Introduction: The Simplest System
A free particle (\(V = 0\) everywhere) seems trivial, but it reveals key quantum features: the continuous spectrum, wave packet spreading, and the relationship between position and momentum uncertainty.
Energy Eigenstates
The TISE: \(-\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2} = E\psi\)
Solutions: \(\psi_k(x) = Ae^{ikx}\) with \(E = \frac{\hbar^2k^2}{2m}\)
Continuous spectrum: any \(k\) (and thus any \(E > 0\)) is allowed.
Wave Packets
Plane waves aren't normalizable. Physical states are wave packets:
\[\psi(x, 0) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} \phi(k)e^{ikx}\, dk\]where \(\phi(k)\) is the momentum-space wavefunction.
Worked Example: Time Evolution
Each \(k\) component evolves with its own frequency \(\omega(k) = \hbar k^2/2m\):
\[\psi(x, t) = \frac{1}{\sqrt{2\pi}}\int \phi(k)e^{i(kx - \omega(k)t)}\, dk\]Since \(\omega \propto k^2\) (not linear), different \(k\) components travel at different speeds → dispersion.
The Quantum Connection
Dispersion causes wave packets to spread over time. A localized electron becomes delocalized—its position uncertainty grows. This is purely quantum: classical particles don't spread. The spreading rate is faster for more localized initial states (smaller \(\Delta x\) means larger \(\Delta k\)).