Introduction: An Infinitely Narrow, Infinitely Deep Well
The delta function potential is an idealization: infinitely deep but infinitely narrow, with finite "strength." Despite being singular, it's exactly solvable and illustrates key quantum concepts with minimal math.
The Potential
\[V(x) = -\alpha\delta(x)\]where \(\alpha > 0\) has units of [energy × length].
Boundary Condition at the Delta
\(\psi\) is continuous at \(x = 0\), but \(\psi'\) has a discontinuity:
\[\Delta\psi' = \psi'(0^+) - \psi'(0^-) = -\frac{2m\alpha}{\hbar^2}\psi(0)\]This follows from integrating the TISE across \(x = 0\).
Key Results
- Exactly one bound state exists
- Binding energy: \(E = -\frac{m\alpha^2}{2\hbar^2}\)
- Wavefunction: \(\psi(x) = \frac{\sqrt{m\alpha}}{\hbar}e^{-m\alpha|x|/\hbar^2}\)
The Quantum Connection
The delta potential models impurities in solids, short-range nuclear forces, and serves as a limiting case for understanding how any attractive potential creates bound states. Its single bound state shows that even the weakest 1D attraction binds a particle.