In this work, we address the quartic quantum oscillator in phase space using two approaches: computational and algebraic methods. In order to achieve such an aim, we built simplistic unitary representations for Galilei group, as a consequence the Schrödinger equation is derived in the phase space. In this context, the amplitudes of quasi-probability are associated with the Wigner function. In a computational way, we apply the techniques of Lie methods. As a result, we determine the solution of the quantum oscillator in the phase space and calculate the corresponding Wigner function. We also calculated the negativity parameter of the analyzed system.

中文翻译：

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相空间中量子四次振子的解析解

在这项工作中，我们使用两种方法解决相空间中的四次量子振荡器：计算方法和代数方法。为了实现这一目标，我们为伽利略群建立了简单的酉表示，因此薛定谔方程是在相空间中导出的。在这种情况下，准概率的幅度与 Wigner 函数相关联。在计算方面，我们应用了李方法的技术。因此，我们确定了相空间中量子振荡器的解，并计算了相应的 Wigner 函数。我们还计算了分析系统的负参数。