arXiv quant-ph/06062462006-06-29quant-phcond-mat.str-elnucl-th

最优可观测演化的随机Schroedinger方程

Stochastic Schroedinger equation from optimal observable evolution

Denis Lacroix

备注：: 13 pages, 1 figure
发表信息：: AnnalsPhys.322:2055-2076,2007
DOI：: 10.1016/j.aop.2006.09.003

摘要：

在本文中，我们考虑一组由$| Q \right>$表示的试验波函数和一组相关的运算符$A_α$，这些运算符生成连接这些试验状态的变换。利用变分原理，我们证明了我们总是可以得到一个量子Monte-Carlo方法，其中系统的量子演化被$D = |Q_a> <Q_b|$形式的密度矩阵之间的跳跃所取代，并且其中$A_α$的矩的平均演化直到给定的$k$阶，即$< A_{α_1} >$，$< A_{α_1} A_{α_2} >$，. $< A_{α_1} ... A_{α_k} >$被约束为在沿着每个随机轨迹的每个时间步长处遵循精确的Eklfest演化。然后，一组越来越详细的随机近似的量子问题的方法时，越来越多的约束条件，即当$k$增加的精确解。如果应用于试验状态的哈密顿量$H$可以写成$A_α$的多项式，则蒙特卡罗过程甚至可能变得精确。这种形式将希尔伯特空间中的量子跃迁与相空间动力学联系起来。我们表明，随机薛定谔方程的推导可以大大简化，利用存在的这个层次的近似和它的连接到Escherifest定理。举例说明了几个例子：自由波包展开，克尔振荡器，克尔振荡器的广义版本，以及相互作用玻色子或费米子。

英文摘要：

In this article, we consider a set of trial wave-functions denoted by $| Q \right>$ and an associated set of operators $A_α$ which generate transformations connecting those trial states. Using variational principles, we show that we can always obtain a quantum Monte-Carlo method where the quantum evolution of a system is replaced by jumps between density matrices of the form $D = |Q_a> <Q_b|$, and where the average evolutions of the moments of $A_α$ up to a given order $k$, i.e. $< A_{α_1} >$, $< A_{α_1} A_{α_2} >$,..., $< A_{α_1} ... A_{α_k} >$, are constrained to follow the exact Ehrenfest evolution at each time step along each stochastic trajectory. Then, a set of more and more elaborated stochastic approximations of a quantum problem is obtained which approach the exact solution when more and more constraints are imposed, i.e. when $k$ increases. The Monte-Carlo process might even become exact if the Hamiltonian $H$ applied on the trial state can be written as a polynomial of $A_α$. The formalism makes a natural connection between quantum jumps in Hilbert space and phase-space dynamics. We show that the derivation of stochastic Schroedinger equations can be greatly simplified by taking advantage of the existence of this hierarchy of approximations and its connection to the Ehrenfest theorem. Several examples are illustrated: the free wave-packet expansion, the Kerr oscillator, a generalized version of the Kerr oscillator, as well as interacting bosons or fermions.

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摘要：

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