Wave Dispersion

Equation (4.56) indicates that, as a wave packet propagates, its envelope remains the same shape. Actually, this result is misleading, and is only obtained because of the neglect of second-order terms in the expansion (4.51). If we keep more terms in this expansion then we can show that the wave packet does actually change shape as it propagates. However, this demonstration is most readily effected by means of the following simple argument. The packet extends in Fourier space from $k_0-{\mit\Delta} k/2$ to $k_0+{\mit\Delta} k/2$. Thus, part of the packet propagates at the velocity $v_g(k_0-{\mit\Delta} k/2)$, and part at the velocity $v_g(k_0+{\mit\Delta} k/2)$. Consequently, the packet spreads out as it propagates, because some parts of it move faster than others. Roughly speaking, the spatial extent of the packet in real space grows as

$${\mit\Delta} x\sim ({\mit\Delta} x)_0+ \left[v_g(k_0+{\mit\Delta}... ...} k/2)\right] t\sim ({\mit\Delta} x)_0+\frac{dv_g(k_0)}{dk}\,{\mit\Delta} k\,t,$$ (4.61)

where $({\mit\Delta} x)_0\sim ({\mit\Delta} k)^{-1}$ is the extent of the packet at $t=0$. Hence, from Equation (4.53),

$${\mit\Delta} x \sim ({\mit\Delta} x)_0 + \frac{d^{2}\omega(k_0)}{dk^{2}}\,\frac{t}{({\mit\Delta} x)_0}.$$ (4.62)

We, thus, conclude that the spatial extent of the packet grows linearly in time, at a rate proportional to the second derivative of $\omega(k)$ with respect to $k$ (evaluated at the packet's central wavenumber). This effect is known as wave dispersion. Furthermore, it is clear that the relation $\omega=\omega(k)$ governs the degree of wave dispersion, which explains why it is called a dispersion relation.

Note that electromagnetic wave packets, which are governed by the linear dispersion relation $\omega=k\,c$, do not disperse as they propagate, because $d^2\omega/dk^2=0$. Particle wave packets, on the other hand, are governed by the quadratic dispersion relation (4.23) and, therefore, disperse as they propagate. In fact, it follows from Equations (4.23) and (4.62) that the width of a particle wave packet grows in time as

$${\mit\Delta} x \simeq ({\mit\Delta} x)_0 + \frac{\hbar}{m}\,\frac{t}{({\mit\Delta} x)_0}.$$ (4.63)

For example, if an electron wave packet is initially localized in a region of atomic dimensions (i.e., ${\mit\Delta} x\sim 10^{-10}\,{\rm m}$) then the width of the packet doubles in about $10^{-16}\,{\rm s}$.