Introduction: A More Realistic Well
Real potentials aren't infinitely deep. The finite square well has bound states that tunnel into classically forbidden regions—wavefunctions extend beyond the walls.
The Potential
\[V(x) = \begin{cases} -V_0 & |x| < a \\ 0 & |x| > a \end{cases}\]Bound states: \(-V_0 < E < 0\)
Solution Structure
Inside well (\(|x| < a\)): \(\psi'' = -k^2\psi\) with \(k = \sqrt{2m(E + V_0)}/\hbar\)
Solutions: \(\sin(kx)\) or \(\cos(kx)\)
Outside well (\(|x| > a\)): \(\psi'' = \kappa^2\psi\) with \(\kappa = \sqrt{-2mE}/\hbar\)
Solutions: \(e^{-\kappa|x|}\) (decaying)
Key Features
- Finite number of bound states (depends on \(V_0\) and \(a\))
- Wavefunction penetrates into walls (evanescent tail)
- At least one bound state always exists for any \(V_0 > 0\)
- Energies found from transcendental equations (next lesson)
The Quantum Connection
The finite well is the first step toward realistic potentials. The exponential decay outside the well is key to tunneling. Semiconductor quantum wells and nuclear potentials are modeled this way.