Demos.DoubleWell.Bound1D

Bound states well below the barrier separating the two (symmetric) wells occur as doublets with the energetic gaps being proportional to the tunneling rate. Bound states well above the barrier essentially behave like those of a single well potential. Wavefunctions localized around the barrier are found at intermediate energies. For illustration we use the Razavy double well potential for which 40 analytical solutions were published in our work on conditional quasi-exact solvability, but only for specifc values of the parameters. For the choice used here, the lowest seven eigenenergies are analytic, i.e. three doublets (below the barrier) and the first single state (slightly above the barrier), see following table (for β=−0.1)

n	En	parity
0	−9.00200125	even
1	−9.00200115	odd
2	−4.01170828	even
3	−4.01105034	odd
4	−1.15974458	even
5	−0.95694851	odd
6	+0.21345411	even

Note that the Razavy problem is anti-isospectral to the generalized planar pendulum which is also conditionally quasi-exactly solvable.

Fourier Grid method

One way to solve the time-independent Schrödinger equation is by using an evenly spaced grid. This allows us to apply the efficient Fourier grid method for calculating the matrix elements of the Fourier Grid Hamiltonian (FGH). Here a symmetry-adapted variant is used, yielding only states of even parity.

Matlab version	C++ version
Animated sequence of bound state wavefunctions	Animation
Input data file	C++ input and the same as Python script with some plotting
Logfile output	Logfile output

Gauss-Hermite DVR scheme

For the special case of a double well potential, a DVR expansion in eigenstates of the harmonic oscillator is often a good idea. The matrix elements of the Hamiltonian are only nonzero for harmonic oscillator states whose quantum numbers differ by not more than 2. For the example here, we get the same accuracy by employing only half the number of grid points, which can greatly speed up the diagonalisation. Also here a symmetry-adapted variant is used, yielding only states of even parity.

• Animated sequence of bound state wavefunctions
• Input data file
• Logfile output