### D.44 De­riva­tion of group ve­loc­ity

The ob­jec­tive of this note is to de­rive the wave func­tion for a wave packet if time is large.

To shorten the writ­ing, the Fourier in­te­gral (7.64) for will be ab­bre­vi­ated as:

where it will be as­sumed that is a well be­haved func­tions of and at least twice con­tin­u­ously dif­fer­en­tiable. Note that the wave num­ber at which the group ve­loc­ity equals is a sta­tion­ary point for . That is the key to the math­e­mat­i­cal analy­sis.

The so-called “method of sta­tion­ary phase” says that the in­te­gral is neg­li­gi­bly small as long as there are no sta­tion­ary points 0 in the range of in­te­gra­tion. Phys­i­cally that means that the wave func­tion is zero at large time po­si­tions that can­not be reached with any group ve­loc­ity within the range of the packet. It there­fore im­plies that the wave packet prop­a­gates with the group ve­loc­ity, within the vari­a­tion that it has.

To see why the in­te­gral is neg­li­gi­ble if there are no sta­tion­ary points, just in­te­grate by parts:

This is small of or­der 1 for large times. And if is cho­sen to smoothly be­come zero at the edges of the wave packet, rather than abruptly, you can keep in­te­grat­ing by parts to show that the wave func­tion is much smaller still. That is im­por­tant if you have to plot a wave packet for some book on quan­tum me­chan­ics and want to have its sur­round­ings free of vis­i­ble per­tur­ba­tions.

For large time po­si­tions with val­ues within the range of packet group ve­loc­i­ties, there will be a sta­tion­ary point to . The wave num­ber at the sta­tion­ary point will be in­di­cated by , and the value of and its sec­ond de­riv­a­tive by and . (Note that the sec­ond de­riv­a­tive is mi­nus the first de­riv­a­tive of the group ve­loc­ity, and will be as­sumed to be nonzero in the analy­sis. If it would be zero, non­triv­ial mod­i­fi­ca­tions would be needed.)

Now split the ex­po­nen­tial in the in­te­gral into two,

It is con­ve­nient to write the dif­fer­ence in in terms of a new vari­able :

By Tay­lor se­ries ex­pan­sion it can be seen that is a well be­haved mo­not­o­nous func­tion of . The in­te­gral be­comes in terms :

Now split func­tion apart as in

The part within brack­ets pro­duces an in­te­gral

and in­te­gra­tion by parts shows that to be small of or­der 1/.

That leaves the first part, , which pro­duces

Change to a new in­te­gra­tion vari­able

Note that since time is large, the lim­its of in­te­gra­tion will be ap­prox­i­mately and un­less the sta­tion­ary point is right at an edge of the wave packet. The in­te­gral be­comes

where is the sign of . The re­main­ing in­te­gral is a Fres­nel in­te­gral that can be looked up in a ta­ble book. Away from the edges of the wave packet, the in­te­gra­tion range can be taken as all , and then

Con­vert back to the orig­i­nal vari­ables and there you have the claimed ex­pres­sion for the large time wave func­tion.

Right at the edges of the wave packet, mod­i­fied in­te­gra­tion lim­its for must be used, and the re­sult above is not valid. In par­tic­u­lar it can be seen that the wave packet spreads out a dis­tance of or­der be­yond the stated wave packet range; how­ever, for large times is small com­pared to the size of the wave packet, which is pro­por­tional to .

For the math­e­mat­i­cally picky: the treat­ment above as­sumes that the wave packet mo­men­tum range is not small in an as­ymp­totic sense, (i.e. it does not go to zero when be­comes in­fi­nite.) It is just small in the sense that the group ve­loc­ity must be mo­not­o­nous. How­ever, Kaplun’s ex­ten­sion the­o­rem im­plies that the packet size can be al­lowed to be­come zero at least slowly. And the analy­sis is read­ily ad­justed for faster con­ver­gence to­wards zero in any case.