无限方势阱作为有限方势阱在最小长度情形下的极限情形
An infinite square-well potential as a limiting case of a finite square-well potential in a minimal-length scenario
摘要：
无限深方势阱是量子力学中研究最广泛的问题之一。在最小长度的情况下，其研究需要额外的照顾，因为在井壁的边界条件没有很好地固定。为了避免这种情况，我们解决了有限的方势阱的边界条件是固定的，即使在最小长度的情况下，然后我们采取的限制的潜力走向无穷大，找到本征函数和能量方程的无限方势阱。虽然能量本征值的一阶修正与文献中的相同，但我们的结果表明本征函数在方阱壁处具有一阶导数连续性，这与前人的工作是不一致的。这是因为在文献中，作者忽略了双曲型解，并假定了无限深方势阱的本征函数的一阶导数在阱壁上不连续，这是不正确的。正如我们所表明的，本征函数在方阱壁处的一阶导数的连续性保证了几率流密度的连续性和时间演化的么正性
英文摘要：
One of the most widely problem studied in quantum mechanics is of an infinite square-well potential. In a minimal-length scenario its study requires additional care because the boundary conditions at the walls of the well are not well fixed. In order to avoid this we solve the finite square-well potential whose the boundary conditions are well fixed, even in a minimal-length scenario, and then we take the limit of the potential going to infinity to find the eigenfunctions and the energy equation for the infinite square-well potential. Although the first correction for the energy eigenvalues is the same one has been found in the literature, our result shows that the eigenfunctions have the first derivative continuous at the square-well walls what is in disagreement those previous work. That is because in the literature the authors have neglected the hyperbolic solutions and have assumed the discontinuity of the first derivative of the eigenfunctions at the walls of the infinite square-well which is not correct. As we show, the continuity of the first derivative of the eigenfunctions at the square-well walls guarantees the continuity of the probability current density and the unitarity of the time evolution
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