量子统计力学作为Lennard-Jones氦原子的精确经典展开及其结果
Quantum Statistical Mechanics as an Exact Classical Expansion with Results for Lennard-Jones Helium
摘要：
给出了代表经典相空间的量子态，并利用这些量子态将量子统计力学表述为经典统计力学的形式上精确的双微扰展开。一系列的量子贡献来自于位置和动量算符的非对易性。虽然量子态的表述不同，但目前位置算符和动量算符的单独平均值的结果与Wigner（1932）和Kirkwood（1933）一致。第二个级数由波函数对称化产生，用$l$ -粒子置换环的无限阶再求和形式给出。该级数解析地给出了量子理想气体的所有阶数的已知精确结果。领先的修正纠正了柯克伍德给出的修正。用超网链闭包方法计算了Lennard-Jones流体巨势的前四个量子修正。对于氦的液相分支等温线，修正量从几倍到1%不等，非对易性的影响明显大于波函数对称化的影响。所有的修正被认为是可以忽略不计的密度和温度下的氩气研究。计算是计算平凡的方法，避免了计算本征函数，本征值和数值对称化。
英文摘要：
The quantum states representing classical phase space are given, and these are used to formulate quantum statistical mechanics as a formally exact double perturbation expansion about classical statistical mechanics. One series of quantum contributions arises from the non-commutativity of the position and momentum operators. Although the formulation of the quantum states differs, the present results for separate averages of position operators and of momentum operators agree with Wigner (1932) and Kirkwood (1933). The second series arises from wave function symmetrization, and is given in terms of $l$-particle permutation loops in an infinite order re-summation. The series gives analytically the known exact result for the quantum ideal gas to all orders. The leading correction corrects a correction given by Kirkwood. The first four quantum corrections to the grand potential are calculated for a Lennard-Jones fluid using the hypernetted chain closure. For helium on liquid branch isotherms, the corrections range from several times to 1\% of the total classical pressure, with the effects of non-commutativity being significantly larger in magnitude than those of wave function symmetrization. All corrections are found to be negligible for argon at the densities and temperatures studied. The calculations are computationally trivial as the method avoids having to compute eigenfunctions, eigenvalues, and numerical symmetrization.
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