**Exact Finite Differences for Quantum Mechanics** Exact Finite Differences for Quantum Mechanics

DOI: 10.5772/intechopen.71956

Armando Martínez-Pérez and Gabino Torres-Vega Armando Martínez-Pérez and

Additional information is available at the end of the chapter Gabino Torres-Vega

http://dx.doi.org/10.5772/intechopen.71956 Additional information is available at the end of the chapter

### Abstract

[74] Hung ND, Plaza P, Martin M, Meyer YH. Generation of tunable subpicosecond pulses using low-Q dye cavities. Applied Optics. 1992;31:7046-7058. DOI: 10.1364/AO.31.007046

[75] Pham MH, Cadatal-Raduban M, Luong MV, Le HH, Yamanoi K, Nakazato T, Shimizu T, Sarukura N, Nguyen HD. Numerical simulation of ultraviolet picosecond Ce:LiCAF laser emission by optimized resonator transients. Japanese Journal of Applied Physics. 2014;53:

[76] Ono S, Suzuki Y, Kozeki T, Murakami H, Ohtake H, Sarukura N, Sato H, Machida S, Shimamura K, Fukuda T. High-energy, all-solid-state, ultraviolet laser power-amplifier module design and its output-energy scaling principle. Applied Optics. 2002;41:7556-

062701. DOI: 10.7567/JJAP.53.062701

162 Numerical Simulations in Engineering and Science

7560. DOI: 10.1364/AO.41.007556

We introduce a finite difference derivative, on a non-uniform partition, with the characteristic that the derivative of the exponential function is the exponential function itself, times a constant, which is similar to what happens in the continuous variable case. Aside from its application to perform numerical computations, this is particularly useful in defining a quantum mechanical discrete momentum operator.

Keywords: exact finite differences derivative, discrete quantum mechanical momentum operator, time operator
