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 .

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DETAILS

To discover the quasi-exact problem, begin with . It is then found that
,
which is identified as .
The orthonormalized wavefunctions are:
,
,
.
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