2.11: Evolution of Wave-Packets

We have seen, in Equation \ref{e2.45}, how to write the wavefunction of a particle that is initially localized in \(x\)-space. Let us examine how this wavefunction evolves in time. According to Equation \ref{e2.42}, we have

\[ \psi(x,t) = \int_{-\infty}^{\infty} \bar{\psi}(k)\,\rm e^{\,{\rm i}\,\phi(k)}\,dk, \label{e2.56} \]

where

\[\phi(k) = k\,x - \omega(k)\,t. \nonumber \]

The function \(\bar{\psi}(k)\) is obtained by Fourier transforming the wavefunction at \(t=0\). [See Equations \ref{e2.42a} and \ref{e2.51}.] According to Equation \ref{e2.53}, \(|\bar{\psi}(k)|\) is strongly peaked around \(k=k_0\). Thus, it is a reasonable approximation to Taylor expand \(\phi(k)\) about \(k_0\). Keeping terms up to second order in \(k-k_0\), we obtain

\[\psi(x,t)\propto \int_{-\infty}^{\infty} \bar{\psi}(k)\, \exp\!\left[\,{\rm i}\left\{\phi_0+ \phi_0'\,(k-k_0) + \frac{1}{2}\,\phi_0''\,(k-k_0)^2\right\}\right], \nonumber \]

where

\[\begin{align} \phi_0 &= \phi(k_0) = k_0\,x-\omega_0\,t,\\[4pt] \phi_0'&= \frac{d\phi(k_0)}{dk} = x - v_g\,t,\\[4pt] \phi_0''&=\frac{d^2\phi(k_0)}{dk^2} = - \alpha\,t,\end{align} \nonumber \]

with

\[\begin{align} \omega_0 &= \omega(k_0),\\[4pt] v_g &= \frac{d\omega(k_0)}{dk},\\[4pt] \alpha &= \frac{d^2\omega(k_0)}{dk^2}.\label{e2.64}\end{align} \]

Substituting from Equation \ref{e2.51}, rearranging, and then changing the variable of integration to \(y=(k-k_0)/(2\,\Delta k)\), we get

\[\psi(x,t)\propto \rm e^{\,{\rm i}\,(k_0\,x-\omega_0\,t)} \int_{-\infty}^{\infty}\rm e^{\, {\rm i}\,\beta_1\,y-(1+{\rm i}\,\beta_2)\,y^2}\,dy, \nonumber \]

where

\[\begin{align} \beta_1 &= 2\,\Delta k\,(x-x_0-v_g\,t),\\[4pt] \beta_2&= 2\,\alpha\,(\Delta k)^2\,t.\end{align} \nonumber \]

Incidentally, \({\mit\Delta k}=1/(2\,\Delta x)\), where \(\Delta x\) is the initial width of the wave-packet. The previous expression can be rearranged to give

\[ \psi(x,t)\propto \rm e^{\,{\rm i}\,(k_0\,x-\omega_0\,t)-(1+{\rm i}\,\beta_2)\,\beta^2/4}\int_{-\infty}^\infty \rm e^{-(1+{\rm i}\,\beta_2)\,(y-y_0)^2}\,dy, \label{e2.69} \]

where \(y_0={\rm i}\,\beta/2\) and \(\beta=\beta_1/(1+{\rm i}\,\beta_2)\). Again changing the variable of integration to \(z=(1+{\rm i}\,\beta_2)^{1/2}\,(y-y_0)\), we get

\[\psi(x,t)\propto (1+{\rm i}\,\beta_2)^{-1/2}\,\rm e^{\,{\rm i}\,(k_0\,x-\omega_0\,t)-(1+{\rm i}\,\beta_2)\,\beta^2/4}\int_{-\infty}^\infty \rm e^{-z^2}\,dz. \nonumber \]

