# Numerov Solutions for Single- and Double-Well Potentials

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In quantum mechanics, energy states of a one-particle system are given by solutions of the time-independent Schrödinger equation

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Contributed by: Eric R. Bittner (March 2011)

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

Snapshot 1: ground state of double-well potential

In this case quantum mechanical tunneling produces a delocalization of the wavefunction between the two wells. One can clearly see that a considerable portion of is "underneath the barrier".

Snapshot 2: Numerov solution for the first excited state of a harmonic oscillator at

Notice that stationary solutions occur only when when obeys the boundary conditions that it vanish as .

In the next two snapshots, when is slightly above or below this value, diverges as becomes large.

For an elementary overview of quantum mechanics and the use of the Numerov approach to compute stationary solutions, the interested viewer is referred to I. Levine, *Quantum Chemistry, *5th ed*.*, Upper Saddle River, New Jersey: Prentice Hall, 1999.

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