# 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

## Snapshots

## 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