The integral now just reduces to a number. Hence, we obtain

\[ \psi(x,t)\propto\frac{\exp\left[\,{\rm i}\,(k_0\,x-\omega_0\,t) - (x-x_0-v_g\,t)^2\,\{1-{\rm i}\,2\,\alpha\,(\Delta k)^2\,t\}/(4\,\sigma^2)\right]}{\left[1+{\rm i}\,2\,\alpha\,(\Delta k)^2\,t\right]^{1/2}}, \label{exxx} \]

where

\[ \sigma^2(t) = (\Delta x)^2 + \frac{\alpha^2\,t^2}{4\,(\Delta x)^2}. \label{e2.70} \]

Note that the previous wavefunction is identical to our original wavefunction \ref{e2.45} at \(t=0\). This justifies the approximation that we made earlier by Taylor expanding the phase factor \(\phi(k)\) about \(k=k_0\).

According to Equation \ref{exxx}, the probability density of our particle as a function of time is written

\[|\psi(x,t)|^2 \propto \sigma^{\,-1}(t)\exp\left[-\frac{(x-x_0-v_g\,t)^2}{2\,\sigma^2(t)}\right]. \nonumber \]

Hence, the probability distribution is a Gaussian, of characteristic width \(\sigma(t)\), that peaks at \(x=x_0+v_g\,t\). The most likely position of our particle coincides with the peak of the distribution function. Thus, the particle’s most likely position is given by

\[x = x_0+v_g\,t. \nonumber \]

It can be seen that the particle effectively moves at the uniform velocity

\[v_g = \frac{d\omega}{dk}, \nonumber \]

which is known as the group-velocity. In other words, a plane-wave travels at the phase-velocity, \(v_p=\omega/k\), whereas a wave-packet travels at the group-velocity, \(v_g=d\omega/dt\). It follows from the dispersion relation \ref{e2.38} for particle waves that

\[v_g = \frac{p}{m}. \nonumber \]

However, it can be seen from Equation \ref{e2.31} that this is identical to the classical particle velocity. Hence, the dispersion relation \ref{e2.38} turns out to be consistent with classical physics, after all, as soon as we realize that individual particles must be identified with wave-packets rather than plane-waves. In fact, a plane-wave is usually interpreted as a continuous stream of particles propagating in the same direction as the wave.

According to Equation \ref{e2.70}, the width of our wave-packet grows as time progresses. Indeed, it follows from Equations \ref{e2.38} and \ref{e2.64} that the characteristic time for a wave-packet of original width \(\Delta x\) to double in spatial extent is

\[t_2 \sim \frac{m\,(\Delta x)^2}{\hbar}. \nonumber \]

For instance, if an electron is originally localized in a region of atomic scale (i.e., \(\Delta x\sim 10^{-10}\,{\rm m}\)) then the doubling time is only about \(10^{-16}\,{\rm s}\). Evidently, particle wave-packets (for freely-moving particles) spread very rapidly.

Note, from the previous analysis, that the rate of spreading of a wave-packet is ultimately governed by the second derivative of \(\omega(k)\) with respect to \(k\). [See Equations \ref{e2.64} and \ref{e2.70}.] This explains why a functional relationship between \(\omega\) and \(k\) is generally known as a dispersion relation—it governs how fast wave-packets disperse as time progresses. However, for the special case where \(\omega\) is a linear function of \(k\), the second derivative of \(\omega\) with respect to \(k\) is zero, and, hence, there is no dispersion of wave-packets: that is, wave-packets propagate without changing shape. The dispersion relation \ref{e2.7} for light-waves is linear in \(k\). It follows that light pulses propagate through a vacuum without spreading. Another property of linear dispersion relations is that the phase-velocity, \(v_p=\omega/k\), and the group-velocity, \(v_g=d\omega/dk\), are identical. Thus, plane light-waves and light pulses both propagate through a vacuum at the characteristic speed \(c=3\times 10^8\,{\rm m/s}\). Of course, the dispersion relation \ref{e2.38} for particle waves is not linear in \(k\). Hence, particle plane-waves and particle wave-packets propagate at different velocities, and particle wave-packets also gradually disperse as time progresses.