This paper employs the finite difference method to solve the timeindependent Schrdinger equation for a particle with mass m in a finite square well. Through Python programming, the energy levels and wave functions are numerically computed, with convergence and accuracy analyzed. By this method, we elucidate the roles of the well depth and width, discuss the conditions under which a finite well can be approximated as an infinite well. The tunneling rate versus energy level is also presented. Combining theoretical derivation with numerical results, this research is suitable for teaching quantum mechanics. The applied method can be extended to more complex systems, such as the harmonic oscillator and delta potential.