耦合杨-米尔斯振子的冒险：I。半经典展开
Adventures of the Coupled Yang-Mills Oscillators: I. Semiclassical Expansion
摘要：
研究了在Yang-Mills（YM）方程空间齐次极限下，含$n=2,3$的$x^2y^2$势中的量子力学运动.这些系统在经典极限（$\hbar = 0$）下表现出很强的随机性，并表现出量子力学约束特征。我们利用Wigner-Kirkwood（WK）方法，通过半经典展开计算了超越T-F近似的配分函数$Z(t)$.我们推导出一种新的紧凑形式的微分方程的WK功能。在分离了等势面面通道中的运动和中心区域中的运动之后，我们证明了，如果我们通过绝热分离自由度正确地处理双曲通道中的量子运动，那么在$\hbar$中，TF项的高阶修正在八阶之前消失.最后，我们得到了配分函数关于参数$g^2\hbar^4t^3$的渐近展开式。
英文摘要：
We study the quantum mechanical motion in the $x^2y^2$ potentials with $n=2,3$, which arise in the spatially homogeneous limit of the Yang-Mills (YM) equations. These systems show strong stochasticity in the classical limit ($\hbar = 0$) and exhibit a quantum mechanical confinement feature. We calculate the partition function $Z(t)$ going beyond the Thomas-Fermi (TF) approximation by means of the semiclassical expansion using the Wigner-Kirkwood (WK) method. We derive a novel compact form of the differential equation for the WK function. After separating the motion in the channels of the equipotential surface from the motion in the central region, we show that the leading higher-order corrections to the TF term vanish up to eighth order in $\hbar$, if we treat the quantum motion in the hyperbolic channels correctly by adiabatic separation of the degrees of freedom. Finally, we obtain an asymptotic expansion of the partition function in terms of the parameter $g^2\hbar^4t^3$.
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