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.

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DETAILS

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 the eigenvalue problem of the Hamiltonian (these are solutions of the stationary Schrödinger equation): , where is the reduced Planck's constant (, is the mass of the atom, is the slope of the linear potential , , and ( = 1, 2, ...) are the zeros of the Airy function . All of these lie on the negative part of the axis.
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