Evolution of Wave Packets

where

(98) |

(99) |

(100) | |||

(101) | |||

(102) |

with

Substituting from Eq. (93), rearranging, and then changing the variable of integration to , we get

(106) |

(107) | |||

(108) |

Incidentally, , where is the initial width of the wave packet. The above expression can be rearranged to give

where and . Again changing the variable of integration to , we get

(110) |

where

Note that the above wavefunction is identical to our original wavefunction (86) at . This, justifies the approximation which we made earlier by Taylor expanding the phase factor about .

According to Eq. (111), the probability density of our particle
as a function of time is written

(113) |

(114) |

(115) |

(116) |

According to Eq. (112), the width of our wave packet
grows as time progresses. Indeed, it follows from Eqs. (79) and (105)
that the characteristic time for a wave packet of original width
to double in spatial extent is

(117) |

Note, from the previous analysis, that the rate of spreading of a wave packet is ultimately
governed by the second derivative of
with respect to . See Eqs. (105) and (112). This is why a functional relationship between and
is generally known as a *dispersion relation*: *i.e.*, because it
governs how wave packets disperse as time progresses.
However, for the special case where is a *linear* function
of , the second derivative of with respect to is zero,
and, hence, there is no dispersion of wave packets: *i.e.*, wave packets
propagate without changing shape. Now, the dispersion relation
(50) for light waves is linear in . It follows that light pulses
propagate through a vacuum without spreading. Another property
of linear dispersion relations is that the phase velocity, , and
the group velocity,
, are *identical*. Thus, both plane light waves
and light pulses propagate through a vacuum at the characteristic
speed
. Of course, the dispersion relation (79) for particle waves is *not* linear in . Hence, particle
plane waves and particle wave packets propagate at different velocities,
and particle wave packets also gradually disperse as time progresses.