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 $\Psi$ will be ab­bre­vi­ated as:

\begin{displaymath}
\Psi = \int_{k_1}^{k_2} f(k) e^{{\rm i}\varphi t} {\,\rm d}...
... = \frac{x}{t} - v_{\rm {g}}
\quad \varphi'' = - v_{\rm {g}}'
\end{displaymath}

where it will be as­sumed that $\varphi$ is a well be­haved func­tions of $k$ and $f$ at least twice con­tin­u­ously dif­fer­en­tiable. Note that the wave num­ber $k_0$ at which the group ve­loc­ity equals $x$$\raisebox{.5pt}{$/$}$$t$ is a sta­tion­ary point for $\varphi$. 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 $\varphi'$ $\vphantom0\raisebox{1.5pt}{$=$}$ 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:

\begin{displaymath}
\Psi = \frac{f(k)}{{\rm i}\varphi't} e^{{\rm i}\varphi t}\b...
...k)}{{\rm i}\varphi't}\right)'
e^{{\rm i}\varphi t} {\,\rm d}k
\end{displaymath}

This is small of or­der 1$\raisebox{.5pt}{$/$}$$t$ for large times. And if $\overline{\Phi}_0(p)$ 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 $x$$\raisebox{.5pt}{$/$}$$t$ val­ues within the range of packet group ve­loc­i­ties, there will be a sta­tion­ary point to $\varphi$. The wave num­ber at the sta­tion­ary point will be in­di­cated by $k_0$, and the value of $\varphi$ and its sec­ond de­riv­a­tive by $\varphi_0$ and $\varphi_0''$. (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,

\begin{displaymath}
\Psi = e^{{\rm i}\varphi_0 t} \int_{k_1}^{k_2} f(k)
e^{{\rm i}(\varphi-\varphi_0)t} {\,\rm d}k
\end{displaymath}

It is con­ve­nient to write the dif­fer­ence in $\varphi$ in terms of a new vari­able $\overline{k}$:

\begin{displaymath}
\varphi-\varphi_0 = {\textstyle\frac{1}{2}} \varphi_0'' \ov...
... \qquad \overline{k} \sim k - k_0\quad\mbox{for}\quad k\to k_0
\end{displaymath}

By Tay­lor se­ries ex­pan­sion it can be seen that $\overline{k}$ is a well be­haved mo­not­o­nous func­tion of $k$. The in­te­gral be­comes in terms $\overline{k}$:

\begin{displaymath}
\Psi = e^{{\rm i}\varphi_0 t} \int_{\overline{k}_1}^{\overl...
...ad g(\overline{k}) = f(k) \frac{{\rm d}k}{{\rm d}\overline{k}}
\end{displaymath}

Now split func­tion $g$ apart as in

\begin{displaymath}
g(\overline{k}) = g(0) + [g(\overline{k})-g(0)]
\end{displaymath}

The part within brack­ets pro­duces an in­te­gral

\begin{displaymath}
e^{{\rm i}\varphi_0 t} \int_{\overline{k}_1}^{\overline{k}_...
...rm i}\frac12\varphi_0''\overline{k}^2t}
{\,\rm d}\overline{k}
\end{displaymath}

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

That leaves the first part, $g(0)$ $\vphantom0\raisebox{1.5pt}{$=$}$ $f(k_0)$, which pro­duces

\begin{displaymath}
\Psi = e^{{\rm i}\varphi_0 t} f(k_0) \int_{\overline{k}_1}^...
...\rm i}\frac12\varphi_0''\overline{k}^2t} {\,\rm d}\overline{k}
\end{displaymath}

Change to a new in­te­gra­tion vari­able

\begin{displaymath}
u \equiv \sqrt{\frac{\vert\varphi_0''\vert t}{2}} \overline{k}
\end{displaymath}

Note that since time is large, the lim­its of in­te­gra­tion will be ap­prox­i­mately $u_1$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\vphantom0\raisebox{1.5pt}{$-$}$$\infty$ and $u_2$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\infty$ un­less the sta­tion­ary point is right at an edge of the wave packet. The in­te­gral be­comes

\begin{displaymath}
\Psi = e^{{\rm i}\varphi_0 t} f(k_0) \sqrt{\frac{2}{\vert\varphi_0''\vert t}}
\int_{u_1}^{u_2} e^{\pm{\rm i}u^2} {\,\rm d}u
\end{displaymath}

where $\pm$ is the sign of $\varphi_0''$. 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 $u$, and then

\begin{displaymath}
\Psi = e^{{\rm i}\varphi_0 t} e^{\pm{\rm i}\pi/4} f(k_0)
\sqrt{\frac{2\pi}{\vert\varphi_0''\vert t}}
\end{displaymath}

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 $u$ 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 $\sqrt{t}$ be­yond the stated wave packet range; how­ever, for large times $\sqrt{t}$ is small com­pared to the size of the wave packet, which is pro­por­tional to $t$.

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 $t$ 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.