Quasi-exact Solution for a Double-Well Potential

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

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 .