Praise be to Allah.
This is one of the issues on which the scholars (may Allaah have mercy on them) have differed. Some of them think that placing the right hand on the left forearm when standing up from rukoo is sunnah mustahabbah (preferable) because of the general meaning of the hadeeth of Sahl ibn Sad al-Saaidi (may Allaah be pleased with him), who said: The people were told that a man should place his right hand on his left forearm in prayer. They said: the general meaning of the phrase in prayer makes it clear that when standing up from rukoo, one should place ones hands thus, because the qiyaam (standing) includes the standing both before rukoo and after, so a person should place his right hand on his left forearm in both qiyaams.
They said that during rukoo, the hands should be on the knees, in sujood they should be on the ground, in juloos (sitting) they should be on the thighs, and in qiyaam (standing) they should be on the chest. This is where the hands should be during the qiyaam before rukoo and after. The attitude of humility and proper conduct before the Lord requires this in both qiyaams, and this position is the furthest away from fidgeting. Other scholars think that the right hand does not have to be placed on the left forearm after standing up from rukoo, and their evidence is the fact that no specific report on this matter has been narrated, as well as the fact that the general meaning of qiyaam does not include standing up from rukoo, because this is a straightening of the back, not a standing (qiyaam), so the person should let his hands drop to his sides.
Some scholars say that a person has the choice of placing his hands on his chest or not, because nothing clear has been reported in the Sunnah on this point. This is the opinion of Imaam Ahmad, may Allaah have mercy on him.
The first opinion is more likely to be correct, in sha Allaah, but in any case, either way is valid. Differences in such matters should not lead to reprehensible disputes and conflict, or accusations of bidah if someone does this, or of neglecting the Sunnah if he does not. Such comments can only be made in cases where there is a clear saheeh text and definite evidence.
For more information on this matter, see:
Al-Furoo by Ibn Muflih, 1/433; al-Insaaf by al-Mirdaawi, 2/63; Sharh Muntaha al-Iraadaat, 1/185; al-Sharh al-Mumti by Ibn Uthaymeen, 3/146. The hadeeth scholar Shaykh Badee al-Deen al-Sindi wrote a paper specifically on this topic in which he supported the opinion that it is sunnah to place the right hand on the left forearm after standing up from rukoo: Ziyaadat al-Khushoo bi wad al-Yumna alal-Yusra bad al-Rukoo. Shaykh Abd al-Azeez ibn Abd-Allaah ibn Baaz also wrote a paper on this topic, in which he supported the idea of putting the hands on the chest after standing up from rukoo.
And Allaah knows best.