Introduction: The Most Important Wave Packet
The Gaussian wave packet is analytically tractable and represents the minimum uncertainty state. It's the prototype for understanding all quantum wave packets.
Initial Gaussian
A normalized Gaussian centered at \(x_0\) with momentum \(p_0\):
\[\psi(x, 0) = \left(\frac{1}{2\pi\sigma^2}\right)^{1/4} e^{-(x-x_0)^2/4\sigma^2} e^{ip_0 x/\hbar}\]Width: \(\Delta x = \sigma\), and by Fourier: \(\Delta p = \hbar/2\sigma\).
Key Properties
- Saturates uncertainty bound: \(\Delta x \cdot \Delta p = \hbar/2\)
- Gaussian in both position AND momentum space
- Remains Gaussian under free evolution (though it spreads)
Worked Example
Expectation values:
- \(\langle x\rangle = x_0 + (p_0/m)t\) — moves classically
- \(\langle p\rangle = p_0\) — constant (free particle)
- Position uncertainty grows: \(\Delta x(t) = \sigma\sqrt{1 + (\hbar t/2m\sigma^2)^2}\)
The Quantum Connection
The Gaussian wave packet is used to model particles in realistic situations: electrons in metals, photons in pulses, atoms in traps. Its analytic tractability makes it the "hydrogen atom" of wave packet physics.