²é¿´: 1007  |  »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¥
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] 0703»¯Ñ§µ÷¼Á +10 ÄÝÄÝninicgb 2026-03-15 14/700 2026-03-19 22:59 by ѧԱ8dgXkO
[¿¼ÑÐ] ¡¾¿¼Ñе÷¼Á¡¿»¯Ñ§×¨Òµ 281·Ö£¬Ò»Ö¾Ô¸ËÄ´¨´óѧ£¬³ÏÐÄÇóµ÷¼Á +5 ³Ô³Ô³Ô²ÅÓÐÒâÒå 2026-03-19 5/250 2026-03-19 16:18 by 30660438
[¿¼ÑÐ] 271²ÄÁϹ¤³ÌÇóµ÷¼Á +6 .6lL 2026-03-18 6/300 2026-03-19 15:41 by haoshis
[¿¼ÑÐ] 085600²ÄÁÏÓ뻯¹¤Çóµ÷¼Á +6 Ð÷ÐÒÓë×Ó 2026-03-17 6/300 2026-03-19 13:27 by houyaoxu
[¿¼ÑÐ] Ò»Ö¾Ô¸Äϲý´óѧ£¬327·Ö£¬²ÄÁÏÓ뻯¹¤085600 +3 Ncdx123456 2026-03-19 3/150 2026-03-19 13:18 by houyaoxu
[¿¼ÑÐ] ²ÄÁÏ080500µ÷¼ÁÇóÊÕÁô +4 Ò»¿Åmeteor 2026-03-13 4/200 2026-03-19 10:32 by 30660438
[¿¼ÑÐ] ±¾¿ÆÖ£ÖÝ´óѧÎïÀíѧԺ£¬Ò»Ö¾Ô¸»ª¿Æ070200ѧ˶£¬346Çóµ÷¼Á +4 ÎÒ²»ÊÇÒ»¸ù´Ð 2026-03-18 4/200 2026-03-19 09:11 by ¸¡ÔÆ166
[¿¼ÑÐ] 0703»¯Ñ§ 305Çóµ÷¼Á +4 FY_yy 2026-03-14 4/200 2026-03-19 05:54 by anny19840123
[¿¼ÑÐ] 304Çóµ÷¼Á +12 СÐÜjoy 2026-03-14 13/650 2026-03-18 12:34 by Linda Hu
[¿¼ÑÐ] 0854£¬¼ÆËã»úÀàÕÐÊÕµ÷¼Á +3 ºúÀ±ÌÀ·ÅÌÇ 2026-03-15 6/300 2026-03-18 12:09 by Éϰ¶Éϰ¶¡­¡­..
[¿¼ÑÐ] 280Çóµ÷¼Á +6 ¹¾ààÏþÏþ 2026-03-18 7/350 2026-03-18 11:25 by Î޼ʵIJÝÔ­
[»ù½ðÉêÇë] ±»ÎÒÑÔÖУºÐÂÄ£°å²»Ç¿µ÷¸ñʽÁË£¬¼Ùר¼Ò¿ªÊ¼¹Ü¸ñʽÁË +4 beefly 2026-03-14 4/200 2026-03-17 22:04 by »ÆÄñÓÚ·ÉChao
[¿¼ÑÐ] 277µ÷¼Á +5 ×ÔÓɼå±ý¹û×Ó 2026-03-16 6/300 2026-03-17 19:26 by Àîleezz
[¿¼ÑÐ] ²ÄÁÏר˶326Çóµ÷¼Á +6 Ä«ìÏæ¦Ý· 2026-03-15 7/350 2026-03-17 17:10 by ruiyingmiao
[¿¼ÑÐ] 332Çóµ÷¼Á +6 Zz°æ 2026-03-13 6/300 2026-03-17 17:03 by ruiyingmiao
[¿¼ÑÐ] 290Çóµ÷¼Á +3 p asserby. 2026-03-15 4/200 2026-03-17 16:35 by wangkm
[»ù½ðÉêÇë] ½ñÄêµÄ¹ú»ù½ðÊÇ´ò·ÖÖÆÂ𣿠50+3 zhanghaozhu 2026-03-14 3/150 2026-03-16 17:07 by ±±¾©À³ÒðÈóÉ«
[¿¼ÑÐ] 0703 ÎïÀí»¯Ñ§µ÷¼Á +3 ÎÒ¿ÉÒÔÉϰ¶µÄ¶Ô 2026-03-13 5/250 2026-03-16 10:50 by ÎÒ¿ÉÒÔÉϰ¶µÄ¶ÔÂ
[¿¼ÑÐ] ±¾¿ÆÄϾ©´óѧһ־Ը´¨´óҩѧ327 +3 ÂóÌï¸ûÕß 2026-03-14 3/150 2026-03-14 20:04 by ÍâÐÇÎÄÃ÷
[¿¼ÑÐ] Öпƴó²ÄÁÏר˶319Çóµ÷¼Á +3 ÃÏöβÄÁÏ 2026-03-13 3/150 2026-03-14 18:10 by houyaoxu
ÐÅÏ¢Ìáʾ
ÇëÌî´¦ÀíÒâ¼û