Introduction: Quantum Diffusion
Free quantum particles don't stay localized. Even without any forces, the wavefunction spreads. This is a purely quantum effect with no classical analog.
The Mathematics of Spreading
For a Gaussian with initial width \(\sigma_0\), the width at time \(t\):
\[\sigma(t) = \sigma_0\sqrt{1 + \left(\frac{\hbar t}{2m\sigma_0^2}\right)^2}\]At long times: \(\sigma(t) \approx \frac{\hbar t}{2m\sigma_0}\) (linear growth)
The Spreading Time Scale
Define characteristic time \(\tau = 2m\sigma_0^2/\hbar\):
- For \(t \ll \tau\): width ≈ constant
- For \(t \gg \tau\): width grows linearly with time
Worked Example: Electron vs Baseball
Electron (\(m = 10^{-30}\) kg, \(\sigma_0 = 10^{-9}\) m):
\(\tau \approx 2 \times 10^{-16}\) s — spreads almost instantly
Baseball (\(m = 0.15\) kg, \(\sigma_0 = 10^{-3}\) m):
\(\tau \approx 3 \times 10^{26}\) s — older than the universe!
The Quantum Connection
This explains why baseballs don't spread but electrons do. For macroscopic objects, \(\tau\) is astronomically large. Quantum spreading is negligible at human scales but dominant at atomic scales. The uncertainty principle is responsible: narrower \(\sigma_0\) means larger \(\Delta p\), which causes faster spreading.