Charged Particle Motion in Electromagnetic Fields

where and are the

(3.72) | ||

(3.73) |

Let us assume that expression (3.71) is also the correct quantum mechanical Hamiltonian for a charged particle moving in electromagnetic fields. Obviously, in quantum mechanics, we must treat , , and as operators that do not necessarily commute.

The Heisenberg equations of motion for the components of are

(3.74) |

However,

(3.75) |

where use has been made of Equations (3.33) and (3.71). It follows that

where

(3.77) |

Here, is referred to as the

It is easily seen that

(3.78) |

However,

(3.79) |

where we have employed Equation (3.34). Thus, we obtain

because [from Equation (3.73)]

(3.81) |

Here, is the

We can write the Hamiltonian (3.71) in the form

(3.82) |

The Heisenberg equation of motions for the components of

(3.83) |

Here, we have taken into account the fact that depends explicitly on time through its dependence on . However,

(3.84) |

where use has been made of Equation (3.80). Moreover,

(3.85) | ||

(3.86) |

where we have employed Equation (3.33). The previous five equations yield

which can be combined with Equation (3.76) to give

This equation of motion is a generalization of the Ehrenfest theorem that takes electromagnetic fields into account. The fact that Equation (3.88) is analogous in form to the corresponding classical equation of motion (given that and commute in classical mechanics) justifies our earlier assumption that Equation (3.71) is the correct quantum mechanical Hamiltonian for a charged particle moving in electromagnetic fields.