In this study, we have discussed the implications of acceleration in quantum mechanics by means of a generalized derivative operator (GDO). A new Schrödinger equation is obtained which depends on the reduced Compton wavelength of the particle. We have discussed its implications in quantum mechanics for different types of potentials mainly the infinite wall potential, the gravitational linear field potential, the Cornell potential and the Coulomb repulsive potential. The corresponding wave functions and discrete energies are modified and differ from the results obtained in the conventional formalism. The major results obtained concerned the large improvement of the ground energy of the electron subject to the gravitational acceleration in addition to Cornell potential and the emergence of quantized electric charge in the theory without including Dirac monopoles or using gauge theories.

中文翻译：

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量子力学中的加速和电荷量子化

在这项研究中，我们通过广义导数算子 (GDO) 讨论了加速在量子力学中的含义。得到一个新的薛定谔方程，它取决于粒子的缩减康普顿波长。我们已经讨论了它在量子力学中对不同类型势能的影响，主要是无限壁势、引力线性场势、康奈尔势和库仑排斥势。相应的波函数和离散能量被修改，与传统形式得到的结果不同。