Energy Levels in a Half-Infinite Linear Well
The energy spectrum of a quantum particle moving in a potential well is discrete. The density of energy eigenstates grows as the potential well's slope decreases. This is similar to the behavior of a free particle.
A wall of infinite height at the origin makes value of
inaccessible to the particle.The
lowest twelve energy values are calculated numerically. The number of the energy levels
is infinite; they can be calculated from solutions of theeigenvalue problem of the Hamiltonian (these are solutions of the stationary Schrödinger equation): ,
is the reduced Planck's constant (
is the mass of the atom,
is the slope of the linear potential
= 1, 2, ...) arethe zeros of the Airy function . All of these lie on the negative part of the axis.