Bound states, scattering states. Examples. The harmonic oscillator

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Modern 2

small displacements from equilibrium

Consider "particles" connected by some attractive potential in equilibrium:

$V(x) = V(x_0) + V'(x_0) (x-x_0) + V''(x_0) (x - x_0) ^ 2 + \dots$

At equilibrium, $V'(x_0) = 0 \$, so

$\Delta V = V'' \Delta x \$ to second order. This quadratic potential is called harmonic becuase the restoring force is linear: hence the particle executes simple harmonic motion when slightly perturbed from equilibrium.

some examples of potentials

Figure 4.5 in the book. A sandwich composed of AlGaAs - GaAs - AlGaAs. The central GaAs section is 6 nm wide.

Regions that do not contain a well

1. There are eigenfunctions for $E > V \$. These eigenfunctions, however, are plane waves and are not square integrable. They cannot represent a single particle, but can represent a constant flux of particles. We calculate transmission and reflection coefficients by comparing fluxes.

Regions that do contain a well

1. There exists an eigenfunction for every $E > E_{min} \$. These eigenfunctions are not square integrable.
2. For $E < E_{min} \$ eigenfunctions exist only for selected eigenvalues. For the finite square well potential we solve for these eigenvalues numerically.