论文标题
恒定磁场中的二维氢原子
Two-dimensional Hydrogen-like Atom in a Constant Magnetic Field
论文作者
论文摘要
考虑了恒定磁场中的二维氢样原子。发现这实际上是两个单独的问题。磁场会引起核与电子之间的有效吸引力，而它引起有效的排斥。两个问题中的每一个都有三个单独的案例，具体取决于能量特征值移动的迹象。对于六种可能性中的两种（负数为负的能量特征值），这表明可以精确地获得前四个溶液。对于六种可能性中的另外两个（阳性能量特征值），这表明可以精确地获得前八种溶液。对于高阶表示，能量特征值是第五或更高阶多项式的根，因此，必须以数值获得特征值。一旦已知能量特征值，也已知径向波方程的解决方案。径向波方程的精确溶液，剩余的两种可能性（零移动特征值）通过递归关系给出了任何所需的顺序。
The two-dimensional hydrogen-like atom in a constant magnetic field is considered. It is found that this is actually two separate problems. One for which the magnetic field causes an effective attraction between the nucleus and the electron and one for which it causes an effective repulsion. Each of the two problems has three separate cases depending on the sign of a shifted energy eigenvalue. For two of the six possibilities (shifted energy eigenvalue that is negative) it is shown that the first four solutions can be obtained exactly. For another two of the six possibilities (shifted energy eigenvalue that is positive) it is shown that the first eight solutions can be obtained exactly. For higher order states the energy eigenvalue is the root of a fifth or higher order polynomial, hence, the eigenvalue must be obtained numerically. Once the energy eigenvalue is known the solution to the radial wave equation is also known. Exact solutions for the radial wave equation, for the remaining two possibilities (shifted energy eigenvalue that is zero), are given to any desired order by means of a recursion relation.
