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arXiv 0707.3510physics.chem-phmath-phmath.MPphysics.atom-ph
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具有电偶极矩分子场中电子薛定谔方程的解析解

Analytic solution of the Schrodinger equation for an electron in the field of a molecule with an electric dipole moment

A. D. Alhaidari

摘要:

我们放松通常的对角约束的矩阵表示的特征值波动方程,允许它是三对角。这导致了一个更大的解空间,其中包含了非中心电偶极子势cos(theta)/r^2的精确解析解,而已知非中心电偶极子势不属于精确可解势。因此,我们能够得到一个精确的解析解的三维时间无关的薛定谔方程的带电粒子在一个点电偶极子,可以携带非零净电荷的领域。这个问题模拟了电子与具有永久电偶极矩的分子(中性或离子化)的相互作用。该解决方案被写为一系列的平方可积函数,支持三对角矩阵表示的波算子的角和径向分量。此外,该解适用于所有能量,离散的(对于束缚态)以及连续的(对于散射态)。波函数的径向和角向分量的展开系数用满足三项递归关系的正交多项式表示。对于库仑自由的情况下,分子是中性的,我们计算其偶极矩的临界值低于不允许电子捕获。这些临界值不仅得到了基态,它同意已经知道的结果,而且也为激发态以及。

英文摘要:

We relax the usual diagonal constraint on the matrix representation of the eigenvalue wave equation by allowing it to be tridiagonal. This results in a larger solution space that incorporates an exact analytic solution for the non-central electric dipole potential cos(theta)/r^2, which was known not to belong to the class of exactly solvable potentials. As a result, we were able to obtain an exact analytic solution of the three-dimensional time-independent Schrodinger equation for a charged particle in the field of a point electric dipole that could carry a nonzero net charge. This problem models the interaction of an electron with a molecule (neutral or ionized) that has a permanent electric dipole moment. The solution is written as a series of square integrable functions that support a tridiagonal matrix representation for the angular and radial components of the wave operator. Moreover, this solution is for all energies, the discrete (for bound states) as well as the continuous (for scattering states). The expansion coefficients of the radial and angular components of the wavefunction are written in terms of orthogonal polynomials satisfying three-term recursion relations. For the Coulomb-free case, where the molecule is neutral, we calculate critical values for its dipole moment below which no electron capture is allowed. These critical values are obtained not only for the ground state, where it agrees with already known results, but also for excited states as well.

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