Znn3bq.jpeg
²é¿´: 1047  |  »Ø¸´: 15
µ±Ç°Ö»ÏÔʾÂú×ãÖ¸¶¨Ìõ¼þµÄ»ØÌû£¬µã»÷ÕâÀï²é¿´±¾»°ÌâµÄËùÓлØÌû

Ê®µãÖӵĿ§·È

½ð³æ (ÕýʽдÊÖ)

[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?
»Ø¸´´ËÂ¥

» ²ÂÄãϲ»¶

ÒÑÔÄ   »Ø¸´´ËÂ¥   ¹Ø×¢TA ¸øTA·¢ÏûÏ¢ ËÍTAºì»¨ TAµÄ»ØÌû

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
ÒÑÔÄ   »Ø¸´´ËÂ¥   ¹Ø×¢TA ¸øTA·¢ÏûÏ¢ ËÍTAºì»¨ TAµÄ»ØÌû
²é¿´È«²¿ 16 ¸ö»Ø´ð

mshwangg

ÖÁ×ðľ³æ (ÕýʽдÊÖ)

¡¾´ð°¸¡¿Ó¦Öú»ØÌû

¡ï
¸Ðл²ÎÓ룬ӦÖúÖ¸Êý +1
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
ÒÑÔÄ   »Ø¸´´ËÂ¥   ¹Ø×¢TA ¸øTA·¢ÏûÏ¢ ËÍTAºì»¨ TAµÄ»ØÌû

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
ÒÑÔÄ   »Ø¸´´ËÂ¥   ¹Ø×¢TA ¸øTA·¢ÏûÏ¢ ËÍTAºì»¨ TAµÄ»ØÌû

walk1997

½ð³æ (ÖøÃûдÊÖ)

¡ï ¡ï ¡ï ¡ï ¡ï ¡ï ¡ï ¡ï ¡ï ¡ï
Ê®µãÖӵĿ§·È(leongoall´ú·¢): ½ð±Ò+10, Ӧ¥Ö÷ÒªÇ󣬽±Àø½ð±Ò£¡ 2014-06-28 19:28:18
Ê®µãÖӵĿ§·È: »ØÌûÖö¥ 2014-06-29 13:40:24
°ÑEq.2ÖÐt+\Delta t »»³Ét', t'±ê¼Ç³Ét''
ÕâÑù¾ÍµÃµ½\rho(t')µÄ»ý·Ö±í´ïʽ
°ÑÕâ1±í´ïʽ´úÈëEq.2ÖеÄÓұߠ ¾ÍµÃµ½3Ïî
µÚÒ»ÏîÒÆ×ó±ß  µÚÈýÏî½»»»Ï»ý·ÖºÍ[]´ÎÐò ¾ÍµÃµ½eq.3
5Â¥2014-06-27 20:58:31
ÒÑÔÄ   »Ø¸´´ËÂ¥   ¹Ø×¢TA ¸øTA·¢ÏûÏ¢ ËÍTAºì»¨ TAµÄ»ØÌû
×î¾ßÈËÆøÈÈÌûÍÆ¼ö [²é¿´È«²¿] ×÷Õß »Ø/¿´ ×îºó·¢±í
[¿¼ÑÐ] ±¾¿ÆÄÏ·½Ò½¿Æ´óѧ Ò»Ö¾Ô¸985 ҩѧѧ˶284·Ö Çóµ÷¼Á +5 ÈõË®ÌýÎÄ 2026-04-09 5/250 2026-04-12 13:16 by shengxi123
[¿¼ÑÐ] 296Çóµ÷¼Á +14 Íô£¡£¿£¡ 2026-04-10 16/800 2026-04-12 10:48 by zhouyuwinner
[¿¼ÑÐ] 326Çóµ÷¼Á +6 Shansyn 2026-04-10 6/300 2026-04-12 09:46 by hammer3
[¿¼ÑÐ] 267Çóµ÷¼Á +8 ÔÙæҲҪ³Ô·¹°¡ 2026-04-09 8/400 2026-04-11 21:42 by cfdbai
[¿¼ÑÐ] 11408¡£358Çóµ÷¼Á +3 TMYzds 2026-04-07 3/150 2026-04-11 17:10 by µªÆøÆøÆø
[¿¼ÑÐ] 298Çóµ÷¼Á +5 ²ÐºÉÐÂÁø 2026-04-07 5/250 2026-04-11 11:02 by ×ÏêØ×ÏÆå
[¿¼ÑÐ] 0854µ÷¼Á +5 ÒôÏñµêÌý»¨¹ÄÏ· 2026-04-10 5/250 2026-04-11 10:49 by qingpingzhu
[¿¼ÑÐ] 281Çóµ÷¼Á +11 ¾õµÃºÃµÄ°É 2026-04-10 11/550 2026-04-11 09:35 by ÄæË®³Ë·ç
[¿¼ÑÐ] 284Çóµ÷¼Á +19 èó@@ 2026-04-06 21/1050 2026-04-10 21:12 by zhouxiaoyu
[¿¼ÑÐ] ¼ÆËã»úÀàÇóµ÷¼Á£¬22408-274·Ö +7 Éϰ¶deС³æ 2026-04-09 8/400 2026-04-10 19:56 by fxue1114
[¿¼ÑÐ] ÖпÆÔº×Ü·Ö315Çóµ÷¼Á +8 lallalh 2026-04-09 8/400 2026-04-10 19:30 by dick_runner
[¿¼ÑÐ] 085800 ÄÜÔ´¶¯Á¦Çóµ÷¼Á +6 °¢biu°¡°¡°¡°¡°¡ 2026-04-10 6/300 2026-04-10 15:03 by hemengdong
[¿¼²©] ²©Ê¿×Ô¼ö +7 ¿É¿ÉСÅÖ 2026-04-08 7/350 2026-04-10 08:28 by kimhero
[¿¼ÑÐ] ¿¼Ñе÷¼Á-²ÄÁÏÀà-284 +28 Ïë»»ÊÖ»ú²»Ïë½âÊ 2026-04-08 28/1400 2026-04-09 20:08 by µ¹Êý321?
[¿¼ÑÐ] ²ÄÁϵ÷¼Á +14 Ò»ÑùYWY 2026-04-05 15/750 2026-04-09 13:36 by ¹ÊÈË??
[¿¼ÑÐ] һ־Ը³¶«´óѧ071000ÉúÎïѧѧ˶³õÊÔ·ÖÊý276Çóµ÷¼Á +3 Ľ¾øcc 2026-04-09 3/150 2026-04-09 09:57 by liuhuiying09
[¿¼ÑÐ] 263·ÖBÇøÇóµ÷¼Á +6 Àînihao 2026-04-08 6/300 2026-04-08 09:38 by ÄÏ¿ªÐ¡ôë
[¿¼ÑÐ] 312Çóµ÷¼Á +18 gtw1 2026-04-06 20/1000 2026-04-07 18:16 by À¶ÔÆË¼Óê
[¿¼ÑÐ] µ÷¼Á +4 mcbbc 2026-04-06 5/250 2026-04-07 12:33 by upczlm1989
[¿¼ÑÐ] 085500»úеר˶³õÊÔ288Çóµ÷¼Á +3 GZJguo666- 2026-04-05 3/150 2026-04-05 18:06 by jkddd
ÐÅÏ¢Ìáʾ
ÇëÌî´¦ÀíÒâ¼û