I was wondering if anyone can give me some assistance on a homework problem. Here it is,(adsbygoogle = window.adsbygoogle || []).push({});

Consider a wave packet defined by

[itex]

\begin{equation}

\vec{A}(\vec{r},t)=\int \hat{\mathcal{A}}(\vec{k}-\vec{k_0})

\frac{e^{i(\vec{k}\cdot\vec{r}-\omega(k)t)}}{(2\pi)^{3/2}}d\vec{k}

\end{equation}

[/itex]

where

[itex]

\hat{\mathcal{A}}(\vec{k}-\vec{k_0})

[/itex]

is a function that is peaked at [itex]\vec{k}=\vec{k_0}[/itex].

(a) Show that this packet can be written in the form

[itex]

\begin{equation}

\vec{A}(\vec{r},t)=e^{i(\vec{k_0}\cdot\vec{r}-\omega(k_0)t)}\mathcal{A}(\vec{r}-v_gt)+\cdots

\end{equation}

[/itex],

where [itex]\vec{v}_g=\vec{v}_{\mathrm{group}}=\vec{\nabla}_k\omega

(k)|_{k_0}[/itex] is the group velocity and [itex]\mathcal{A}(\vec{r}-\vec{v}_g t)[/itex] is a function that is peaked at [itex]\vec{r}=\vec{v}_gt[/itex] Hint: expand [itex]\omega(k)[/itex] around [itex]\vec{k}_0[/itex]

(b) Show that for a wave packet not to "spread", i.e., not change its shape from that given by [itex]\mathcal{A}(\vec{r})[/itex], it is required that [itex]\vec{v}_{\mathrm{group}}=\vec{v}_{\mathrm{phase}}[/itex]. Here [itex]\vec{v}_{\mathrm{phase}}[/itex] is the phase velocity [itex]\vec{v}_{\mathrm{phase}}\equiv\omega/k[/itex].

(c) As a consequence of the condition [itex]\vec{v}_\mathrm{phase}=\vec{v}_\mathrm{group}[/itex] show that [itex]\omega=kc[/itex] which holds for light in a vacuum. Then deduce that the wave equation [itex]\square\vec{A}=0[/itex] follows.

(d) Suppose we had [itex]\omega(k)=bk^2[/itex], where [itex]b[/itex] is some constant. Would the phase and group velocities be the same? What differential equation would you deduce? Would the wave packet maintain its shape?

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Wave Packet problem

**Physics Forums | Science Articles, Homework Help, Discussion**