Wigner phase space distribution via classical adiabatic switching.
Wigner phase space distribution via classical adiabatic switching.
复制标题
通过经典绝热切换的维格纳相空间分布。
作者：
Amartya Bose;N. Makri
由于傅里叶型积分的振荡性质，多自由度系统的维格纳相空间密度的计算是一项极其艰巨的任务。我们提出了一个简单有效的近似程序来生成维格纳分布，避免了与维格纳变换相关的计算困难。从一个合适的零阶哈密顿量开始，其中维格纳密度可用（无论是解析还是数值），相空间分布通过经典轨迹在时间上传播，而扰动逐渐开启。根据经典绝热定理，如果扰动无限缓慢地打开，每个轨迹保持恒定的作用。我们证明了绝热开关过程可以产生谐振子本征态的精确维格纳密度，也可以产生Wentzel-Kramers-Brillouin （WKB）近似内的非调和哈密顿本征态的精确维格纳密度。我们通过引入依赖于每个轨迹能量的密度重标因子，将方法推广到有限温度。时间相关的性质可以简单地通过在完整目标哈密顿量下对每条轨迹进行积分而得到。进一步，通过构造，生成的近似Wigner分布在经典传播下是不变的，从而严格保持热力学性质。对一维和耗散系统的数值试验表明，在较宽的温度范围内，该方法得到的结果与全量子力学方法得到的结果非常吻合。该方法除需要经典轨迹积分所需的力场外，不需要任何输入，具有简单、高效的特点，适用于准经典轨迹计算。
Evaluation of the Wigner phase space density for systems of many degrees of freedom presents an extremely demanding task because of the oscillatory nature of the Fourier-type integral. We propose a simple and efficient, approximate procedure for generating the Wigner distribution that avoids the computational difficulties associated with the Wigner transform. Starting from a suitable zeroth-order Hamiltonian, for which the Wigner density is available (either analytically or numerically), the phase space distribution is propagated in time via classical trajectories, while the perturbation is gradually switched on. According to the classical adiabatic theorem, each trajectory maintains a constant action if the perturbation is switched on infinitely slowly. We show that the adiabatic switching procedure produces the exact Wigner density for harmonic oscillator eigenstates and also for eigenstates of anharmonic Hamiltonians within the Wentzel-Kramers-Brillouin (WKB) approximation. We generalize the approach to finite temperature by introducing a density rescaling factor that depends on the energy of each trajectory. Time-dependent properties are obtained simply by continuing the integration of each trajectory under the full target Hamiltonian. Further, by construction, the generated approximate Wigner distribution is invariant under classical propagation, and thus, thermodynamic properties are strictly preserved. Numerical tests on one-dimensional and dissipative systems indicate that the method produces results in very good agreement with those obtained by full quantum mechanical methods over a wide temperature range. The method is simple and efficient, as it requires no input besides the force fields required for classical trajectory integration, and is ideal for use in quasiclassical trajectory calculations.