Chapter 4 turns the general rules of wave mechanics into a systematic problem-solving method. The potential divides space into regions; the time-independent Schrödinger equation determines the local form of the wave function; boundary and normalizability conditions select the physically allowed solutions.
A bound state is spatially localized and normalizable. Its wave function decays where the energy lies below the asymptotic potential, and its allowed energies form a discrete set.
An unbound state extends to infinity. It represents propagation or scattering and belongs to a continuous energy spectrum.
Every model begins with
The sign of \(E-V\) controls the local behavior: oscillatory when \(E>V\), exponential when \(E<V\). This observation anticipates confinement, penetration and tunneling.
| Model | Central idea |
|---|---|
| Harmonic oscillator | Smooth confinement and equally spaced energies |
| Finite well | Boundary matching and transcendental quantization |
| Attractive well, \(E>0\) | Reflection, transmission and resonances |
| Delta well | Derivative jump at a singular potential |
| Rectangular barrier | Evanescent waves and tunneling |
Begin locally. In a region where the potential is constant, the TISE can be written as
For \(E>V\), \(k\) is real and the independent solutions oscillate. For \(E<V\), write \(k=i\kappa\); the solutions are exponential:
For a bound state, the exponentially growing branch is discarded at each infinity. For scattering, the oscillatory branches are assigned directions and interpreted through current.
For a bound state, solve in every region and ask whether a globally continuous, differentiable and normalizable wave function exists. This occurs only for isolated energies, giving a discrete spectrum. For an unbound state, the incident energy is continuous; matching fixes the reflected and transmitted amplitudes.
Quantization and scattering are not separate rules. Both follow from joining local solutions into one acceptable global wave function. For a bound state, the matching conditions select isolated energies. For a scattering state, they determine amplitude ratios and therefore \(R\) and \(T\).
This page is designed to support short guided exercises on: Bound and unbound states: chapter roadmap.
Use these anchors to design compact exercises. The exercise should be answerable from this page plus standard algebra, while longer derivations, full worked examples and broader context should point back to the original book.