Having developed a formalism for the quantum theory of radiation, we will find it both interesting and profitable to determine what wave function most nearly describes the classical electromagnetic field given by Eo cos (vt •f $) and H0 sin (vt 4-j)- This field is supposedly known with absolute cer tainty, whereas the quantum operators for the electric and magnetic fields are associated with nonzero uncertainties obeying the relation
(symbols defined in Sec. 14-1), just as the position q and momentum p obey the relation Aq Ap ^ We propose that the wave function which corre sponds most closely to the classical field must have minimum uncertainty [equality in (1)] for all time when subject to the appropriate simple harmonic potential. We will find in Appendix H that minimum uncertainty merely at one time is not sufficient, for if the width of the wave packet is initially chosen too small, it becomes too large at a later time, as illustrated in Fig. 15-16. Furthermore, it fails to satisfy the conditions for minimum uncertainty except at isolated points in time. If the right width is chosen for the potential, how ever, the packet bounces back and forth sinusoidally without changing width, as shown in Fig. 15-26. This latter packet “coheres,” always has minimum uncertainty, and resembles (Fig. 15-2a) the classical field as closely as quan tum mechanics permits.