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
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Details
A wall of infinite height at the origin makes value of
inaccessible to the particle.
Thelowest 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.Permanent Citation