Quaternion supersymmetric quantum mechanics and SWKB approximation

A theory of quaternionic supersymmetric quantum mechanics has been developed in terms of quaternion supercharges, superpartner Hamiltonian and energy eigenvalues. We have obtained higher-order Hamiltonians with the help of lowering/raising quaternion operators and superpartner Hamiltonians, so one can arrive at higher-dimensional theories from lower dimensions by using this technique. It has been shown that partner Hamiltonians contain many terms which can indicate the presence of an extra particle, dark matter, or any unanswered physics phenomenon. Supersymmetric WKB approximation has also been tested in the quaternion space for superpotentials V+ and V-, and it is concluded that the SWKB quantization condition in quaternion space is not exact and is slightly different from the usual SWKB quantization condition.