Joan Lindsay Orr

\( \newcommand{\CC}{\mathbb{C}} \newcommand{\NN}{\mathbb{N}} \newcommand{\RR}{\mathbb{R}} \newcommand{\TT}{\mathbb{T}} \newcommand{\ZZ}{\mathbb{Z}} \newcommand{\H}{\mathcal{H}} \newcommand{\e}{\epsilon} \newcommand{\x}{\mathbf{x}} \newcommand{\y}{\mathbf{y}} \)

Phase and Group Velocity

The most general (1-dimensional) wave function is a function of the form \(f(kx - \omega t)\), where \(f:\RR\rightarrow\CC\). Any particular value of \(f\), say \(f(c)\), moves to the right at a speed \(\omega/k\), because this is how \(x\) and \(t\) evolve on the line \(kx - \omega t = c\).

In particular a plane wave, \(e^{i (kx - \omega t)}\), propagates to the right with speed \(\omega/k\). In this case, \(k\) is the wave number, the number of cycles in a spatial interval of length \(2\pi\), and \(\omega\) is the (normalized) frequency, the number of cycles in a temporal interval of length \(2\pi\). From this we have the wavelength \(\lambda = 2\pi/k\) and the frequency \(\nu = \omega / 2\pi\) and recover the formula velocity = frequency \(\times\) wavelength.

Physically, the refractive index of a medium will vary with the wavelength; this is the phenomenon which enables a prism to split a beam of white light. Mathematically we typically make \(\omega\) depend on \(k\) (called the dispersion relation__), and we can assemble a _wave packet of the form: \[ \Psi(x, t) = \int_{-\infty}^\infty c(k) e^{i (kx - \omega(k) t)} dk \]

Now suppose that \(c(k)\) is narrowly peaked about some value \(k_0\). If the peak is narrow enough then \(\omega(k)\) is well-approximated by the first two terms of its Taylor series for all values of \(k\) for which \(c(k)\) is non-zero. In this case \[ \begin{align} \Psi(x, t) &= \int_{-\infty}^\infty c(k) e^{i (kx - (\omega(k_0) + \omega'(k_0)(k-k_0) )t)} dk \\ &= e^{i (k_0x - \omega(k_0)t} \int_{-\infty}^\infty c(k) e^{i ((k-k_0)x - \omega'(k_0)(k-k_0)t)} \\ &= e^{i (k_0x - \omega(k_0)t} \int_{-\infty}^\infty c(k) e^{i (x - \omega'(k_0)t)(k-k_0)} \\ &= e^{i (k_0x - \omega(k_0)t} F(x - \omega'(k_0)t) \end{align} \] The first term of these is a plane wave which propagates with velocity \(\omega(k_0) / k_0\), and this is the phase velocity. This is modulated by the seond term, which propagates with velocity \(\omega'(k_0) / 1 = \omega'(k_0)\). This is the group velocity.

Application to the Schrödinger Equation

For a free particle (i.e. \(V(x) = 0\) for all \(x\)) the time-independent Schrödinger Equation has solutions of the form \[ \Psi(x,t) = e^{-iEt/\hbar} \psi(x) \] where \(\psi\) is a solution of \[ -\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2} = E\psi \] and so \[ \frac{d^2\psi}{dx^2} = -\frac{2mE}{\hbar^2}\psi = - k^2 \psi \] where \[ k^2 = 2mE/\hbar^2. \]

Thus \(\psi(x) = e^{ikx}\) (traveling in the positive direction; there's also a solution traveling in the negative direction of course), and \[ \Psi(x, t) = e^{i(kx - Et/\hbar)} \] It follows that \[ \omega(k) = \frac{E}{\hbar} = \frac{\hbar k^2}{2m} \]

Thus the phase velocity is \[ \frac{\omega(k)}{k} = \frac{\hbar k}{2m} \] and the group velocity is \[ \omega'(k) = \frac{\hbar k}{m} \]

Now the classical (macro) velocity \(v\) satisfies \(E = \frac{1}{2}mv^2\), and so \[ v^2 = \frac{2E}{m} = \frac{\hbar^2 k^2}{m^2} \] Thus we see the group velocity corresponds to the classical velocity of the particle/wave group, while the phase velocity is off by a factor of \(\sqrt{2}\).