Perturbation theory in statistical mechanics
Perturbation theory in statistical mechanics
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DOI：
10.1080/00018735500101254
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发表时间：
1955-10
影响因子：
--
通讯作者：
S. Nakajima
中科院分区：
文献类型：
--
作者：
S. Nakajima
这很明显，如果T是5/f的特征函数。如果T**是任何正交波函数的完备集，它也是有效的，因为迹（即算子的对角矩阵元素的和）在酉变换下是不变的。为了从（1.1）中求出自由能，Jt的所有特征值必须通过求解薛定谔方程来求出，这在简单系统中是不可行的。另一方面，在式（1.2）中，可以使用任意方便的集合{Tt~}。尽管如此，轨迹的精确评估通常是非常困难的，因此已经开发了许多近似方法。其中，早期应用于铁磁性分子场论的I-Iartree方法就是一个简单的例子。Hushimi（1940）对这种方法作了系统的研究。如附录所示，该方法基于变分原理，即最大熵原理。其他近似方法或多或少是基于密度算子exp（—fl~）的展开定理。例如，在fl中直接展开成幂级数经常被用于
This is obvious if the T,~ are the eigenfunctions of 5/f. It is also valid if the T** are any complete set of orthonormalized wave functions since the trace, ie the sum of diagonal matrix elements of an operator, is invariant under unitary transformations. In order to evaluate the free energy from (1.1), all eigenvalues of Jt have to be found by solving the SchrSdinger equation, which is not feasible except for simple systems. In the form (1.2), on the other hand, use can be made of any convenient set {Tt~}. Even so, the exact evaluation of the trace is normally very difficult, and a number of approximate methods have therefore been developed. Among them, the I-Iartree method which at an early stage has been applied in the molecular field theory of ferromagnetism, is a simple example. A systematic study of this method was given by Hushimi (1940). As is shown in the Appendix, the method is based on the variation principle, ie the principle of maximum entropy. Other approximate methods are more or less based on certain expansion theorems of the density operator exp (--fl~). For instance, the straightforward expansion into a power series in fl has often been used in the