论文标题
粒子在带有增量井的环上的量子机械问题
The Quantum Mechanical Problem of a Particle on a Ring with Delta Well
论文作者
论文摘要
带有质量$ m $的无自旋电子的问题，充电$ e $限制在半径$ r_0 $的环上，并具有具有缩放系数（深度）$κ$的非权利性理论的吸引人的Dirac Delta潜力（深度）$ themantivistic Theopical具有封闭的形式分析解决方案。单界状态函数是双曲线余弦的形式，但是包含一个参数$ d> 0 $，这是跨质量方程$ \ coth（d）=λd$的单个积极的真实解决方案，而非零零$λ= \ frac {2} {πκ} $。边界状态的能量特征值$ \ varepsilon = - \ frac {d^2} {2π^2} \ ailt \ of frac {q e m r_0} {2 \ hbar^2} $。此外，存在一套无限制的解决方案，正式的解决方案是通过将$ d \替换为i d $的术语从绑定解决方案获得的$ \ cosh（x）\ Overset {x \ to i x} {\ longrightArrow} \ cos（x）$。
The problem of a spin-free electron with mass $m$, charge $e$ confined onto a ring of radius $R_0$ and with an attractive Dirac delta potential with scaling factor (depth) $κ$ in non-relativistic theory has closed form analytical solutions. The single bound state function is of the form of a hyperbolic cosine that however contains a parameter $d>0$ which is the single positive real solution of the transcendental equation $\coth(d) = λd$ for non zero real $λ=\frac{2}{πκ}$. The energy eigenvalue of the bound state $\varepsilon=-\frac{d^2}{2π^2}\approx \frac{q e m R_0}{2 \hbar^2}$. In addition a discretly infinite set of unbounded solutions exists, formally these solutions are obtained from the terms for the bound solution by substituting $d \to i d $ yielding $\cot(d) = λd$ as characteristic equation with the corresponding set of solutions $d_k, k\in\mathbb{N}$, the respective state functions can be obtained via $\cosh(x)\overset{x \to i x}{\longrightarrow}\cos(x)$.
