where , , and . Now, the Heisenberg equation of motion (3.88) only involves the electric and magnetic fields, and is independent of the vector and scalar potentials. On the other hand, the previous wave equation involves the potentials, but not the fields. As is well known, the vector and scalar potentials are not well defined, in that there are many different potentials that generate the same electric and magnetic fields. To be more exact, a transformation of the form and , where

and is an arbitrary function, leaves the and fields unaffected. Such a transformation is known as a

The previous three equations can be combined to give

Let

(3.93) |

where

(3.94) |

It follows that

(3.95) | ||

(3.96) | ||

(3.97) |

Hence, Equation (3.92) becomes

(3.98) |

which is analogous in form to Equation (3.89). Thus, we deduce that if and then . In other words, a gauge transformation introduces a position- and time-dependent phase-shift, , into the wavefunction.

Now, Equation (3.88) is equivalent to Equations (3.76) and (3.87). If we take the expectation values of the latter two equations then we obtain

(3.99) | ||

(3.100) |

However, the quantities ,

(3.101) | ||

(3.102) |

et cetera. Thus, Equations (3.88) and (3.89) do indeed give consistent results under gauge transformation.