A pedagogical approach to conformable quantum mechanics with applications

We present a straightforward but precise approach to the conformable quantum mechanics using the local conformable derivative introduced in [R. Khalil, M. Al Horani, A. Yousef and M. Sababheh, J. Comput. Appl. Math. 264, 65 (2014)]. The conformable wave equation is defined and used to find some important results for the approach. The conformable Schr?dinger wave equation is then justified. It is then used to solve three one-dimensional illustrative examples, namely, the infinite potential well and the quantum harmonic oscillator, using series solutions in terms of the Mittag-Leffler functions and the singular Coulomb potential using the generalized Laguerre polynomials. The effect of the conformable derivative parameter ? on the wave function, normalized probability densities and energy levels is presented for different states. It is found that there is a gradual transition in the probability densities with the ? . Our results are compared with the corresponding available approaches using different definitions of nonlocal fractional derivative. The resemblance of our approach to that in traditional quantum mechanics makes it more appropriate for students to learn.