论文标题
Anharmonic振荡器：解决方案
Anharmonic oscillator: a solution
论文作者
论文摘要
结果表明，对于具有$ v（x）= x^2+g^2 x^4 $的一维量子Anharmonic振荡器，$ g^2 $（弱耦合体制）的扰动理论（PT）和$ \ hbar $ for Entergies conconcide conconcide conconcide of $ g^2 $（弱耦合体制）的扰动理论（PT）。这与以下事实相关：$ x $ -space和$（gx）$ - 空间中的动力学对应于相同的能量谱，并有效耦合常数$ \ hbar g^2 $。分别得出了两个方程，它们分别得出了这两个空间中的动力学，分别是Riccati-Bloch（RB）和广义Bloch（GB）方程。波动功能的对数衍生物的$ g^2 $的pt导致PT（$ x $系数为$ x $系数），用于RB方程，以及GB方程的$ \ hbar $ $ \ hbar $的真实半经典扩展，这对应于该路径形式的密度矩阵的循环扩展。这两个扩展的2-参数插值导致$ x $ - 空格的波函数均匀近似，并具有前所未有的精度$ \ sim 10^{ - 6} $本地和前所未有的准确性$ \ sim 10^{ - 9} -9} -10} -10} -10} -10} -10} -10} -10^{ - 10} $ $ geq 0 $ geq 0 $ geq 0 $ ge g^2.简要讨论了对径向四分之一振荡器的概括。
It is shown that for the one-dimensional quantum anharmonic oscillator with potential $V(x)= x^2+g^2 x^4$ the Perturbation Theory (PT) in powers of $g^2$ (weak coupling regime) and the semiclassical expansion in powers of $\hbar$ for energies coincide. It is related to the fact that the dynamics in $x$-space and in $(gx)$-space corresponds to the same energy spectrum with effective coupling constant $\hbar g^2$. Two equations, which govern the dynamics in those two spaces, the Riccati-Bloch (RB) and the Generalized Bloch (GB) equations, respectively, are derived. The PT in $g^2$ for the logarithmic derivative of wave function leads to PT (with polynomial in $x$ coefficients) for the RB equation and to the true semiclassical expansion in powers of $\hbar$ for the GB equation, which corresponds to a loop expansion for the density matrix in the path integral formalism. A 2-parametric interpolation of these two expansions leads to a uniform approximation of the wavefunction in $x$-space with unprecedented accuracy $\sim 10^{-6}$ locally and unprecedented accuracy $\sim 10^{-9}-10^{-10}$ in energy for any $g^2 \geq 0$. A generalization to the radial quartic oscillator is briefly discussed.
