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十點(diǎn)鐘的咖啡

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[求助] iteration method for density equations_help_me 已有2人參與

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

$\fracgfwsz7l{dt}\rho(t)=\frac{1}{i\hbar}[V(t),\rho(t)]$ (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樓 2014-06-27 14:01:39

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walk1997

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十點(diǎn)鐘的咖啡(leongoall代發(fā)): 金幣+10, 應(yīng)樓主要求，獎勵金幣！ 2014-06-28 19:28:18
十點(diǎn)鐘的咖啡: 回帖置頂 2014-06-29 13:40:24

把Eq.2中t+\Delta t 換成t', t'標(biāo)記成t''
這樣就得到\rho(t')的積分表達(dá)式
把這1表達(dá)式代入Eq.2中的右邊就得到3項
第一項移左邊第三項交換下積分和[]次序就得到eq.3

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5樓2014-06-27 20:58:31

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mshwangg

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【答案】應(yīng)助回帖

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十點(diǎn)鐘的咖啡: 金幣+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樓2014-06-27 14:27:15

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