AI 中文总结
研究斜壁无限深势阱中粒子的量子定态，通过艾里函数求解能谱，发现大量子数下能级渐近靠近，并在对称斜壁情形下出现基态受限程度反转的异常现象。
AI 中文摘要
研究了斜壁无限深势阱中粒子的量子力学问题。能量本征值由涉及第一类艾里函数的超越方程确定。讨论了两个极限情形。接下来，通过图解法和数值方法求得了允许的能量。图解分析表明，对于大的量子数，相邻能级会任意地靠近。这一能谱的渐近行为确实成立，这通过一个直观论证加以说明，并借助艾里函数的基本性质得到证实。在某种意义上，玻尔对应原理比标准盒子中粒子或谐振子的情形得到更精确的满足。一个态被视为越受限，则在该态的经典禁区内找到粒子的概率越小。对某些定态计算了受限程度，并与谐振子进行了比较，对于谐振子，其基态是所有定态中受限程度最低的。在对称情形下——左右壁斜率相等——发生了意想不到的事情：超过某个临界斜率后，基态不再是受限程度最低的本征态。除了其有趣的物理特性外，这一问题还为学生提供了熟悉艾里函数的机会，而艾里函数并不属于物理本科生标准数学知识范畴。
英文摘要
The quantum mechanical problem of a particle in a infinite potential well with slanted walls is studied. The energy eigenvalues are determined by a transcendental equation involving the Airy function of the first kind. Two limiting cases are discussed. Next, the allowed energies are found by graphical and numerical methods. The graphical analysis hints that for large quantum numbers the consecutive energy levels get arbitrarily close together. This asymptotic behavior of the energy spectrum actually holds, as shown by means of an intuitive argument and confirmed by resorting to basic properties of Airy functions. In a sense, Bohr's correspondence principle is more accurately fulfilled than in the cases of the standard particle in a box or the harmonic oscillator. A state is regarded the more confined the less the probability of finding the particle in the classically forbidden region. For some stationary states the confinement degree is calculated and a comparison is made with the harmonic oscillator, for which the ground state is the least confined of all stationary states. In the symmetric case --- right and left walls of equal slope --- something unexpected happens: beyond a certain critical slope the ground state is no longer the least confined eigenstate. Over and above its interesting physical features, this problem provides students with the opportunity to get acquainted with Airy functions, which do not belong to the standard mathematical repertoire of physics undergraduates.
Comments17 pages, 8 figures, 2 tables. Accepted for publication in Revista Brasileira de Ensino de Física
