规范变换

Also we use the time-dependent SU (2) gauge transformation to diagonalize the Hamilton operator, obtain the berry phase and analytically the time-evolution operator. The particle number difference of species A between the two wells is studied analytically.

此外我们还研究了在双阱玻色-爱因斯坦凝聚中纠缠态的演化，研究发现随着组分间相互作用和随穿率的比值的增加系统演化到Bell态的概率变大，而且组分自身内在的相互作用对形成Bell态的几率没有影响；并且用含时Su（2）规范变换对角化哈密顿量得到了系统的Berry位相和时间演化算符，并研究了量子随穿过程。

As a matter of fact,free particle are automatically placed in the field interaction once local gauge transformation is made on them.

其实 ，对自由粒子做局域规范变换，粒子就已被自动置于场的相互作用之中了。

Then time-dependent invariant is introduced and a kind of method to solve the Schrodinger equation explicitly called Gauge transformation method.

稍后引入含时不变量的定义，以及精确求解薛定谔方程的含时规范变换方法，再次讨论了Aharonov-Bohm效应，Berry相位和绝热近似。

This paper gives a full and explicit analysis for Geometric Phase, from basic theory to familiar calculation modules in articles. We focus on the application of gauge transformation method to Geometric Phase Calculation.

本文对几何相作了一些比较细致全面的分析，从基础理论开始一直到文献中比较常见的模型计算，将重点放在正则规范变换方法在计算几何相中的应用上，着重掌握一种计算几何相的方法。

By performing appropriate time-dependent gauge transformation,closed formulas for the time evolution of quantum states and Berry phases are obtained.

通过适当的含时规范变换得到有限维含时谐振子量子态时间演化的封闭解，并给出量子态的Berry相位。

At the end of this chapter a concrete calculation example of Gauge transformation is given.In chapter 2 of this paper, two example applying gauge transformation for Quantum Geometric Phase in adiabatic condition are given, with explicit calculation process .The two examples is similar in objects to calculate, they are two state quantum system expressed in sphere coordination.

第二章主要讲了两个利用含时规范变换求解绝热近似下的几何相的例子，给出比较具体的计算步骤，它们有一个共同特点就是计算的对象都是二态体系，计算时采用的都是球坐标系，计算的都是只有两个量子态的纯粹的量子系统。

First,basing on the equivalence betweenthe AKNS hierarchy and the cKP hierarchy with k=1,we establish a unifiedframe to solve AKNS hierarchy by two types of gauge transformation.

第一，基于AKNS系列与k＝1的约束KP系列的等价性，我们建立一个用两种类型规范变换求解AKNS系列的统一框架。

We alsodiscuss three chains of successive gauge transformation with two steps,andgenerate sAKNS hierarchy from"free"Lax operator

我们讨论了规范变换的三种递推，以及从"自由"Lax算子生成sAKNS系列。

Third,similar to the case of pure Bose system,for sAKNS hierarchydefined by the constraints of SKP2 hierarchy,we present one unified way tosolve sAKNS hierarchy in terms of two types of gauge transformation.

第三，类似于纯玻色情形，对于由SKP2系列的约束所定义的AKNS系列的超对称化-sAKNS系列，我们提出一个用两种类型规范变换求解它的统一方式。

Further, with the help of Riccati equations, an infinite number of conservation laws for the solton hierarchy are deduced. For the sake of simplicity, taking the general TD hierarchy as an illustrative example, we prove that its 2×2 Lenard pair of operators forms a Hamiltonian pair. Thus the isospectral evolution TD hierarchy is the general Hamiltonian system and possesses the Bi-Hamiltonian structures and Multi-Hamiltonian structures. By using the method of derivation of functional under some constraint condition, a complete one-to-one correspondence between the Hamiltonian functions of the hierarchy and its conservation density functions can be built. These results can also be applied to the isospectral evolution soliton hierarchy of this paper. Finally, there's a gauge transformation between the spectral problem of this paper and the AKNS system. Moreover, the potentials in these spectral problems satisfy the general Miura transformation, the corresponding relationship between the two soliton hierarchies is also given.

进一步本文还通过特征函数的组合关系所满足的Riccati方程，得到了该等谱方程族的无穷多个守恒律；为简便起见，本文以广义TD族为例，由它的2×2 Lenard算子对的性质证明了此算子对为Hamilton算子对，这说明广义TD族是广义Hamilton系统且具有Bi-Hamilton结构和Multi-Hamilton结构；进而利用它的依赖于谱参数的一般守恒密度的积分在约束条件下求泛函导数的方法，得到了广义TD族的Hamilton函数与守恒密度之间的对应关系，这些性质对于由本文提出的2×2谱问题所导出的等谱孤子族仍成立；另外此谱问题与AKNS系统存在着规范变换，位势之间有广义Miura变换，而孤子方程之间也满足一定的等价关系。

The split between the two groups can hardly be papered over.

这两个团体间的分歧难以掩饰。

This approach not only encourages a greater number of responses, but minimizes the likelihood of stale groupthink.

这种做法不仅鼓励了更多的反应，而且减少跟风的可能性。