This closing page connects the models developed in the chapter without replacing their derivations. The common starting point is the stationary Schrödinger equation; the physical interpretation comes from the potential and the boundary conditions of each problem.
For the harmonic oscillator and the finite potential well, the wave function must describe a localized particle. This normalizability requirement selects only particular energies from the formal solutions of the differential equation.
In the oscillator, the selected energies are equally spaced. In the finite well, the matching of the oscillatory interior solution to the decaying exterior tails selects the allowed levels.
A bound state need not vanish sharply at the classical limits. In the finite well the wave function has exponentially decaying tails outside the well, so there is a nonzero probability of finding the particle in those regions.
For positive-energy scattering, the solution is described in terms of incident, reflected, and transmitted waves. The amplitudes are fixed by imposing continuity conditions at each change in the potential.
The same procedure applies to the finite well, the delta potential, and the rectangular step. What changes from one model to another is the form of the solution in each region.
Reflection and transmission express how the incident probability current is divided by the potential. For the stationary scattering problems treated in this chapter, conservation of probability gives
At a step or barrier, a region that is classically forbidden can still contain an evanescent wave. This is the basis for the tunneling behavior discussed in the chapter.
Boundary conditions are not merely algebraic constraints. In bound-state problems they turn formal solutions into a discrete spectrum. In scattering problems they relate the amplitudes on the two sides of a potential and thereby determine reflection and transmission.
Use the individual pages for the equations and definitions of each model: the harmonic oscillator, finite potential well, attractive finite well, delta well, and rectangular step potential. The detailed matching calculations remain in their respective sections.
A potential determines the local form of the Schrödinger equation; boundary conditions determine which of its solutions describe a physical state. This is why the chapter contains both discrete bound-state energies and continuous scattering energies, together with reflection, transmission, penetration, and tunneling.