  The concept of out-of-time-ordered correlation (OTOC) function is treated as
a very strong theoretical probe of quantum randomness, using which one can
study both chaotic and non-chaotic phenomena in the context of quantum
statistical mechanics. In this paper, we define a general class of OTOC, which
can perfectly capture quantum randomness phenomena in a better way. Further we
demonstrate an equivalent formalism of computation using a general time
independent Hamiltonian having well defined eigenstate representation for
integrable supersymmetric quantum systems. We found that one needs to consider
two new correlators apart from the usual one to have a complete quantum
description. To visualize the impact of the given formalism we consider the two
well known models viz. Harmonic Oscillator and one dimensional potential well
within the framework of supersymmetry. For the Harmonic Oscillator case we
obtain similar periodic time dependence but dissimilar parameter dependences
compared to the results obtained from both micro-canonical and canonical
ensembles in quantum mechanics without supersymmetry. On the other hand, for
one dimensional potential well problem we found significantly different time
scale and the other parameter dependence compared to the results obtained from
non-supersymmetric quantum mechanics. Finally, to establish the consistency of
the prescribed formalism in the classical limit, we demonstrate the phase space
averaged version of the classical version of OTOCs from a model independent
Hamiltonian along with the previously mentioned these well cited models.
