Question

2. Using the Baker-Campbell-Hausdorff (BCH) Identity from the lecture notes B A +B+ }(A,B)+((A,(A,B]]+[B,B,A]])-.. Show that these are special cases (all 3 are graded): (a) eta'Pekp' X = exp' Xe*:'Peta'p' (b) et(pX+x'P) = eka'Petp'xe-ska'p' (c) ep' X Pe-kp' X = P-pl

Answer 1

Ma.Solution :- Given that data using the Barker- Campbell - H (BCH) identity fooms the lecture note dala Barker- - Campbell - Hausdor CM eMeBo A+B+{[A, B] ++ (A, [418])+(6,B,A)). . - L0 ala) ; alp pla Cea] --ih (c, ã) -0 [2, f] = in [ciê] =0 [sc, is the consonant] e" e" -er zero] GO) = 0 (2] =0 etalet och plantes 'p (@, KAD tetxa dela where ab ġ the constant

216) :- - we have from egin » & xp 1 ple et apte pe + 2 ( xlp, & ple] + Álfhe de alre, 14 p'a) + (ylipla, [y/pla, %2'p] 97 set upttr'*** 05 ) alp' [lit] = exp[ # a'p this plant(%) *alpl (-16)] = exp [xp plant is alp? - exp [ť (alp+ pla)] explisalp) I from eqin > exy [tolat total] - et ole tipler in apa or ( became Laddition of in communitate retraite als ehr als 4 vign because because HCMD

glitt et pilot et plans et plag (1-fp2 4 )*+----) retplace-i spatli mpat---). = (143 clar (part - ) (* #roa Pecin - t-kt4 + (-) + x - 1) = 3**/***) [pos] = P-pl ( all these concreten arl Zero]