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[ÇóÖú] iteration method for density equations_help_me ÒÑÓÐ2È˲ÎÓë

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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mshwangg

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ÒýÓûØÌû:
8Â¥: Originally posted by walk1997 at 2014-06-27 22:24:54
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9Â¥2014-06-28 00:29:21
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mshwangg

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Ê®µãÖӵĿ§·È: ½ð±Ò+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
2Â¥2014-06-27 14:27:15
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Ê®µãÖӵĿ§·È

½ð³æ (ÕýʽдÊÖ)

ÒýÓûØÌû:
2Â¥: 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)?
3Â¥2014-06-27 14:48:47
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Ê®µãÖӵĿ§·È

½ð³æ (ÕýʽдÊÖ)

ÒýÓûØÌû:
2Â¥: 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:

4Â¥2014-06-27 15:00:09
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