The solution to Mathieu's equation was modified and combined to only one single sine function. In this form, the cumulative probability functions for a fixed RF-phase can be determined as arcsine distributions. The probability of a normalized position $\mathit{u}/u_{\text{max}}$ or normalized velocity $\dot{\mathit{u}}/{\dot{u}}_{\text{max}}$ is relative to $1/\sqrt{1-{\left(\mathit{u}/u_{\text{max}}\right)}^2}$, $\dot{\mathit{u}}/{\dot{u}}_{\text{max}}$ respectively. This corrects reference descriptions present in the literature, which stated the probability to be proportional to $\sqrt{1-{\left(\mathit{u}/u_{\text{max}}\right)}^2}$. Resulting probability plots show that ions that are heavy on the relative mass scale of quadrupole ion traps, are more likely to be found close to the maximum of their oscillation amplitude. Light ions are more likely to be found at the center of their oscillation. The velocity distributions show that the likeliest velocity converges for low q-values to the mean-square velocity but splits into two likely velocity regions with increasing q-value. It is further emphasized, that these distributions describe the probability of a single ion and in order to describe the behavior of an ensemble of ions, it is inevitably needed to define a distribution of oscillation amplitudes $u_{\text{max}}$.

Mathieu 方程的解被修改并合并为一个单一的正弦函数。在这种形式中，固定 RF 相位的累积概率函数可以确定为反正弦分布。归一化位置的概率$\mathit{你}/{你}_{\text{最大限度}}$ 或归一化速度 $\dot{\mathit{你}}/{\dot{你}}_{\text{最大限度}}$ 是相对于 $1/\sqrt{1-{\left(\mathit{你}/{你}_{\text{最大限度}}\right)}^2}$, $\dot{\mathit{你}}/{\dot{你}}_{\text{最大限度}}$分别。这更正了文献中的参考描述，该描述指出概率与$\sqrt{1-{\left(\mathit{你}/{你}_{\text{最大限度}}\right)}^2}$. 结果的概率图显示，在四极杆离子阱的相对质量标度上较重的离子更有可能在其振荡幅度的最大值附近被发现。轻离子更有可能在其振荡的中心被发现。速度分布表明，对于低q 值，最可能的速度收敛到均方速度，但随着q值的增加，分裂成两个可能的速度区域。需要进一步强调的是，这些分布描述了单个离子的概率，为了描述一组离子的行为，不可避免地需要定义振荡幅度的分布${你}_{\text{最大限度}}$.