Time Development of a Gaussian Wave Packet *

So far, we have performed our Fourier Transforms at $t=0$ and looked at the result only at $t=0$. We will now put time back into the wave function and look at the wave packet at later times. We will see that the behavior of photons and non-relativistic electrons is quite different.

Assume we start with our Gaussian (minimum uncertainty) wavepacket $A(k)=e^{-\alpha (k-k_0)^2}$ at $t=0$. We can do the Fourier Transform to position space, including the time dependence.

$$\bgroup\color{black}\psi(x,t)=\int\limits_{-\infty}^\infty\; A(k) e^{i(kx-\omega(k)t)}\; dk\egroup$$

We write explicitly that $\omega$ depends on $k$. For our free particle, this just means that the energy depends on the momentum. For a photon, $E=pc$, so $\hbar\omega=\hbar kc$, and hence $\omega=kc$. For an non-relativistic electron, $E={p^2\over 2m}$, so $\hbar\omega={\hbar^2 k^2\over 2m}$, and hence $\omega={\hbar k^2\over 2m}$.

To cover the general case, lets expand $\omega(k)$ around the center of the wave packet in k-space.

$$\bgroup\color{black}\omega(k)=\omega(k_0)+\left.{d\omega\over... ...2}\left.{d^2\omega\over dk^2}\right\vert _{k_0}(k-k_0)^2\egroup$$

We anticipate the outcome a bit and name the coefficients.

$$\bgroup\color{black}\omega(k)=\omega_0+v_g (k-k_0)+\beta (k-k_0)^2\egroup$$

For the photon, $v_g=c$ and $\beta=0$. For the NR electron, $v_g={\hbar k_0\over m}$ and $\beta={\hbar\over 2m}$.

Performing the Fourier Transform, we get

$$\begin{eqnarray*} \psi(x,t)&=&\sqrt{\pi\over\alpha+i\beta t}\; e^{i(k_0x-\omega_... ...2 t^2}}\; e^{-\alpha(x-v_gt)^2\over 2(\alpha^2+\beta^2 t^2)}.\\ \end{eqnarray*}$$

We see that the photon will move with the velocity of light and that the wave packet will not disperse, because $\beta=0$.

For the NR electron, the wave packet moves with the correct group velocity, $v_g={p\over m}$, but the wave packet spreads with time. The RMS width is $\sigma=\sqrt{\alpha^2+\left({\hbar t\over 2m}\right)^2}$.

A wave packet naturally spreads because it contains waves of different momenta and hence different velocities. Wave packets that are very localized in space spread rapidly.