Mathematical Methods for Physics and Engineering - Matematica.NET

19.3 EXERCISESNow evaluate the expectation value using the eigenvalue properties of H, namelyH|r〉 = E r |r〉, and deduce the sum rule for oscillation strengths,∞∑r=0(E r − E 0 )|〈r | x | 0〉| 2 = N22m .19.9 By considering the functionF(λ) =exp(λA)B exp(−λA),where A and B are linear operators and λ is a parameter, and finding itsderivatives with respect to λ, prove thate A Be −A = B + [ A, B ] + 1 2! [ A, [ A, B ]]+ 1 [ A, [ A, [ A, B ]]]+ ··· .3!Use this result to express( ) ( )iLx θ−iLx θexp L y expas a linear combination of the angular momentum operators L x , L y and L z .19.10 For a system containing more than one particle, the total angular momentum Jand its components are represented by operators that have completely analogouscommutation relations to those for the operators for a single particle, i.e. J 2 haseigenvalue j(j +1) 2 and J z has eigenvalue m j for the state |j,m j 〉. The usualorthonormality relationship 〈j ′ ,m ′ j | j,m j〉 = δ j ′ j δ m ′jm jis also valid.A system consists of two (distinguishable) particles A and B. ParticleA is inan l = 3 state and can have state functions of the form |A, 3,m A 〉, whilst B isin an l = 2 state with possible state functions |B,2,m B 〉. The range of possiblevalues for j is |3 − 2| ≤j ≤|3+2|, i.e.1≤ j ≤ 5, and the overall state functioncanbewrittenas|j,m j 〉 =∑m A m B| A, 3,m A 〉|B,2,m B 〉.m A +m B =m jC jm jThe numerical coefficients C jm jm A m Bare known as Clebsch–Gordon coefficients.Assume (as can be shown) that the ladder operators U(AB) andD(AB) forthe system can be written as U(A) +U(B) andD(A) +D(B), respectively, andthat they lead to relationships equivalent to (19.34) and (19.35) with l replacedby j and m by m j .(a) Apply the operators to the (obvious) relationship|AB, 5, 5〉 = |A, 3, 3〉|B,2, 2〉to show that√√6|AB, 5, 4〉 = |A, 3, 2〉|B,2, 2〉 + 4|A, 3, 3〉|B,2, 1〉.10 10(b) Find, to within an overall sign, the real coefficients c and d in the expansion|AB, 4, 4〉 = c|A, 3, 2〉|B,2, 2〉 + d|A, 3, 3〉|B,2, 1〉by requiring it to be orthogonal to |AB, 5, 4〉. Check your answer by consideringU(AB)|AB, 4, 4〉.(c) Find, to within an overall sign, and as efficiently as possible, an expressionfor |AB, 4, −3〉 as a sum of products of the form |A, 3,m A 〉|B,2,m B 〉.673