There is significant literature on Schrödinger differential equation (SDE) solutions, where the fractional derivatives are stated in terms of Caputo derivative (CD). There is hardly any work on analytical and numerical SDE solutions involving conformable fractional derivative (CFD). For the reasons stated above, we are required to solve the SDE in the form of CFD. The main goal of this research is to offer a novel combined computational approach by using conformable natural transform (CNT) and the homotopy perturbation method (HPM) for extracting analytical and numerical solutions of the time-fractional conformable Schrödinger equation (TFCSE) with zero and nonzero trapping potential. We call it the conformable natural transform homotopy perturbation method (CNTHPM). The relative, recurrence, and absolute errors of the problems are analyzed to evaluate the efficiency and consistency of the CNTHPM. The error analysis has confirmed the higher degree of accuracy and convergence rates, which indicates the effectiveness and reliability of the suggested method. Furthermore, 2D and 3D graphs compare the exact and approximate solutions. The procedure is quick, precise, and easy to implement, and it yields outstanding results. In addition, numerical results are also compared with other methods such as the differential transform method (DTM), split-step finite difference method (SSFDM), homotopy analysis method (HAM), homotopy perturbation method (HPM), Adomian decomposition method (ADM), and two-dimensional differential transform method (TDDTM). The comparison shows excellent agreement with these methods, which means that CNTHPM is a suitable alternative tool to the methods based on CD for the solutions of the time-fractional SDE. Moreover, we can conclude that the CFD is a suitable alternative to the CD in the modeling of time-fractional SDE. The Banach fixed point theory was also used to test the uniqueness of the solution, convergence, and error analysis.

有大量关于薛定谔微分方程 (SDE) 解的文献，其中分数导数用 Caputo 导数 (CD) 表示。几乎没有关于涉及一致分数导数 (CFD) 的解析和数值 SDE 解决方案的工作。由于上述原因，我们需要以 CFD 的形式求解 SDE。本研究的主要目标是提供一种新颖的组合计算方法，通过使用一致自然变换 (CNT) 和同伦摄动方法 (HPM) 来提取零和时间分数一致薛定谔方程 (TFCSE) 的解析和数值解。非零诱捕势。我们称之为顺应自然变换同伦摄动法（CNTHPM）。相对，复发，分析问题的绝对误差，以评估 CNTHPM 的效率和一致性。误差分析证实了更高程度的准确性和收敛速度，这表明了所提出方法的有效性和可靠性。此外，2D 和 3D 图比较了精确解和近似解。该程序快速、精确且易于实施，并产生出色的结果。此外，数值结果还与微分变换法（DTM）、分步有限差分法（SSFDM）、同伦分析法（HAM）、同伦摄动法（HPM）、阿多米安分解法（ADM）等方法进行了比较。 ) 和二维差分变换方法 (TDDTM)。比较表明与这些方法非常一致，这意味着 CNTHPM 是基于 CD 方法的一种合适的替代工具，用于求解时间分数 SDE。此外，我们可以得出结论，在时间分数 SDE 建模中，CFD 是 CD 的合适替代方案。Banach 不动点理论也被用来检验解的唯一性、收敛性和误差分析。