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小木虫论坛-学术科研互动平台 » 专业学科区 » 物理 » 原子、分子&光学 » iteration method for density equations_help_me
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十点钟的咖啡

金虫 (正式写手)

[求助] iteration method for density equations_help_me 已有2人参与

hi there; I need some help with the following formulas
In the interaction picture.

     (1)

Then

   (2)

This equation can be iterated. and it is

   (3)



I can understand the eq.(2), but not the eq. (3).
Is anybody know how to get the equation (3). and why do we want to do such calculation?
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1楼 2014-06-27 14:01:39
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mshwangg

至尊木虫 (正式写手)
引用回帖:
12楼: Originally posted by walk1997 at 2014-06-28 12:44:00
我算法差不多一点都不会呀  只会用mathematica做数值积分 : (
我感觉新版本的mathematica数值积分速度已经快追上c的包了
以前是调用cernlib中的积分库(fortran下)
现在mathematica下也可以调用c写的cuba包  和自 ...

你太猛，跟不上你的节奏
自己写的算法真心比不上那些函数库和软件
找时间研究下你说的
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13楼2014-06-28 15:49:42
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mshwangg

至尊木虫 (正式写手)

【答案】应助回帖

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感谢参与，应助指数 +1
十点钟的咖啡: 金币+1 2014-06-27 14:50:51
What your aim is to solve equation 1.
However, the statement in eq.2 and eq3 is only the method using Euler's formula. The precision of this method is the order of h^3 (h is the step).
In many cases, the h^3 precision is not sufficient for meeting your expectation.
In my opinion, it's better to use the Runge-Kutta method or use some software such as Matlab or Mathematica
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2楼2014-06-27 14:27:15
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十点钟的咖啡

金虫 (正式写手)
引用回帖:
2楼: Originally posted by mshwangg at 2014-06-27 14:27:15
What your aim is to solve equation 1.
However, the statement in eq.2 and eq3 is only the method using Euler's formula. The precision of this method is the order of h^3 (h is the step).
In many case ...

My equations are about the numerical solution. I just do not get the analytic solution (3). How can we derive the eq.(3) for (1) and (2)?
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3楼2014-06-27 14:48:47
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十点钟的咖啡

金虫 (正式写手)
引用回帖:
2楼: Originally posted by mshwangg at 2014-06-27 14:27:15
What your aim is to solve equation 1.
However, the statement in eq.2 and eq3 is only the method using Euler's formula. The precision of this method is the order of h^3 (h is the step).
In many case ...

This is not the Euler's method. There is always a fact of (1/2!) when we use the second order Euler' method.

and when I try to get the second derivative of the density operator. I get:
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4楼2014-06-27 15:00:09
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