations of the form

(432) |

(433) |

(434) |

(435) |

(436) |