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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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walk1997

金虫 (著名写手)
这样做的目的 上面已经解释了
在有限的\Delta 时 更精确....
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6楼2014-06-27 21:01:35
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mshwangg

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【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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