Energy Levels in a Half-Infinite Linear Well

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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.

Contributed by: Reinhard Tiebel (March 2011)
Open content licensed under CC BY-NC-SA



A wall of infinite height at the origin makes value of

inaccessible to the particle.


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): ,


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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