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十点钟的咖啡

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[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.

$\frac{d}{dt}\rho(t)=\frac{1}{i\hbar}[V(t),\rho(t)]$ (1)

Then

$\rho(t+\Delta t) = \rho(t)+\frac{1}{i\hbar}\int_t^{t+\Delta t} dt' [V(t'),\rho(t')]$ (2)

This equation can be iterated. and it is

$\Delta \rho(t)=\frac{1}{i\hbar}\int_t^{t+\Delta t} dt' [V(t'),\rho(t')]+(\frac{1}{i\hbar})^2 \int_t^{t+\Delta t} dt' \int_t^{t'}dt''[\underbrace{V(t'),[V(t'')}_{Note\;t'\;and\;t''},\rho(t'')]]$ (3)

$\Delta \rho(t) = \rho(t+\Delta t) - \rho(t)$

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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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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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:

$\frac{d^2}{dt^2}\rho(t)=(\frac{1}{i\hbar})^2([\frac{dV(t)}{dt}, \rho(t)]+[V(t),\frac{d \rho(t)}{dt}])$

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4楼2014-06-27 15:00:09

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十点钟的咖啡(leongoall代发): 金币+10, 应楼主要求，奖励金币！ 2014-06-28 19:28:18
十点钟的咖啡: 回帖置顶 2014-06-29 13:40:24

把Eq.2中t+\Delta t 换成t', t'标记成t''
这样就得到\rho(t')的积分表达式
把这1表达式代入Eq.2中的右边就得到3项
第一项移左边第三项交换下积分和[]次序就得到eq.3

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5楼2014-06-27 20:58:31

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