Wolfram Demonstrations Project

13,000+ Open Interactive Demonstrations
Powered by Notebook Technology »

Quasi-exact Solution for a Double-Well Potential

Initializing live version
Download to Desktop

Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram Player or other Wolfram Language products.

If a Schrödinger equation admits exact analytic solutions only for certain values of the parameters in the Hamiltonian, the problem is said to be quasi-exact. We have found that the bimodal Gaussian function

[more]

is a quasi-exact solution of the Schrödinger equation for the double-well potential

, ,

which closely approximates a pair of harmonic oscillator potentials with origins near .

The ground-state energy is given by .

There is no other analytic solution for this potential. The first excited state can be surmised to have the approximate form

.

The first excited state energy is approximated by . The second excited state is represented by a function orthogonal to both and . The energy is approximated by .

The graphic displays the energy as a blue horizontal line on a plot of . The wavefunction is plotted on the right. Select the appropriate checkboxes to display the wavefunctions and and the corresponding energies and .

[less]

Contributed by: S. M. Blinder (August 2022)
Open content licensed under CC BY-NC-SA

Snapshots

Details

$Failed
Embed Code

Take advantage of the Wolfram Notebook Emebedder for the recommended user experience.

Related Demonstrations More by Author
Browse all topics
Feedback (field required)
Email (field required) Name
Occupation Organization
Note: Your message & contact information may be shared with the author of any specific Demonstration for which you give feedback.
Send